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Introduction To Algorithms 4th Thomas H Cormen Charles E Leiserson

Introduction To Algorithms 4th Thomas H Cormen Charles E Leiserson Introduction To Algorithms 4th Thomas H Cormen Charles E Leiserson Introduction To Algorithms 4th Thomas H Cormen Charles E Leiserson

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Introduction to Algorithms Fourth Edition
Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Fourth Edition The MIT Press Cambridge, Massachusetts London, England
c 2022 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. The MIT Press would like to thank the anonymous peer reviewers who provided comments on drafts of this book. The generous work of academic experts is essential for establishing the authority and quality of our publications. We acknowledge with gratitude the contributions of these otherwise uncredited readers. This book was set in Times Roman and MathTime Professional II by the authors. Names: Cormen, Thomas H., author. jLeiserson, Charles Eric, author. j Rivest, Ronald L., author. jStein, Clifford, author. Title: Introduction to algorithms / Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Description: Fourth edition. jCambridge, Massachusetts : The MIT Press, [2022] jIncludes bibliographical references and index. Identiûers: LCCN 2021037260 jISBN 9780262046305 Subjects: LCSH: Computer programming. jComputer algorithms. Classiûcation: LCC QA76.6 .C662 2022 jDDC 005.13--dc23 LC record available at http://lccn.loc.gov/2021037260 10 9 8 7 6 5 4 3 2 1
Contents Preface xiii I Foundations Introduction 3 1 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 12 2 Getting Started 17 2.1 Insertion sort 17 2.2 Analyzing algorithms 25 2.3 Designing algorithms 34 3 Characterizing Running Times 49 3.1 O-notation, �-notation, and ‚-notation 50 3.2 Asymptotic notation: formal deûnitions 53 3.3 Standard notations and common functions 63 4 Divide-and-Conquer 76 4.1 Multiplying square matrices 80 4.2 Strassen’s algorithm for matrix multiplication 85 4.3 The substitution method for solving recurrences 90 4.4 The recursion-tree method for solving recurrences 95 4.5 The master method for solving recurrences 101 ? 4.6 Proof of the continuous master theorem 107 ? 4.7 Akra-Bazzi recurrences 115
vi Contents 5 Probabilistic Analysis and Randomized Algorithms 126 5.1 The hiring problem 126 5.2 Indicator random variables 130 5.3 Randomized algorithms 134 ? 5.4 Probabilistic analysis and further uses of indicator random variables 140 II Sorting and Order Statistics Introduction 157 6 Heapsort 161 6.1 Heaps 161 6.2 Maintaining the heap property 164 6.3 Building a heap 167 6.4 The heapsort algorithm 170 6.5 Priority queues 172 7 Quicksort 182 7.1 Description of quicksort 183 7.2 Performance of quicksort 187 7.3 A randomized version of quicksort 191 7.4 Analysis of quicksort 193 8 Sorting in Linear Time 205 8.1 Lower bounds for sorting 205 8.2 Counting sort 208 8.3 Radix sort 211 8.4 Bucket sort 215 9 Medians and Order Statistics 227 9.1 Minimum and maximum 228 9.2 Selection in expected linear time 230 9.3 Selection in worst-case linear time 236 III Data Structures Introduction 249 10 Elementary Data Structures 252 10.1 Simple array-based data structures: arrays, matrices, stacks, queues 252 10.2 Linked lists 258 10.3 Representing rooted trees 265
Contents vii 11 Hash Tables 272 11.1 Direct-address tables 273 11.2 Hash tables 275 11.3 Hash functions 282 11.4 Open addressing 293 11.5 Practical considerations 301 12 Binary Search Trees 312 12.1 What is a binary search tree? 312 12.2 Querying a binary search tree 316 12.3 Insertion and deletion 321 13 Red-Black Trees 331 13.1 Properties of red-black trees 331 13.2 Rotations 335 13.3 Insertion 338 13.4 Deletion 346 IV Advanced Design and Analysis Techniques Introduction 361 14 Dynamic Programming 362 14.1 Rod cutting 363 14.2 Matrix-chain multiplication 373 14.3 Elements of dynamic programming 382 14.4 Longest common subsequence 393 14.5 Optimal binary search trees 400 15 Greedy Algorithms 417 15.1 An activity-selection problem 418 15.2 Elements of the greedy strategy 426 15.3 Huffman codes 431 15.4 Ofüine caching 440 16 Amortized Analysis 448 16.1 Aggregate analysis 449 16.2 The accounting method 453 16.3 The potential method 456 16.4 Dynamic tables 460
viii Contents V Advanced Data Structures Introduction 477 17 Augmenting Data Structures 480 17.1 Dynamic order statistics 480 17.2 How to augment a data structure 486 17.3 Interval trees 489 18 B-Trees 497 18.1 Deûnition of B-trees 501 18.2 Basic operations on B-trees 504 18.3 Deleting a key from a B-tree 513 19 Data Structures for Disjoint Sets 520 19.1 Disjoint-set operations 520 19.2 Linked-list representation of disjoint sets 523 19.3 Disjoint-set forests 527 ? 19.4 Analysis of union by rank with path compression 531 VI Graph Algorithms Introduction 547 20 Elementary Graph Algorithms 549 20.1 Representations of graphs 549 20.2 Breadth-ûrst search 554 20.3 Depth-ûrst search 563 20.4 Topological sort 573 20.5 Strongly connected components 576 21 Minimum Spanning Trees 585 21.1 Growing a minimum spanning tree 586 21.2 The algorithms of Kruskal and Prim 591 22 Single-Source Shortest Paths 604 22.1 The Bellman-Ford algorithm 612 22.2 Single-source shortest paths in directed acyclic graphs 616 22.3 Dijkstra’s algorithm 620 22.4 Difference constraints and shortest paths 626 22.5 Proofs of shortest-paths properties 633
Contents ix 23 All-Pairs Shortest Paths 646 23.1 Shortest paths and matrix multiplication 648 23.2 The Floyd-Warshall algorithm 655 23.3 Johnson’s algorithm for sparse graphs 662 24 Maximum Flow 670 24.1 Flow networks 671 24.2 The Ford-Fulkerson method 676 24.3 Maximum bipartite matching 693 25 Matchings in Bipartite Graphs 704 25.1 Maximum bipartite matching (revisited) 705 25.2 The stable-marriage problem 716 25.3 The Hungarian algorithm for the assignment problem 723 VII Selected Topics Introduction 745 26 Parallel Algorithms 748 26.1 The basics of fork-join parallelism 750 26.2 Parallel matrix multiplication 770 26.3 Parallel merge sort 775 27 Online Algorithms 791 27.1 Waiting for an elevator 792 27.2 Maintaining a search list 795 27.3 Online caching 802 28 Matrix Operations 819 28.1 Solving systems of linear equations 819 28.2 Inverting matrices 833 28.3 Symmetric positive-deûnite matrices and least-squares approximation 838 29 Linear Programming 850 29.1 Linear programming formulations and algorithms 853 29.2 Formulating problems as linear programs 860 29.3 Duality 866 30 Polynomials and the FFT 877 30.1 Representing polynomials 879 30.2 The DFT and FFT 885 30.3 FFT circuits 894
x Contents 31 Number-Theoretic Algorithms 903 31.1 Elementary number-theoretic notions 904 31.2 Greatest common divisor 911 31.3 Modular arithmetic 916 31.4 Solving modular linear equations 924 31.5 The Chinese remainder theorem 928 31.6 Powers of an element 932 31.7 The RSA public-key cryptosystem 936 ? 31.8 Primality testing 942 32 String Matching 957 32.1 The naive string-matching algorithm 960 32.2 The Rabin-Karp algorithm 962 32.3 String matching with ûnite automata 967 ? 32.4 The Knuth-Morris-Pratt algorithm 975 32.5 Sufûx arrays 985 33 Machine-Learning Algorithms 1003 33.1 Clustering 1005 33.2 Multiplicative-weights algorithms 1015 33.3 Gradient descent 1022 34 NP-Completeness 1042 34.1 Polynomial time 1048 34.2 Polynomial-time veriûcation 1056 34.3 NP-completeness and reducibility 1061 34.4 NP-completeness proofs 1072 34.5 NP-complete problems 1080 35 Approximation Algorithms 1104 35.1 The vertex-cover problem 1106 35.2 The traveling-salesperson problem 1109 35.3 The set-covering problem 1115 35.4 Randomization and linear programming 1119 35.5 The subset-sum problem 1124 VIII Appendix: Mathematical Background Introduction 1139 A Summations 1140 A.1 Summation formulas and properties 1140 A.2 Bounding summations 1145
Contents xi B Sets, Etc. 1153 B.1 Sets 1153 B.2 Relations 1158 B.3 Functions 1161 B.4 Graphs 1164 B.5 Trees 1169 C Counting and Probability 1178 C.1 Counting 1178 C.2 Probability 1184 C.3 Discrete random variables 1191 C.4 The geometric and binomial distributions 1196 ? C.5 The tails of the binomial distribution 1203 D Matrices 1214 D.1 Matrices and matrix operations 1214 D.2 Basic matrix properties 1219 Bibliography 1227 Index 1251
Preface Not so long ago, anyone who had heard the word <algorithm= was almost certainly a computer scientist or mathematician. With computers having become prevalent in our modern lives, however, the term is no longer esoteric. If you look around your home, you’ll ûnd algorithms running in the most mundane places: your microwave oven, your washing machine, and, of course, your computer. You ask algorithms to make recommendations to you: what music you might like or what route to take when driving. Our society, for better or for worse, asks algorithms to suggest sentences for convicted criminals. You even rely on algorithms to keep you alive, or at least not to kill you: the control systems in your car or in medical equipment.1 The word <algorithm= appears somewhere in the news seemingly every day. Therefore, it behooves you to understand algorithms not just as a student or practitioner of computer science, but as a citizen of the world. Once you understand algorithms, you can educate others about what algorithms are, how they operate, and what their limitations are. This book provides a comprehensive introduction to the modern study of com- puter algorithms. It presents many algorithms and covers them in considerable depth, yet makes their design accessible to all levels of readers. All the analyses are laid out, some simple, some more involved. We have tried to keep explanations clear without sacriûcing depth of coverage or mathematical rigor. Each chapter presents an algorithm, a design technique, an application area, or a related topic. Algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The book contains 231 ûgures4many with multiple parts4illustrating how the algorithms work. Since we emphasize efficiency as a design criterion, we include careful analyses of the running times of the algorithms. 1 To understand many of the ways in which algorithms inüuence our daily lives, see the book by Fry [162].
xiv Preface The text is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Because it discusses engineering issues in algorithm design, as well as mathematical aspects, it is equally well suited for self-study by technical professionals. In this, the fourth edition, we have once again updated the entire book. The changes cover a broad spectrum, including new chapters and sections, color illus- trations, and what we hope you’ll ûnd to be a more engaging writing style. To the teacher We have designed this book to be both versatile and complete. You should ûnd it useful for a variety of courses, from an undergraduate course in data structures up through a graduate course in algorithms. Because we have provided considerably more material than can ût in a typical one-term course, you can select the material that best supports the course you wish to teach. You should ûnd it easy to organize your course around just the chapters you need. We have made chapters relatively self-contained, so that you need not worry about an unexpected and unnecessary dependence of one chapter on an- other. Whereas in an undergraduate course, you might use only some sections from a chapter, in a graduate course, you might cover the entire chapter. We have included 931 exercises and 162 problems. Each section ends with exer- cises, and each chapter ends with problems. The exercises are generally short ques- tions that test basic mastery of the material. Some are simple self-check thought exercises, but many are substantial and suitable as assigned homework. The prob- lems include more elaborate case studies which often introduce new material. They often consist of several parts that lead the student through the steps required to ar- rive at a solution. As with the third edition of this book, we have made publicly available solutions to some, but by no means all, of the problems and exercises. You can ûnd these so- lutions on our website, http://mitpress.mit.edu/algorithms/. You will want to check this site to see whether it contains the solution to an exercise or problem that you plan to assign. Since the set of solutions that we post might grow over time, we recommend that you check the site each time you teach the course. We have starred (?) the sections and exercises that are more suitable for graduate students than for undergraduates. A starred section is not necessarily more difû- cult than an unstarred one, but it may require an understanding of more advanced mathematics. Likewise, starred exercises may require an advanced background or more than average creativity.
Preface xv To the student We hope that this textbook provides you with an enjoyable introduction to the ûeld of algorithms. We have attempted to make every algorithm accessible and inter- esting. To help you when you encounter unfamiliar or difûcult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms and sup- porting ûgures to help you visualize what is going on. Since this book is large, your class will probably cover only a portion of its material. Although we hope that you will ûnd this book helpful to you as a course textbook now, we have also tried to make it comprehensive enough to warrant space on your future professional bookshelf. What are the prerequisites for reading this book?  You need some programming experience. In particular, you should understand recursive procedures and simple data structures, such as arrays and linked lists (although Section 10.2 covers linked lists and a variant that you may ûnd new).  You should have some facility with mathematical proofs, and especially proofs by mathematical induction. A few portions of the book rely on some knowledge of elementary calculus. Although this book uses mathematics throughout, Part I and Appendices A–D teach you all the mathematical techniques you will need. Our website, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises. Feel free to check your solutions against ours. We ask, however, that you not send your solutions to us. To the professional The wide range of topics in this book makes it an excellent handbook on algo- rithms. Because each chapter is relatively self-contained, you can focus on the topics most relevant to you. Since most of the algorithms we discuss have great practical utility, we address implementation concerns and other engineering issues. We often provide practical alternatives to the few algorithms that are primarily of theoretical interest. If you wish to implement any of the algorithms, you should ûnd the transla- tion of our pseudocode into your favorite programming language to be a fairly straightforward task. We have designed the pseudocode to present each algorithm clearly and succinctly. Consequently, we do not address error handling and other software-engineering issues that require speciûc assumptions about your program- ming environment. We attempt to present each algorithm simply and directly with- out allowing the idiosyncrasies of a particular programming language to obscure its essence. If you are used to 0-origin arrays, you might ûnd our frequent practice of
xvi Preface indexing arrays from 1 a minor stumbling block. You can always either subtract 1 from our indices or just overallocate the array and leave position 0 unused. We understand that if you are using this book outside of a course, then you might be unable to check your solutions to problems and exercises against solutions provided by an instructor. Our website, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises so that you can check your work. Please do not send your solutions to us. To our colleagues We have supplied an extensive bibliography and pointers to the current literature. Each chapter ends with a set of chapter notes that give historical details and ref- erences. The chapter notes do not provide a complete reference to the whole ûeld of algorithms, however. Though it may be hard to believe for a book of this size, space constraints prevented us from including many interesting algorithms. Despite myriad requests from students for solutions to problems and exercises, we have adopted the policy of not citing references for them, removing the temp- tation for students to look up a solution rather than to discover it themselves. Changes for the fourth edition As we said about the changes for the second and third editions, depending on how you look at it, the book changed either not much or quite a bit. A quick look at the table of contents shows that most of the third-edition chapters and sections appear in the fourth edition. We removed three chapters and several sections, but we have added three new chapters and several new sections apart from these new chapters. We kept the hybrid organization from the ûrst three editions. Rather than organizing chapters only by problem domains or only according to techniques, this book incorporates elements of both. It contains technique-based chapters on divide-and-conquer, dynamic programming, greedy algorithms, amortized analy- sis, augmenting data structures, NP-completeness, and approximation algorithms. But it also has entire parts on sorting, on data structures for dynamic sets, and on algorithms for graph problems. We ûnd that although you need to know how to ap- ply techniques for designing and analyzing algorithms, problems seldom announce to you which techniques are most amenable to solving them. Some of the changes in the fourth edition apply generally across the book, and some are speciûc to particular chapters or sections. Here is a summary of the most signiûcant general changes:  We added 140 new exercises and 22 new problems. We also improved many of the old exercises and problems, often as the result of reader feedback. (Thanks to all readers who made suggestions.)
Preface xvii  We have color! With designers from the MIT Press, we selected a limited palette, devised to convey information and to be pleasing to the eye. (We are delighted to display red-black trees in4get this4red and black!) To enhance readability, deûned terms, pseudocode comments, and page numbers in the in- dex are in color.  Pseudocode procedures appear on a tan background to make them easier to spot, and they do not necessarily appear on the page of their ûrst reference. When they don’t, the text directs you to the relevant page. In the same vein, nonlocal references to numbered equations, theorems, lemmas, and corollaries include the page number.  We removed topics that were rarely taught. We dropped in their entirety the chapters on Fibonacci heaps, van Emde Boas trees, and computational geom- etry. In addition, the following material was excised: the maximum-subarray problem, implementing pointers and objects, perfect hashing, randomly built binary search trees, matroids, push-relabel algorithms for maximum üow, the iterative fast Fourier transform method, the details of the simplex algorithm for linear programming, and integer factorization. You can ûnd all the removed material on our website, http://mitpress.mit.edu/algorithms/.  We reviewed the entire book and rewrote sentences, paragraphs, and sections to make the writing clearer, more personal, and gender neutral. For example, the <traveling-salesman problem= in the previous editions is now called the <traveling-salesperson problem.= We believe that it is critically important for engineering and science, including our own ûeld of computer science, to be welcoming to everyone. (The one place that stumped us is in Chapter 13, which requires a term for a parent’s sibling. Because the English language has no such gender-neutral term, we regretfully stuck with <uncle.=)  The chapter notes, bibliography, and index were updated, reüecting the dra- matic growth of the ûeld of algorithms since the third edition.  We corrected errors, posting most corrections on our website of third-edition errata. Those that were reported while we were in full swing preparing this edition were not posted, but were corrected in this edition. (Thanks again to all readers who helped us identify issues.) The speciûc changes for the fourth edition include the following:  We renamed Chapter 3 and added a section giving an overview of asymptotic notation before delving into the formal deûnitions.  Chapter 4 underwent substantial changes to improve its mathematical founda- tion and make it more robust and intuitive. The notion of an algorithmic re- currence was introduced, and the topic of ignoring üoors and ceilings in recur-
xviii Preface rences was addressed more rigorously. The second case of the master theorem incorporates polylogarithmic factors, and a rigorous proof of a <continuous= version of the master theorem is now provided. We also present the powerful and general Akra-Bazzi method (without proof).  The deterministic order-statistic algorithm in Chapter 9 is slightly different, and the analyses of both the randomized and deterministic order-statistic algorithms have been revamped.  In addition to stacks and queues, Section 10.1 discusses ways to store arrays and matrices.  Chapter 11 on hash tables includes a modern treatment of hash functions. It also emphasizes linear probing as an efûcient method for resolving collisions when the underlying hardware implements caching to favor local searches.  To replace the sections on matroids in Chapter 15, we converted a problem in the third edition about ofüine caching into a full section.  Section 16.4 now contains a more intuitive explanation of the potential func- tions to analyze table doubling and halving.  Chapter 17 on augmenting data structures was relocated from Part III to Part V, reüecting our view that this technique goes beyond basic material.  Chapter 25 is a new chapter about matchings in bipartite graphs. It presents algorithms to ûnd a matching of maximum cardinality, to solve the stable- marriage problem, and to ûnd a maximum-weight matching (known as the <as- signment problem=).  Chapter 26, on task-parallel computing, has been updated with modern termi- nology, including the name of the chapter.  Chapter 27, which covers online algorithms, is another new chapter. In an online algorithm, the input arrives over time, rather than being available in its entirety at the start of the algorithm. The chapter describes several examples of online algorithms, including determining how long to wait for an elevator before taking the stairs, maintaining a linked list via the move-to-front heuristic, and evaluating replacement policies for caches.  In Chapter 29, we removed the detailed presentation of the simplex algorithm, as it was math heavy without really conveying many algorithmic ideas. The chapter now focuses on the key aspect of how to model problems as linear programs, along with the essential duality property of linear programming.  Section 32.5 adds to the chapter on string matching the simple, yet powerful, structure of sufûx arrays.
Preface xix  Chapter 33, on machine learning, is the third new chapter. It introduces sev- eral basic methods used in machine learning: clustering to group similar items together, weighted-majority algorithms, and gradient descent to ûnd the mini- mizer of a function.  Section 34.5.6 summarizes strategies for polynomial-time reductions to show that problems are NP-hard.  The proof of the approximation algorithm for the set-covering problem in Sec- tion 35.3 has been revised. Website You can use our website, http://mitpress.mit.edu/algorithms/, to obtain supplemen- tary information and to communicate with us. The website links to a list of known errors, material from the third edition that is not included in the fourth edition, solutions to selected exercises and problems, Python implementations of many of the algorithms in this book, a list explaining the corny professor jokes (of course), as well as other content, which we may add to. The website also tells you how to report errors or make suggestions. How we produced this book Like the previous three editions, the fourth edition was produced in L A TEX 2". We used the Times font with mathematics typeset using the MathTime Professional II fonts. As in all previous editions, we compiled the index using Windex, a C pro- gram that we wrote, and produced the bibliography using BIBTEX. The PDF ûles for this book were created on a MacBook Pro running macOS 10.14. Our plea to Apple in the preface of the third edition to update MacDraw Pro for macOS 10 went for naught, and so we continued to draw illustrations on pre-Intel Macs running MacDraw Pro under the Classic environment of older versions of macOS 10. Many of the mathematical expressions appearing in illustrations were laid in with the psfrag package for L A TEX 2". Acknowledgments for the fourth edition We have been working with the MIT Press since we started writing the ûrst edi- tion in 1987, collaborating with several directors, editors, and production staff. Throughout our association with the MIT Press, their support has always been out- standing. Special thanks to our editors Marie Lee, who put up with us for far too long, and Elizabeth Swayze, who pushed us over the ûnish line. Thanks also to Director Amy Brand and to Alex Hoopes.
xx Preface As in the third edition, we were geographically distributed while producing the fourth edition, working in the Dartmouth College Department of Computer Science; the MIT Computer Science and Artiûcial Intelligence Laboratory and the MIT Department of Electrical Engineering and Computer Science; and the Columbia University Department of Industrial Engineering and Operations Re- search, Department of Computer Science, and Data Science Institute. During the COVID-19 pandemic, we worked largely from home. We thank our respective universities and colleagues for providing such supportive and stimulating environ- ments. As we complete this book, those of us who are not retired are eager to return to our respective universities now that the pandemic seems to be abating. Julie Sussman, P.P.A., came to our rescue once again with her technical copy- editing under tremendous time pressure. If not for Julie, this book would be riddled with errors (or, let’s say, many more errors than it has) and would be far less read- able. Julie, we will be forever indebted to you. Errors that remain are the responsi- bility of the authors (and probably were inserted after Julie read the material). Dozens of errors in previous editions were corrected in the process of creating this edition. We thank our readers4too many to list them all4who have reported errors and suggested improvements over the years. We received considerable help in preparing some of the new material in this edition. Neville Campbell (unafûliated), Bill Kuszmaul of MIT, and Chee Yap of NYU provided valuable advice regarding the treatment of recurrences in Chapter 4. Yan Gu of the University of California, Riverside, provided feedback on parallel algorithms in Chapter 26. Rob Shapire of Microsoft Research altered our approach to the material on machine learning with his detailed comments on Chapter 33. Qi Qi of MIT helped with the analysis of the Monty Hall problem (Problem C-1). Molly Seaman and Mary Reilly of the MIT Press helped us select the color palette in the illustrations, and Wojciech Jarosz of Dartmouth College suggested design improvements to our newly colored ûgures. Yichen (Annie) Ke and Linda Xiao, who have since graduated from Dartmouth, aided in colorizing the illus- trations, and Linda also produced many of the Python implementations that are available on the book’s website. Finally, we thank our wives4Wendy Leiserson, Gail Rivest, Rebecca Ivry, and the late Nicole Cormen4and our families. The patience and encouragement of those who love us made this project possible. We affectionately dedicate this book to them. THOMAS H. CORMEN Lebanon, New Hampshire CHARLES E. LEISERSON Cambridge, Massachusetts RONALD L. RIVEST Cambridge, Massachusetts CLIFFORD STEIN New York, New York June, 2021
Part I Foundations
Introduction When you design and analyze algorithms, you need to be able to describe how they operate and how to design them. You also need some mathematical tools to show that your algorithms do the right thing and do it efûciently. This part will get you started. Later parts of this book will build upon this base. Chapter 1 provides an overview of algorithms and their place in modern com- puting systems. This chapter deûnes what an algorithm is and lists some examples. It also makes a case for considering algorithms as a technology, alongside tech- nologies such as fast hardware, graphical user interfaces, object-oriented systems, and networks. In Chapter 2, we see our ûrst algorithms, which solve the problem of sorting a sequence of n numbers. They are written in a pseudocode which, although not directly translatable to any conventional programming language, conveys the struc- ture of the algorithm clearly enough that you should be able to implement it in the language of your choice. The sorting algorithms we examine are insertion sort, which uses an incremental approach, and merge sort, which uses a recursive tech- nique known as <divide-and-conquer.= Although the time each requires increases with the value of n, the rate of increase differs between the two algorithms. We determine these running times in Chapter 2, and we develop a useful <asymptotic= notation to express them. Chapter 3 precisely deûnes asymptotic notation. We’ll use asymptotic notation to bound the growth of functions4most often, functions that describe the running time of algorithms4from above and below. The chapter starts by informally deûn- ing the most commonly used asymptotic notations and giving an example of how to apply them. It then formally deûnes ûve asymptotic notations and presents conven- tions for how to put them together. The rest of Chapter 3 is primarily a presentation of mathematical notation, more to ensure that your use of notation matches that in this book than to teach you new mathematical concepts.
4 Part I Foundations Chapter 4 delves further into the divide-and-conquer method introduced in Chapter 2. It provides two additional examples of divide-and-conquer algorithms for multiplying square matrices, including Strassen’s surprising method. Chapter 4 contains methods for solving recurrences, which are useful for describing the run- ning times of recursive algorithms. In the substitution method, you guess an answer and prove it correct. Recursion trees provide one way to generate a guess. Chap- ter 4 also presents the powerful technique of the <master method,= which you can often use to solve recurrences that arise from divide-and-conquer algorithms. Al- though the chapter provides a proof of a foundational theorem on which the master theorem depends, you should feel free to employ the master method without delv- ing into the proof. Chapter 4 concludes with some advanced topics. Chapter 5 introduces probabilistic analysis and randomized algorithms. You typically use probabilistic analysis to determine the running time of an algorithm in cases in which, due to the presence of an inherent probability distribution, the running time may differ on different inputs of the same size. In some cases, you might assume that the inputs conform to a known probability distribution, so that you are averaging the running time over all possible inputs. In other cases, the probability distribution comes not from the inputs but from random choices made during the course of the algorithm. An algorithm whose behavior is determined not only by its input but by the values produced by a random-number generator is a randomized algorithm. You can use randomized algorithms to enforce a probability distribution on the inputs4thereby ensuring that no particular input always causes poor performance4or even to bound the error rate of algorithms that are allowed to produce incorrect results on a limited basis. Appendices A–D contain other mathematical material that you will ûnd helpful as you read this book. You might have seen much of the material in the appendix chapters before having read this book (although the speciûc deûnitions and nota- tional conventions we use may differ in some cases from what you have seen in the past), and so you should think of the appendices as reference material. On the other hand, you probably have not already seen most of the material in Part I. All the chapters in Part I and the appendices are written with a tutorial üavor.
1 The Role of Algorithms in Computing What are algorithms? Why is the study of algorithms worthwhile? What is the role of algorithms relative to other technologies used in computers? This chapter will answer these questions. 1.1 Algorithms Informally, an algorithm is any well-deûned computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output in a ûnite amount of time. An algorithm is thus a sequence of computational steps that transform the input into the output. You can also view an algorithm as a tool for solving a well-speciûed computa- tional problem. The statement of the problem speciûes in general terms the desired input/output relationship for problem instances, typically of arbitrarily large size. The algorithm describes a speciûc computational procedure for achieving that in- put/output relationship for all problem instances. As an example, suppose that you need to sort a sequence of numbers into mono- tonically increasing order. This problem arises frequently in practice and provides fertile ground for introducing many standard design techniques and analysis tools. Here is how we formally deûne the sorting problem: Input: A sequence of n numbers ha1; a2; : : : ; ani. Output: A permutation (reordering) ha0 1; a0 2; : : : ; a0 ni of the input sequence such that a0 1 හ a0 2 හ    හ a0 n. Thus, given the input sequence h31; 41; 59; 26; 41; 58i, a correct sorting algorithm returns as output the sequence h26; 31; 41; 41; 58; 59i. Such an input sequence is
6 Chapter 1 The Role of Algorithms in Computing called an instance of the sorting problem. In general, an instance of a problem1 consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem. Because many programs use it as an intermediate step, sorting is a fundamental operation in computer science. As a result, you have a large number of good sort- ing algorithms at your disposal. Which algorithm is best for a given application depends on4among other factors4the number of items to be sorted, the extent to which the items are already somewhat sorted, possible restrictions on the item values, the architecture of the computer, and the kind of storage devices to be used: main memory, disks, or even4archaically4tapes. An algorithm for a computational problem is correct if, for every problem in- stance provided as input, it halts4ûnishes its computing in ûnite time4and out- puts the correct solution to the problem instance. A correct algorithm solves the given computational problem. An incorrect algorithm might not halt at all on some input instances, or it might halt with an incorrect answer. Contrary to what you might expect, incorrect algorithms can sometimes be useful, if you can control their error rate. We’ll see an example of an algorithm with a controllable error rate in Chapter 31 when we study algorithms for ûnding large prime numbers. Ordi- narily, however, we’ll concern ourselves only with correct algorithms. An algorithm can be speciûed in English, as a computer program, or even as a hardware design. The only requirement is that the speciûcation must provide a precise description of the computational procedure to be followed. What kinds of problems are solved by algorithms? Sorting is by no means the only computational problem for which algorithms have been developed. (You probably suspected as much when you saw the size of this book.) Practical applications of algorithms are ubiquitous and include the follow- ing examples:  The Human Genome Project has made great progress toward the goals of iden- tifying all the roughly 30,000 genes in human DNA, determining the sequences of the roughly 3 billion chemical base pairs that make up human DNA, stor- ing this information in databases, and developing tools for data analysis. Each of these steps requires sophisticated algorithms. Although the solutions to the various problems involved are beyond the scope of this book, many methods to solve these biological problems use ideas presented here, enabling scientists to accomplish tasks while using resources efûciently. Dynamic programming, as 1 Sometimes, when the problem context is known, problem instances are themselves simply called <problems.=
1.1 Algorithms 7 in Chapter 14, is an important technique for solving several of these biological problems, particularly ones that involve determining similarity between DNA sequences. The savings realized are in time, both human and machine, and in money, as more information can be extracted by laboratory techniques.  The internet enables people all around the world to quickly access and retrieve large amounts of information. With the aid of clever algorithms, sites on the internet are able to manage and manipulate this large volume of data. Exam- ples of problems that make essential use of algorithms include ûnding good routes on which the data travels (techniques for solving such problems appear in Chapter 22), and using a search engine to quickly ûnd pages on which par- ticular information resides (related techniques are in Chapters 11 and 32).  Electronic commerce enables goods and services to be negotiated and ex- changed electronically, and it depends on the privacy of personal informa- tion such as credit card numbers, passwords, and bank statements. The core technologies used in electronic commerce include public-key cryptography and digital signatures (covered in Chapter 31), which are based on numerical algo- rithms and number theory.  Manufacturing and other commercial enterprises often need to allocate scarce resources in the most beneûcial way. An oil company might wish to know where to place its wells in order to maximize its expected proût. A political candidate might want to determine where to spend money buying campaign ad- vertising in order to maximize the chances of winning an election. An airline might wish to assign crews to üights in the least expensive way possible, mak- ing sure that each üight is covered and that government regulations regarding crew scheduling are met. An internet service provider might wish to determine where to place additional resources in order to serve its customers more effec- tively. All of these are examples of problems that can be solved by modeling them as linear programs, which Chapter 29 explores. Although some of the details of these examples are beyond the scope of this book, we do give underlying techniques that apply to these problems and problem areas. We also show how to solve many speciûc problems, including the following:  You have a road map on which the distance between each pair of adjacent in- tersections is marked, and you wish to determine the shortest route from one intersection to another. The number of possible routes can be huge, even if you disallow routes that cross over themselves. How can you choose which of all possible routes is the shortest? You can start by modeling the road map (which is itself a model of the actual roads) as a graph (which we will meet in Part VI and Appendix B). In this graph, you wish to ûnd the shortest path from one vertex to another. Chapter 22 shows how to solve this problem efûciently.
8 Chapter 1 The Role of Algorithms in Computing  Given a mechanical design in terms of a library of parts, where each part may include instances of other parts, list the parts in order so that each part appears before any part that uses it. If the design comprises n parts, then there are nŠ possible orders, where nŠ denotes the factorial function. Because the factorial function grows faster than even an exponential function, you cannot feasibly generate each possible order and then verify that, within that order, each part appears before the parts using it (unless you have only a few parts). This prob- lem is an instance of topological sorting, and Chapter 20 shows how to solve this problem efûciently.  A doctor needs to determine whether an image represents a cancerous tumor or a benign one. The doctor has available images of many other tumors, some of which are known to be cancerous and some of which are known to be benign. A cancerous tumor is likely to be more similar to other cancerous tumors than to benign tumors, and a benign tumor is more likely to be similar to other be- nign tumors. By using a clustering algorithm, as in Chapter 33, the doctor can identify which outcome is more likely.  You need to compress a large ûle containing text so that it occupies less space. Many ways to do so are known, including <LZW compression,= which looks for repeating character sequences. Chapter 15 studies a different approach, <Huff- man coding,= which encodes characters by bit sequences of various lengths, with characters occurring more frequently encoded by shorter bit sequences. These lists are far from exhaustive (as you again have probably surmised from this book’s heft), but they exhibit two characteristics common to many interesting algorithmic problems: 1. They have many candidate solutions, the overwhelming majority of which do not solve the problem at hand. Finding one that does, or one that is <best,= with- out explicitly examining each possible solution, can present quite a challenge. 2. They have practical applications. Of the problems in the above list, ûnding the shortest path provides the easiest examples. A transportation ûrm, such as a trucking or railroad company, has a ûnancial interest in ûnding shortest paths through a road or rail network because taking shorter paths results in lower labor and fuel costs. Or a routing node on the internet might need to ûnd the shortest path through the network in order to route a message quickly. Or a person wishing to drive from New York to Boston might want to ûnd driving directions using a navigation app. Not every problem solved by algorithms has an easily identiûed set of candi- date solutions. For example, given a set of numerical values representing samples of a signal taken at regular time intervals, the discrete Fourier transform converts
1.1 Algorithms 9 the time domain to the frequency domain. That is, it approximates the signal as a weighted sum of sinusoids, producing the strength of various frequencies which, when summed, approximate the sampled signal. In addition to lying at the heart of signal processing, discrete Fourier transforms have applications in data compres- sion and multiplying large polynomials and integers. Chapter 30 gives an efûcient algorithm, the fast Fourier transform (commonly called the FFT), for this problem. The chapter also sketches out the design of a hardware FFT circuit. Data structures This book also presents several data structures. A data structure is a way to store and organize data in order to facilitate access and modiûcations. Using the appro- priate data structure or structures is an important part of algorithm design. No sin- gle data structure works well for all purposes, and so you should know the strengths and limitations of several of them. Technique Although you can use this book as a <cookbook= for algorithms, you might some- day encounter a problem for which you cannot readily ûnd a published algorithm (many of the exercises and problems in this book, for example). This book will teach you techniques of algorithm design and analysis so that you can develop al- gorithms on your own, show that they give the correct answer, and analyze their ef- ûciency. Different chapters address different aspects of algorithmic problem solv- ing. Some chapters address speciûc problems, such as ûnding medians and order statistics in Chapter 9, computing minimum spanning trees in Chapter 21, and de- termining a maximum üow in a network in Chapter 24. Other chapters introduce techniques, such as divide-and-conquer in Chapters 2 and 4, dynamic programming in Chapter 14, and amortized analysis in Chapter 16. Hard problems Most of this book is about efûcient algorithms. Our usual measure of efûciency is speed: how long does an algorithm take to produce its result? There are some problems, however, for which we know of no algorithm that runs in a reasonable amount of time. Chapter 34 studies an interesting subset of these problems, which are known as NP-complete. Why are NP-complete problems interesting? First, although no efûcient algo- rithm for an NP-complete problem has ever been found, nobody has ever proven that an efûcient algorithm for one cannot exist. In other words, no one knows whether efûcient algorithms exist for NP-complete problems. Second, the set of
10 Chapter 1 The Role of Algorithms in Computing NP-complete problems has the remarkable property that if an efûcient algorithm exists for any one of them, then efûcient algorithms exist for all of them. This re- lationship among the NP-complete problems makes the lack of efûcient solutions all the more tantalizing. Third, several NP-complete problems are similar, but not identical, to problems for which we do know of efûcient algorithms. Computer scientists are intrigued by how a small change to the problem statement can cause a big change to the efûciency of the best known algorithm. You should know about NP-complete problems because some of them arise sur- prisingly often in real applications. If you are called upon to produce an efûcient algorithm for an NP-complete problem, you are likely to spend a lot of time in a fruitless search. If, instead, you can show that the problem is NP-complete, you can spend your time developing an efûcient approximation algorithm, that is, an algorithm that gives a good, but not necessarily the best possible, solution. As a concrete example, consider a delivery company with a central depot. Each day, it loads up delivery trucks at the depot and sends them around to deliver goods to several addresses. At the end of the day, each truck must end up back at the depot so that it is ready to be loaded for the next day. To reduce costs, the company wants to select an order of delivery stops that yields the lowest overall distance traveled by each truck. This problem is the well-known <traveling-salesperson problem,= and it is NP-complete.2 It has no known efûcient algorithm. Under certain assumptions, however, we know of efûcient algorithms that compute overall distances close to the smallest possible. Chapter 35 discusses such <approximation algorithms.= Alternative computing models For many years, we could count on processor clock speeds increasing at a steady rate. Physical limitations present a fundamental roadblock to ever-increasing clock speeds, however: because power density increases superlinearly with clock speed, chips run the risk of melting once their clock speeds become high enough. In or- der to perform more computations per second, therefore, chips are being designed to contain not just one but several processing <cores.= We can liken these multi- core computers to several sequential computers on a single chip. In other words, they are a type of <parallel computer.= In order to elicit the best performance from multicore computers, we need to design algorithms with parallelism in mind. Chapter 26 presents a model for =task-parallel= algorithms, which take advantage of multiple processing cores. This model has advantages from both theoretical and 2 To be precise, only decision problems4those with a <yes/no= answer4can be NP-complete. The decision version of the traveling salesperson problem asks whether there exists an order of stops whose distance totals at most a given amount.
1.1 Algorithms 11 practical standpoints, and many modern parallel-programming platforms embrace something similar to this model of parallelism. Most of the examples in this book assume that all of the input data are available when an algorithm begins running. Much of the work in algorithm design makes the same assumption. For many important real-world examples, however, the input actually arrives over time, and the algorithm must decide how to proceed without knowing what data will arrive in the future. In a data center, jobs are constantly arriving and departing, and a scheduling algorithm must decide when and where to run a job, without knowing what jobs will be arriving in the future. Trafûc must be routed in the internet based on the current state, without knowing about where trafûc will arrive in the future. Hospital emergency rooms make triage decisions about which patients to treat ûrst without knowing when other patients will be arriving in the future and what treatments they will need. Algorithms that receive their input over time, rather than having all the input present at the start, are online algorithms, which Chapter 27 examines. Exercises 1.1-1 Describe your own real-world example that requires sorting. Describe one that requires ûnding the shortest distance between two points. 1.1-2 Other than speed, what other measures of efûciency might you need to consider in a real-world setting? 1.1-3 Select a data structure that you have seen, and discuss its strengths and limitations. 1.1-4 How are the shortest-path and traveling-salesperson problems given above similar? How are they different? 1.1-5 Suggest a real-world problem in which only the best solution will do. Then come up with one in which <approximately= the best solution is good enough. 1.1-6 Describe a real-world problem in which sometimes the entire input is available before you need to solve the problem, but other times the input is not entirely available in advance and arrives over time.
12 Chapter 1 The Role of Algorithms in Computing 1.2 Algorithms as a technology If computers were inûnitely fast and computer memory were free, would you have any reason to study algorithms? The answer is yes, if for no other reason than that you would still like to be certain that your solution method terminates and does so with the correct answer. If computers were inûnitely fast, any correct method for solving a problem would do. You would probably want your implementation to be within the bounds of good software engineering practice (for example, your implementation should be well designed and documented), but you would most often use whichever method was the easiest to implement. Of course, computers may be fast, but they are not inûnitely fast. Computing time is therefore a bounded resource, which makes it precious. Although the saying goes, <Time is money,= time is even more valuable than money: you can get back money after you spend it, but once time is spent, you can never get it back. Memory may be inexpensive, but it is neither inûnite nor free. You should choose algorithms that use the resources of time and space efûciently. Efûciency Different algorithms devised to solve the same problem often differ dramatically in their efûciency. These differences can be much more signiûcant than differences due to hardware and software. As an example, Chapter 2 introduces two algorithms for sorting. The ûrst, known as insertion sort, takes time roughly equal to c1n2 to sort n items, where c1 is a constant that does not depend on n. That is, it takes time roughly proportional to n2. The second, merge sort, takes time roughly equal to c2n lg n, where lg n stands for log2 n and c2 is another constant that also does not depend on n. Inser- tion sort typically has a smaller constant factor than merge sort, so that c1 < c2. We’ll see that the constant factors can have far less of an impact on the running time than the dependence on the input size n. Let’s write insertion sort’s running time as c1n  n and merge sort’s running time as c2n  lg n. Then we see that where insertion sort has a factor of n in its running time, merge sort has a factor of lg n, which is much smaller. For example, when n is 1000, lg n is approximately 10, and when n is 1,000,000, lg n is approximately only 20. Although insertion sort usu- ally runs faster than merge sort for small input sizes, once the input size n becomes large enough, merge sort’s advantage of lg n versus n more than compensates for the difference in constant factors. No matter how much smaller c1 is than c2, there is always a crossover point beyond which merge sort is faster.
1.2 Algorithms as a technology 13 For a concrete example, let us pit a faster computer (computer A) running inser- tion sort against a slower computer (computer B) running merge sort. They each must sort an array of 10 million numbers. (Although 10 million numbers might seem like a lot, if the numbers are eight-byte integers, then the input occupies about 80 megabytes, which ûts in the memory of even an inexpensive laptop com- puter many times over.) Suppose that computer A executes 10 billion instructions per second (faster than any single sequential computer at the time of this writing) and computer B executes only 10 million instructions per second (much slower than most contemporary computers), so that computer A is 1000 times faster than computer B in raw computing power. To make the difference even more dramatic, suppose that the world’s craftiest programmer codes insertion sort in machine lan- guage for computer A, and the resulting code requires 2n2 instructions to sort n numbers. Suppose further that just an average programmer implements merge sort, using a high-level language with an inefûcient compiler, with the resulting code taking 50 n lg n instructions. To sort 10 million numbers, computer A takes 2  .107/2 instructions 1010 instructions/second D 20,000 seconds (more than 5:5 hours) ; while computer B takes 50  107 lg 107 instructions 107 instructions/second  1163 seconds (under 20 minutes) : By using an algorithm whose running time grows more slowly, even with a poor compiler, computer B runs more than 17 times faster than computer A! The ad- vantage of merge sort is even more pronounced when sorting 100 million numbers: where insertion sort takes more than 23 days, merge sort takes under four hours. Although 100 million might seem like a large number, there are more than 100 mil- lion web searches every half hour, more than 100 million emails sent every minute, and some of the smallest galaxies (known as ultra-compact dwarf galaxies) con- tain about 100 million stars. In general, as the problem size increases, so does the relative advantage of merge sort. Algorithms and other technologies The example above shows that you should consider algorithms, like computer hard- ware, as a technology. Total system performance depends on choosing efûcient algorithms as much as on choosing fast hardware. Just as rapid advances are being made in other computer technologies, they are being made in algorithms as well. You might wonder whether algorithms are truly that important on contemporary computers in light of other advanced technologies, such as
14 Chapter 1 The Role of Algorithms in Computing  advanced computer architectures and fabrication technologies,  easy-to-use, intuitive, graphical user interfaces (GUIs),  object-oriented systems,  integrated web technologies,  fast networking, both wired and wireless,  machine learning,  and mobile devices. The answer is yes. Although some applications do not explicitly require algorith- mic content at the application level (such as some simple, web-based applications), many do. For example, consider a web-based service that determines how to travel from one location to another. Its implementation would rely on fast hardware, a graphical user interface, wide-area networking, and also possibly on object ori- entation. It would also require algorithms for operations such as ûnding routes (probably using a shortest-path algorithm), rendering maps, and interpolating ad- dresses. Moreover, even an application that does not require algorithmic content at the application level relies heavily upon algorithms. Does the application rely on fast hardware? The hardware design used algorithms. Does the application rely on graphical user interfaces? The design of any GUI relies on algorithms. Does the application rely on networking? Routing in networks relies heavily on algorithms. Was the application written in a language other than machine code? Then it was processed by a compiler, interpreter, or assembler, all of which make extensive use of algorithms. Algorithms are at the core of most technologies used in contempo- rary computers. Machine learning can be thought of as a method for performing algorithmic tasks without explicitly designing an algorithm, but instead inferring patterns from data and thereby automatically learning a solution. At ûrst glance, machine learning, which automates the process of algorithmic design, may seem to make learning about algorithms obsolete. The opposite is true, however. Machine learning is itself a collection of algorithms, just under a different name. Furthermore, it cur- rently seems that the successes of machine learning are mainly for problems for which we, as humans, do not really understand what the right algorithm is. Promi- nent examples include computer vision and automatic language translation. For algorithmic problems that humans understand well, such as most of the problems in this book, efûcient algorithms designed to solve a speciûc problem are typically more successful than machine-learning approaches. Data science is an interdisciplinary ûeld with the goal of extracting knowledge and insights from structured and unstructured data. Data science uses methods
Problems for Chapter 1 15 from statistics, computer science, and optimization. The design and analysis of algorithms is fundamental to the ûeld. The core techniques of data science, which overlap signiûcantly with those in machine learning, include many of the algo- rithms in this book. Furthermore, with the ever-increasing capacities of computers, we use them to solve larger problems than ever before. As we saw in the above comparison be- tween insertion sort and merge sort, it is at larger problem sizes that the differences in efûciency between algorithms become particularly prominent. Having a solid base of algorithmic knowledge and technique is one characteristic that deûnes the truly skilled programmer. With modern computing technology, you can accomplish some tasks without knowing much about algorithms, but with a good background in algorithms, you can do much, much more. Exercises 1.2-1 Give an example of an application that requires algorithmic content at the applica- tion level, and discuss the function of the algorithms involved. 1.2-2 Suppose that for inputs of size n on a particular computer, insertion sort runs in 8n2 steps and merge sort runs in 64 n lg n steps. For which values of n does insertion sort beat merge sort? 1.2-3 What is the smallest value of n such that an algorithm whose running time is 100n2 runs faster than an algorithm whose running time is 2n on the same machine? Problems 1-1 Comparison of running times For each function f .n/ and time t in the following table, determine the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes f .n/ microseconds.
16 Chapter 1 The Role of Algorithms in Computing 1 1 1 1 1 1 1 second minute hour day month year century lg n pn n n lg n n2 n3 2n nŠ Chapter notes There are many excellent texts on the general topic of algorithms, including those by Aho, Hopcroft, and Ullman [5, 6], Dasgupta, Papadimitriou, and Vazirani [107], Edmonds [133], Erickson [135], Goodrich and Tamassia [195, 196], Kleinberg and Tardos [257], Knuth [259, 260, 261, 262, 263], Levitin [298], Louridas [305], Mehlhorn and Sanders [325], Mitzenmacher and Upfal [331], Neapolitan [342], Roughgarden [385, 386, 387, 388], Sanders, Mehlhorn, Dietzfelbinger, and De- mentiev [393], Sedgewick and Wayne [402], Skiena [414], Soltys-Kulinicz [419], Wilf [455], and Williamson and Shmoys [459]. Some of the more practical as- pects of algorithm design are discussed by Bentley [49, 50, 51], Bhargava [54], Kochenderfer and Wheeler [268], and McGeoch [321]. Surveys of the ûeld of al- gorithms can also be found in books by Atallah and Blanton [27, 28] and Mehta and Sahhi [326]. For less technical material, see the books by Christian and Grifûths [92], Cormen [104], Erwig [136], MacCormick [307], and V¨ ocking et al. [448]. Overviews of the algorithms used in computational biology can be found in books by Jones and Pevzner [240], Elloumi and Zomaya [134], and Marchisio [315].
2 Getting Started This chapter will familiarize you with the framework we’ll use throughout the book to think about the design and analysis of algorithms. It is self-contained, but it does include several references to material that will be introduced in Chapters 3 and 4. (It also contains several summations, which Appendix A shows how to solve.) We’ll begin by examining the insertion sort algorithm to solve the sorting prob- lem introduced in Chapter 1. We’ll specify algorithms using a pseudocode that should be understandable to you if you have done computer programming. We’ll see why insertion sort correctly sorts and analyze its running time. The analysis introduces a notation that describes how running time increases with the number of items to be sorted. Following a discussion of insertion sort, we’ll use a method called divide-and-conquer to develop a sorting algorithm called merge sort. We’ll end with an analysis of merge sort’s running time. 2.1 Insertion sort Our ûrst algorithm, insertion sort, solves the sorting problem introduced in Chap- ter 1: Input: A sequence of n numbers ha1; a2; : : : ; ani. Output: A permutation (reordering) ha 0 1; a 0 2; : : : ; a 0 ni of the input sequence such that a 0 1 හ a 0 2 හ    හ a 0 n. The numbers to be sorted are also known as the keys. Although the problem is con- ceptually about sorting a sequence, the input comes in the form of an array with n elements. When we want to sort numbers, it’s often because they are the keys associated with other data, which we call satellite data. Together, a key and satel- lite data form a record. For example, consider a spreadsheet containing student records with many associated pieces of data such as age, grade-point average, and number of courses taken. Any one of these quantities could be a key, but when the
18 Chapter 2 Getting Started spreadsheet sorts, it moves the associated record (the satellite data) with the key. When describing a sorting algorithm, we focus on the keys, but it is important to remember that there usually is associated satellite data. In this book, we’ll typically describe algorithms as procedures written in a pseu- docode that is similar in many respects to C, C++, Java, Python,1 or JavaScript. (Apologies if we’ve omitted your favorite programming language. We can’t list them all.) If you have been introduced to any of these languages, you should have little trouble understanding algorithms <coded= in pseudocode. What separates pseudocode from real code is that in pseudocode, we employ whatever expres- sive method is most clear and concise to specify a given algorithm. Sometimes the clearest method is English, so do not be surprised if you come across an En- glish phrase or sentence embedded within a section that looks more like real code. Another difference between pseudocode and real code is that pseudocode often ig- nores aspects of software engineering4such as data abstraction, modularity, and error handling4in order to convey the essence of the algorithm more concisely. We start with insertion sort, which is an efûcient algorithm for sorting a small number of elements. Insertion sort works the way you might sort a hand of playing cards. Start with an empty left hand and the cards in a pile on the table. Pick up the ûrst card in the pile and hold it with your left hand. Then, with your right hand, remove one card at a time from the pile, and insert it into the correct position in your left hand. As Figure 2.1 illustrates, you ûnd the correct position for a card by comparing it with each of the cards already in your left hand, starting at the right and moving left. As soon as you see a card in your left hand whose value is less than or equal to the card you’re holding in your right hand, insert the card that you’re holding in your right hand just to the right of this card in your left hand. If all the cards in your left hand have values greater than the card in your right hand, then place this card as the leftmost card in your left hand. At all times, the cards held in your left hand are sorted, and these cards were originally the top cards of the pile on the table. The pseudocode for insertion sort is given as the procedure INSERTION-SORT on the facing page. It takes two parameters: an array A containing the values to be sorted and the number n of values of sort. The values occupy positions AŒ1� through AŒn� of the array, which we denote by AŒ1 W n�. When the INSERTION- SORT procedure is ûnished, array AŒ1 W n� contains the original values, but in sorted order. 1 If you’re familiar with only Python, you can think of arrays as similar to Python lists.
2.1 Insertion sort 19 2 ♥ ♥ ♥ 2 ♥ 4 ♥♥ ♥ ♥ ♥ 4 ♥ 5 ♥♥ ♥ ♥ ♥ 5 ♥ ♥ 7 ♥ ♥ ♥ ♥ ♥ ♥ ♥ 7 ♥ ♥ 10 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ 1 0 ♥ Figure 2.1 Sorting a hand of cards using insertion sort. INSERTION-SORT.A; n/ 1 for i D 2 to n 2 key D AŒi� 3 / / Insert AŒi� into the sorted subarray AŒ1 Wi  1�. 4 j D i  1 5 while j > 0 and AŒj � > key 6 AŒj C 1� D AŒj � 7 j D j  1 8 AŒj C 1� D key Loop invariants and the correctness of insertion sort Figure 2.2 shows how this algorithm works for an array A that starts out with the sequence h5; 2; 4; 6; 1; 3i. The index i indicates the <current card= being inserted into the hand. At the beginning of each iteration of the for loop, which is indexed by i, the subarray (a contiguous portion of the array) consisting of elements AŒ1 Wi 1� (that is, AŒ1� through AŒi 1�) constitutes the currently sorted hand, and the remaining subarray AŒi C 1 Wn� (elements AŒi C 1� through AŒn�) corresponds to the pile of cards still on the table. In fact, elements AŒ1 Wi  1� are the elements originally in positions 1 through i  1, but now in sorted order. We state these properties of AŒ1 Wi  1� formally as a loop invariant:
20 Chapter 2 Getting Started 1 2 3 4 5 6 5 2 4 6 1 3 (a) 1 2 3 4 5 6 2 5 4 6 1 3 (b) 1 2 3 4 5 6 2 4 5 6 1 3 (c) 1 2 3 4 5 6 2 4 5 6 1 3 (d) 1 2 3 4 5 6 2 4 5 6 1 3 (e) 1 2 3 4 5 6 2 4 5 6 1 3 (f) Figure 2.2 The operation of INSERTION-SORT.A; n/, where A initially contains the sequence h5; 2; 4; 6; 1; 3i and n D 6. Array indices appear above the rectangles, and values stored in the array positions appear within the rectangles. (a)–(e) The iterations of the for loop of lines 138. In each iteration, the blue rectangle holds the key taken from AŒi�, which is compared with the values in tan rectangles to its left in the test of line 5. Orange arrows show array values moved one position to the right in line 6, and blue arrows indicate where the key moves to in line 8. (f) The ûnal sorted array. At the start of each iteration of the for loop of lines 138, the subarray AŒ1 Wi  1� consists of the elements originally in AŒ1 Wi  1�, but in sorted order. Loop invariants help us understand why an algorithm is correct. When you’re using a loop invariant, you need to show three things: Initialization: It is true prior to the ûrst iteration of the loop. Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration. Termination: The loop terminates, and when it terminates, the invariant4usually along with the reason that the loop terminated4gives us a useful property that helps show that the algorithm is correct. When the ûrst two properties hold, the loop invariant is true prior to every iteration of the loop. (Of course, you are free to use established facts other than the loop invariant itself to prove that the loop invariant remains true before each iteration.) A loop-invariant proof is a form of mathematical induction, where to prove that a property holds, you prove a base case and an inductive step. Here, showing that the invariant holds before the ûrst iteration corresponds to the base case, and showing that the invariant holds from iteration to iteration corresponds to the inductive step. The third property is perhaps the most important one, since you are using the loop invariant to show correctness. Typically, you use the loop invariant along with the condition that caused the loop to terminate. Mathematical induction typically applies the inductive step inûnitely, but in a loop invariant the <induction= stops when the loop terminates.
2.1 Insertion sort 21 Let’s see how these properties hold for insertion sort. Initialization: We start by showing that the loop invariant holds before the ûrst loop iteration, when i D 2.2 The subarray AŒ1 Wi  1� consists of just the single element AŒ1�, which is in fact the original element in AŒ1�. Moreover, this subarray is sorted (after all, how could a subarray with just one value not be sorted?), which shows that the loop invariant holds prior to the ûrst iteration of the loop. Maintenance: Next, we tackle the second property: showing that each iteration maintains the loop invariant. Informally, the body of the for loop works by moving the values in AŒi  1�, AŒi  2�, AŒi  3�, and so on by one position to the right until it ûnds the proper position for AŒi� (lines 437), at which point it inserts the value of AŒi� (line 8). The subarray AŒ1 Wi� then consists of the elements originally in AŒ1 Wi�, but in sorted order. Incrementing i (increasing its value by 1) for the next iteration of the for loop then preserves the loop invariant. A more formal treatment of the second property would require us to state and show a loop invariant for the while loop of lines 537. Let’s not get bogged down in such formalism just yet. Instead, we’ll rely on our informal analysis to show that the second property holds for the outer loop. Termination: Finally, we examine loop termination. The loop variable i starts at 2 and increases by 1 in each iteration. Once i’s value exceeds n in line 1, the loop terminates. That is, the loop terminates once i equals n C 1. Substituting n C1 for i in the wording of the loop invariant yields that the subarray AŒ1 Wn� consists of the elements originally in AŒ1 Wn�, but in sorted order. Hence, the algorithm is correct. This method of loop invariants is used to show correctness in various places throughout this book. Pseudocode conventions We use the following conventions in our pseudocode.  Indentation indicates block structure. For example, the body of the for loop that begins on line 1 consists of lines 238, and the body of the while loop that 2 When the loop is a for loop, the loop-invariant check just prior to the ûrst iteration occurs immedi- ately after the initial assignment to the loop-counter variable and just before the ûrst test in the loop header. In the case of INSERTION-SORT, this time is after assigning 2 to the variable i but before the ûrst test of whether i හ n.
22 Chapter 2 Getting Started begins on line 5 contains lines 637 but not line 8. Our indentation style applies to if-else statements3 as well. Using indentation instead of textual indicators of block structure, such as begin and end statements or curly braces, reduces clutter while preserving, or even enhancing, clarity.4  The looping constructs while, for, and repeat-until and the if-else conditional construct have interpretations similar to those in C, C++, Java, Python, and JavaScript.5 In this book, the loop counter retains its value after the loop is exited, unlike some situations that arise in C++ and Java. Thus, immediately after a for loop, the loop counter’s value is the value that ûrst exceeded the for loop bound.6 We used this property in our correctness argument for insertion sort. The for loop header in line 1 is for i D 2 to n, and so when this loop terminates, i equals nC1. We use the keyword to when a for loop increments its loop counter in each iteration, and we use the keyword downto when a for loop decrements its loop counter (reduces its value by 1 in each iteration). When the loop counter changes by an amount greater than 1, the amount of change follows the optional keyword by.  The symbol </ /= indicates that the remainder of the line is a comment.  Variables (such as i, j , and key) are local to the given procedure. We won’t use global variables without explicit indication.  We access array elements by specifying the array name followed by the index in square brackets. For example, AŒi� indicates the ith element of the array A. Although many programming languages enforce 0-origin indexing for arrays (0 is the smallest valid index), we choose whichever indexing scheme is clearest for human readers to understand. Because people usually start counting at 1, not 0, most4but not all4of the arrays in this book use 1-origin indexing. To be 3 In an if-else statement, we indent else at the same level as its matching if. The ûrst executable line of an else clause appears on the same line as the keyword else. For multiway tests, we use elseif for tests after the ûrst one. When it is the ûrst line in an else clause, an if statement appears on the line following else so that you do not misconstrue it as elseif. 4 Each pseudocode procedure in this book appears on one page so that you do not need to discern levels of indentation in pseudocode that is split across pages. 5 Most block-structured languages have equivalent constructs, though the exact syntax may differ. Python lacks repeat-until loops, and its for loops operate differently from the for loops in this book. Think of the pseudocode line <for i D 1 to n= as equivalent to <for i in range(1, n+1)= in Python. 6 In Python, the loop counter retains its value after the loop is exited, but the value it retains is the value it had during the ûnal iteration of the for loop, rather than the value that exceeded the loop bound. That is because a Python for loop iterates through a list, which may contain nonnumeric values.
2.1 Insertion sort 23 clear about whether a particular algorithm assumes 0-origin or 1-origin index- ing, we’ll specify the bounds of the arrays explicitly. If you are implementing an algorithm that we specify using 1-origin indexing, but you’re writing in a programming language that enforces 0-origin indexing (such as C, C++, Java, Python, or JavaScript), then give yourself credit for being able to adjust. You can either always subtract 1 from each index or allocate each array with one extra position and just ignore position 0. The notation <W= denotes a subarray. Thus, AŒi Wj � indicates the subarray of A consisting of the elements AŒi�; AŒi C 1�; : : : ; AŒj �.7 We also use this notation to indicate the bounds of an array, as we did earlier when discussing the array AŒ1 Wn�.  We typically organize compound data into objects, which are composed of attributes. We access a particular attribute using the syntax found in many object-oriented programming languages: the object name, followed by a dot, followed by the attribute name. For example, if an object x has attribute f , we denote this attribute by x:f. We treat a variable representing an array or object as a pointer (known as a reference in some programming languages) to the data representing the array or object. For all attributes f of an object x, setting y D x causes y:f to equal x:f. Moreover, if we now set x:f D 3, then afterward not only does x:f equal 3, but y:f equals 3 as well. In other words, x and y point to the same object after the assignment y D x. This way of treating arrays and objects is consistent with most contemporary programming languages. Our attribute notation can <cascade.= For example, suppose that the attribute f is itself a pointer to some type of object that has an attribute g. Then the notation x:f:g is implicitly parenthesized as .x:f/:g. In other words, if we had assigned y D x:f, then x:f:g is the same as y:g. Sometimes a pointer refers to no object at all. In this case, we give it the special value NIL.  We pass parameters to a procedure by value: the called procedure receives its own copy of the parameters, and if it assigns a value to a parameter, the change is not seen by the calling procedure. When objects are passed, the pointer to the data representing the object is copied, but the object’s attributes are not. For example, if x is a parameter of a called procedure, the assignment x D y within 7 If you’re used to programming in Python, bear in mind that in this book, the subarray AŒi Wj� includes the element AŒj�. In Python, the last element of AŒi Wj� is AŒj 1�. Python allows negative indices, which count from the back end of the list. This book does not use negative array indices.
24 Chapter 2 Getting Started the called procedure is not visible to the calling procedure. The assignment x:f D 3, however, is visible if the calling procedure has a pointer to the same object as x. Similarly, arrays are passed by pointer, so that a pointer to the array is passed, rather than the entire array, and changes to individual array elements are visible to the calling procedure. Again, most contemporary programming languages work this way.  A return statement immediately transfers control back to the point of call in the calling procedure. Most return statements also take a value to pass back to the caller. Our pseudocode differs from many programming languages in that we allow multiple values to be returned in a single return statement without having to create objects to package them together.8  The boolean operators <and= and <or= are short circuiting. That is, evaluate the expression <x and y= by ûrst evaluating x. If x evaluates to FALSE, then the entire expression cannot evaluate to TRUE, and therefore y is not evaluated. If, on the other hand, x evaluates to TRUE, y must be evaluated to determine the value of the entire expression. Similarly, in the expression <x or y= the ex- pression y is evaluated only if x evaluates to FALSE. Short-circuiting operators allow us to write boolean expressions such as <x ¤ NIL and x:f D y= without worrying about what happens upon evaluating x:f when x is NIL.  The keyword error indicates that an error occurred because conditions were wrong for the procedure to have been called, and the procedure immediately terminates. The calling procedure is responsible for handling the error, and so we do not specify what action to take. Exercises 2.1-1 Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on an array initially containing the sequence h31; 41; 59; 26; 41; 58i. 2.1-2 Consider the procedure SUM-ARRAY on the facing page. It computes the sum of the n numbers in array AŒ1 Wn�. State a loop invariant for this procedure, and use its initialization, maintenance, and termination properties to show that the SUM- ARRAY procedure returns the sum of the numbers in AŒ1 Wn�. 8 Python’s tuple notation allows return statements to return multiple values without creating objects from a programmer-deûned class.
2.2 Analyzing algorithms 25 SUM-ARRAY.A; n/ 1 sum D 0 2 for i D 1 to n 3 sum D sum C AŒi� 4 return sum 2.1-3 Rewrite the INSERTION-SORT procedure to sort into monotonically decreasing in- stead of monotonically increasing order. 2.1-4 Consider the searching problem: Input: A sequence of n numbers ha1; a2; : : : ; ani stored in array AŒ1 Wn� and a value x. Output: An index i such that x equals AŒi� or the special value NIL if x does not appear in A. Write pseudocode for linear search, which scans through the array from begin- ning to end, looking for x. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulûlls the three necessary properties. 2.1-5 Consider the problem of adding two n-bit binary integers a and b, stored in two n-element arrays AŒ0 Wn  1� and BŒ0 Wn  1�, where each element is either 0 or 1, a D Pn1 iD0 AŒi�  2i, and b D Pn1 iD0 BŒi�  2i . The sum c D a C b of the two integers should be stored in binary form in an .n C 1/-element array C Œ0 Wn�, where c D Pn iD0 CŒi�  2i. Write a procedure ADD-BINARY-INTEGERS that takes as input arrays A and B, along with the length n, and returns array C holding the sum. 2.2 Analyzing algorithms Analyzing an algorithm has come to mean predicting the resources that the algo- rithm requires. You might consider resources such as memory, communication bandwidth, or energy consumption. Most often, however, you’ll want to measure computational time. If you analyze several candidate algorithms for a problem,
26 Chapter 2 Getting Started you can identify the most efûcient one. There might be more than just one viable candidate, but you can often rule out several inferior algorithms in the process. Before you can analyze an algorithm, you need a model of the technology that it runs on, including the resources of that technology and a way to express their costs. Most of this book assumes a generic one-processor, random-access ma- chine (RAM) model of computation as the implementation technology, with the understanding that algorithms are implemented as computer programs. In the RAM model, instructions execute one after another, with no concurrent operations. The RAM model assumes that each instruction takes the same amount of time as any other instruction and that each data access4using the value of a variable or storing into a variable4takes the same amount of time as any other data access. In other words, in the RAM model each instruction or data access takes a constant amount of time4even indexing into an array.9 Strictly speaking, we should precisely deûne the instructions of the RAM model and their costs. To do so, however, would be tedious and yield little insight into al- gorithm design and analysis. Yet we must be careful not to abuse the RAM model. For example, what if a RAM had an instruction that sorts? Then you could sort in just one step. Such a RAM would be unrealistic, since such instructions do not appear in real computers. Our guide, therefore, is how real computers are de- signed. The RAM model contains instructions commonly found in real computers: arithmetic (such as add, subtract, multiply, divide, remainder, üoor, ceiling), data movement (load, store, copy), and control (conditional and unconditional branch, subroutine call and return). The data types in the RAM model are integer, üoating point (for storing real- number approximations), and character. Real computers do not usually have a separate data type for the boolean values TRUE and FALSE. Instead, they often test whether an integer value is 0 (FALSE) or nonzero (TRUE), as in C. Although we typically do not concern ourselves with precision for üoating-point values in this book (many numbers cannot be represented exactly in üoating point), precision is crucial for most applications. We also assume that each word of data has a limit on the number of bits. For example, when working with inputs of size n, we typically 9 We assume that each element of a given array occupies the same number of bytes and that the elements of a given array are stored in contiguous memory locations. For example, if array AŒ1 Wn� starts at memory address 1000 and each element occupies four bytes, then element AŒi� is at address 1000 C4.i 1/. In general, computing the address in memory of a particular array element requires at most one subtraction (no subtraction for a 0-origin array), one multiplication (often implemented as a shift operation if the element size is an exact power of 2), and one addition. Furthermore, for code that iterates through the elements of an array in order, an optimizing compiler can generate the address of each element using just one addition, by adding the element size to the address of the preceding element.
2.2 Analyzing algorithms 27 assume that integers are represented by c log2 n bits for some constant c  1. We require c  1 so that each word can hold the value of n, enabling us to index the individual input elements, and we restrict c to be a constant so that the word size does not grow arbitrarily. (If the word size could grow arbitrarily, we could store huge amounts of data in one word and operate on it all in constant time4an unrealistic scenario.) Real computers contain instructions not listed above, and such instructions rep- resent a gray area in the RAM model. For example, is exponentiation a constant- time instruction? In the general case, no: to compute xn when x and n are general integers typically takes time logarithmic in n (see equation (31.34) on page 934), and you must worry about whether the result ûts into a computer word. If n is an exact power of 2, however, exponentiation can usually be viewed as a constant-time operation. Many computers have a <shift left= instruction, which in constant time shifts the bits of an integer by n positions to the left. In most computers, shifting the bits of an integer by 1 position to the left is equivalent to multiplying by 2, so that shifting the bits by n positions to the left is equivalent to multiplying by 2n. Therefore, such computers can compute 2n in 1 constant-time instruction by shift- ing the integer 1 by n positions to the left, as long as n is no more than the number of bits in a computer word. We’ll try to avoid such gray areas in the RAM model and treat computing 2n and multiplying by 2n as constant-time operations when the result is small enough to ût in a computer word. The RAM model does not account for the memory hierarchy that is common in contemporary computers. It models neither caches nor virtual memory. Sev- eral other computational models attempt to account for memory-hierarchy effects, which are sometimes signiûcant in real programs on real machines. Section 11.5 and a handful of problems in this book examine memory-hierarchy effects, but for the most part, the analyses in this book do not consider them. Models that include the memory hierarchy are quite a bit more complex than the RAM model, and so they can be difûcult to work with. Moreover, RAM-model analyses are usually excellent predictors of performance on actual machines. Although it is often straightforward to analyze an algorithm in the RAM model, sometimes it can be quite a challenge. You might need to employ mathematical tools such as combinatorics, probability theory, algebraic dexterity, and the ability to identify the most signiûcant terms in a formula. Because an algorithm might behave differently for each possible input, we need a means for summarizing that behavior in simple, easily understood formulas. Analysis of insertion sort How long does the INSERTION-SORT procedure take? One way to tell would be for you to run it on your computer and time how long it takes to run. Of course, you’d
28 Chapter 2 Getting Started ûrst have to implement it in a real programming language, since you cannot run our pseudocode directly. What would such a timing test tell you? You would ûnd out how long insertion sort takes to run on your particular computer, on that particular input, under the particular implementation that you created, with the particular compiler or interpreter that you ran, with the particular libraries that you linked in, and with the particular background tasks that were running on your computer concurrently with your timing test (such as checking for incoming information over a network). If you run insertion sort again on your computer with the same input, you might even get a different timing result. From running just one implementation of insertion sort on just one computer and on just one input, what would you be able to determine about insertion sort’s running time if you were to give it a different input, if you were to run it on a different computer, or if you were to implement it in a different programming language? Not much. We need a way to predict, given a new input, how long insertion sort will take. Instead of timing a run, or even several runs, of insertion sort, we can determine how long it takes by analyzing the algorithm itself. We’ll examine how many times it executes each line of pseudocode and how long each line of pseudocode takes to run. We’ll ûrst come up with a precise but complicated formula for the running time. Then, we’ll distill the important part of the formula using a convenient no- tation that can help us compare the running times of different algorithms for the same problem. How do we analyze insertion sort? First, let’s acknowledge that the running time depends on the input. You shouldn’t be terribly surprised that sorting a thousand numbers takes longer than sorting three numbers. Moreover, insertion sort can take different amounts of time to sort two input arrays of the same size, depending on how nearly sorted they already are. Even though the running time can depend on many features of the input, we’ll focus on the one that has been shown to have the greatest effect, namely the size of the input, and describe the running time of a program as a function of the size of its input. To do so, we need to deûne the terms <running time= and <input size= more carefully. We also need to be clear about whether we are discussing the running time for an input that elicits the worst-case behavior, the best-case behavior, or some other case. The best notion for input size depends on the problem being studied. For many problems, such as sorting or computing discrete Fourier transforms, the most nat- ural measure is the number of items in the input4for example, the number n of items being sorted. For many other problems, such as multiplying two integers, the best measure of input size is the total number of bits needed to represent the input in ordinary binary notation. Sometimes it is more appropriate to describe the size of the input with more than just one number. For example, if the input to an algorithm is a graph, we usually characterize the input size by both the number
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cheaper than in Coblenz, none of them as high as four marks, even with war tax, poor tax, and “wardrobe.” The house was crowded with the serious-minded of all classes, Poles as well as Germans; the actors were of higher histrionic ability than the average American town of the size of Bromberg sees once a year. Yet equally splendid performances were offered here at these slight prices all the year round. As I strolled hotelward with that pleasant sensation of satisfaction that comes from an evening of genuine entertainment, I could not but wonder whether this, and those other undeniable advantages of German Kultur, whatever sins might justly be charged against it, would be kept up after the Poles had taken Bromberg into their own keeping. As to the walking trip through these eastern provinces which I had planned, fate was once more against me. I might, to be sure, have set out on foot toward the region already amputated from the Empire, but in the course of an hour I should have had the privilege of walking back again. The German-Polish front was just six kilometers from Bromberg, and a wandering stranger would have had exactly the same chance of crossing its succession of trenches as of entering Germany from France a year before. The one and only way of reaching the province of Posen was by train from the village of Kreuz, back along the railway by which I had come. The place had all the appearance of an international frontier, a frontier hastily erected and not yet in efficient running order. Arrangements for examining travelers and baggage consisted only of an improvised fence along the station platform, strewn pellmell with a heterogeneous throng bound in both directions, and their multifarious coffers and bundles. The soldiers who patrolled the line of demarkation with fixed bayonets were callow, thin-faced youths, or men past middle age who had plainly reached the stage of uselessness as combat troops. All wore on their collars the silver oak- leaves of the recently formed “frontier guard.” Their manner toward the harassed travelers was either brutal or cringingly friendly. The Germans in civilian garb who examined passports and baggage were cantankerous and gruff, as if they resented the existence of a frontier where the Fatherland had never admitted that a frontier existed. They vented their wrath especially against men of military age who wished to enter Polish territory—and their interpretation of their
duties in that respect was by no means charitable. Among others, a wretched little dwarf past fifty, whom a glance sufficed to recognize as useless from a military point of view, even had his papers not been stamped with the official Untauglich, was wantonly turned back. Many a family was left only the choice of abandoning the attempt to reach its home or of leaving its adult male members behind. The churls allowed me to pass readily enough, but rescinded their action a moment later. Once beyond the barrier, I had paused to photograph the pandemonium that reigned about it. A lieutenant bellowed and a group of soldiers and officials quickly swarmed about me. Did I not know that photography was forbidden at the front? I protested that the station scenes of Kreuz could scarcely be called military information. What of that? I knew that it was within the zone of the armies, did I not? Rules were rules; it was not the privilege of every Tom, Dick, and Harry to interpret them to his own liking. A lean, hawk-faced civilian, who seemed to be in command, ordered me to open my kodak and confiscated the film it contained. If I set great store by the pictures on it, he would have it developed by the military authorities and let me have those that proved harmless, upon my return. I thanked him for his leniency and strolled toward the compartment I had chosen. Before I had reached it he called me back. “Let me see your papers again,” he demanded, in a far gruffer tone. He glanced casually at them, thrust them into a pocket of his coat, and snapped angrily: “Get your baggage off the train! I am not going to let you through.” It was plain that he was acting from personal rather than official motives. Probably he considered my failure to raise my hat and to smile the sycophant smile with which my fellow-passengers addressed him as an affront to his high Prussian caste. Fortunately he was not alone in command. A more even-tempered official without his dyspeptic leanness beckoned him aside and whispered in his ear. Perhaps he called his attention to the importance of my credentials from Wilhelmstrasse. At any rate, he surrendered my papers after some argument, with an angry shrug of the shoulders, and his less hungry-looking companion brought them back to me. “It has all been arranged,” he smirked. “You may take the train.”
This was still manned by a German crew. For every car that left their territory, however, the Poles required that one of the same class and condition be delivered to them in exchange. Several long freight- trains, loaded from end to end with potatoes, rumbled past us on the parallel track. Two hundred thousand tons of tubers were sent to Germany each month in exchange for coal. It was at that date the only commercial intercourse between the two countries, and explained why potatoes were the one foodstuff of comparative abundance even in Berlin. At Biala the station guards were Polish, but there was little indeed to distinguish them from those of Kreuz and Bromberg. Their uniforms, their rifles, every detail of their equipment, were German, except that some of them wore the square and rather clumsy-looking Polish cap or had decorated their round, red-banded fatigue bonnets with the silver double-eagle of the resurrected empire. Many were without even this insignia of their new allegiance, and only the absence of oak-leaves on their collars showed that they were no longer soldiers of the Fatherland. We halted at Wronki for two hours, which made our departure three hours later, for clocks and watches were turned ahead to correspond with Polish time. Frontier formalities were even more leisurely and disorganized than they had been in Kreuz. The Poles seemed to have something of the amiable but headless temperament of the French. Their officers, too, in their impressive new uniforms with broad red or yellow bands, and their rattling sabers, bore a certain resemblance to children on Christmas morning that did not help to expedite matters under their jurisdiction. They were a bit less “snappy” than the more experienced Germans, somewhat inclined to strut and to flirt, and there were suggestions in their manner that they might not have been horrified at the offer of a tip. When at length my turn had come they found my credentials unsatisfactory. Why had they not been viséed by the Polish consul in Berlin, as well as by the Germans at Frankfurt? I had never dreamed that Berlin boasted a Polish consul. Indeed! Who, then, did I suppose handled the interests of their nation there? However, it was all right. As an American and a fellow-Ally they would let me pass. But I must promise to report at a certain office in Posen within twenty-four hours of my arrival.
Barefoot boys were selling huge slabs of bread and generous lengths of sausage through the car windows. All things are relative, and to the travelers from Germany these “ticket-free” viands of doubtful origin seemed a kingly repast. With every mile forward now it was easier to understand why the loss of the province of Posen had been so serious a blow to the hungry Empire. Here were no arid, sandy stretches, but an endless expanse of rich black loam, capable of feeding many times its rather sparse population. If it had been “pumped dry” by the former oppressors, it was already well on the road to recovery. Wheat, corn, and potatoes covered the flat plains to the horizons on either hand. Cattle and sheep were by no means rare; pigs, goats, ducks, and chickens flocked about every village and farm- house, evidently living in democratic equality with the human inhabitants. There were other suggestions that we were approaching the easy-going East. Men in high Russian boots sauntered behind their draft animals with the leisureliness of those who know the world was not built in a day, nor yet in a year. Churches of Oriental aspect, with steep roofs that were still not Gothic, broke the sameness of the prevailing German architecture. There was something softly un-Occidental in the atmosphere of the great city into which we rumbled at sunset, a city which huge new sign-boards on the station platform stridently announced was no longer Posen, but “Poznan.”
T XI AN AMPUTATED MEMBER (Posen under the Poles) he same spirit that had led the Poles to impress so forcibly upon the traveler the fact that the city in which he had just arrived was now called Poznan (pronounced Poznánya) had manifested itself in a thousand other changes. In so far as time had permitted, every official signboard had already been rendered into Polish and the detested German ones cast into outer darkness. Only those familiar with the Slavic tongue of the new rulers could have guessed what all those glitteringly new enameled placards that adorned the still Boche-featured station were commanding them to do or not to do. Every street in town had been baptized into the new faith and gaily boasted that fact on every corner. For a time the names had been announced in both languages, as in Metz; but a month or so before my arrival the radicals had prevailed and the older placards had been abolished. True, in most cases the new ones were merely translations of the old. But what did it help the German resident who had neglected to learn Polish to know that the “Alte Markt” was still the “Old Market” so long as he could not recognize it under the new designation of “Stary Rynek”? Imagine, if you can, the sensation of waking up some morning to find that Main Street has become Ulica Glòwna, or to discover that the street-car you had always taken no longer runs to Forest Park but to Ogrott Lass. Nothing but the few things that defied quick change, such as post- boxes or names deeply cut into stone façades, had escaped the all- embracing renovation. Indeed, many of these had been deliberately defaced. The cast-iron “Haltestelle der Strassenbahn” high up on the
trolley-supports had been daubed with red paint, though they were still recognizable to motormen and would-be passengers. Many business houses had followed the official lead, and private signs were more apt than not to have the German words that had once called attention to the excellence of the wares within crudely effaced or changed to the new tongue. Sometimes it was not merely the language that had been altered, but the whole tenor of the proprietor’s allegiance. A popular underground beer-hall in the heart of town was no longer the “Bismarck Tunnel,” but the “Tunel Wilsona.” German trucks thundering by on their iron tires bore the white eagle of Poland instead of the black Prussian bird of prey. German newspapers were still published, but as the streets they mentioned were nowhere to be found in all Poznan, their advertisements and much of their news were rather pointless. It gave me a curiously helpless feeling to find myself for the first time in years unable to guess a word of the language about me. Fortunately all Poznan still spoke German. Only once during my stay there did I find myself hampered by my ignorance of Polish—when a theater- ticket office proved to be in charge of a pair recently arrived from Warsaw. On more than one occasion my advances were received coldly, sometimes with scowls. But a reply was always forthcoming, and whenever I announced myself an American, who spoke the less welcome of the two tongues by necessity rather than by choice, apology and friendly overtures immediately followed. Having effaced the lingual reminders of their late oppressors, the Poznanians had proceeded to pay their respects to the bronze heroes they had left behind. The Germans, as is their custom, had littered the public squares with statues of their chief sword-brandishers, in gigantic size—tender reminders to the conquered people of the blessings that had been forced upon them. The downfall of these had been sudden and unceremonious. Some had descended so hastily that the allegorical figures at their feet had suffered the fate so often overtaking faithful henchmen of the fallen mighty. The stone image of an old woman representing “Sorrow” looked doubly sorrowful with broken nose, legs, and fingers. Kaiser Friedrich, Doctor Bismarck with his panacea of “blood and iron,” the world-famed Wilhelm, had all left behind them imposing pedestals, like university chairs awaiting exponents of newer and more lasting doctrines. Here
and there a statue had remained, because it was Polish, but these were few and small and tucked away into the more obscure corners. Next to its change of tongue the most striking feature of the new Poznan was its military aspect. The streets swarmed with soldiers even during the day; in the evening the chief gathering-places became pulsating seas of field gray. For it was still the garb of their former servitude that clothed the vast majority of these warriors of the reborn nation. The silver double-eagle on his service-faded cap was all that was needed to turn a wearer of the German uniform into a soldier of Poland. Many still wore their “Gott mit uns” belt-buckles and their Prussian buttons. A scattering minority, officers for the most part, were conspicuous in the full new Polish uniform—double- breasted, with a forest-green tinge. The high, square cap, distinctive only of the province of Poznan, was more widely in evidence; the less cumbersome headgear of military visitors from Warsaw or Galizia now and then broke the red-banded monotony. But the only universal sign of new fealty was the silver double-eagle. This gleamed everywhere. Men in civilian garb wore it on their hats or in their coat lapels; women adorned their bodices with it; boys and girls proudly displayed it in some conspicuous position. It fluttered on a thousand banners; it bedecked every Polish shop-front; it stared from the covers of newly appeared books, pamphlets, music-sheets in the popular tongue; the very church spires had replaced their crosses with it. One could buy the resurrected insignia, of any size or material, in almost any shop—providing one could produce “legitimation papers” or other proof that it would not be used to disguise a German as a Pole. An over-abundance of swords tended to give the new army a comic-opera aspect, but this detail was offset by the genuine military bearing of all but a few of the multitude in uniform. The great majority, of course, had had German training. Now, however, they put the “pep” of a new game into the old forms of soldierly etiquette. Their two-finger salute was rendered with the precision of ambitious recruits and at the same time with the exactitude of “old-timers.” They sprang unfailingly to attention at sight of a superior officer and stood like automatons until he turned away. Yet there seemed to be an un-German comradeship between the rank and file and the commissioned personnel, a democracy of endeavor, a feeling that
they were all embarked together on the same big new adventure. There were, to be sure, some officers and a few men whose sidewalk manners suggested that they had learned Prussian ways a bit too thoroughly, but they were lost in a mass that had something of the easy-going temperament of the East or the South. All classes of the Polish population were represented in the new army from the bulking countryman who ran after me to say that the photograph I had just taken of him would not be a success because he had not been looking at the lens during the operation to the major who granted me special permission to use my kodak in spite of military rules. This officer had been late in reaching his office, and I passed the time in his anteroom in conversation with his sergeant- major. When he entered at last the entire office force sprang to its feet with what in an older army would have been an exaggeration of discipline. The sergeant-major, his middle finger glued to the seams of his trousers, explained my presence and request. The major asked several questions in Polish, which the sergeant repeated to me in German, relaying my replies back to the major in his native tongue. When the latter had nodded his approval and disappeared, and the office force had relaxed into mere human beings, I expressed my surprise that an officer of such high rank knew no German. “Knows no German!” cried the sergeant-major, bursting into laughter. “The major was for nine years a captain in the German army. He is a graduate of the War College in Berlin and was a member of Hindenburg’s staff. But he never lets a word of the accursed tongue pass his lips if he can possibly avoid it.” The new Polish government had established a conscription act as drastic as if it had been taken bodily from the old German statute- books. All males between the ages of seventeen and forty-five were liable to service. Those between eighteen and thirty had already been called to the colors, though thus far German residents had been tacitly exempted. Every afternoon of my stay in Poznan a hundred or two of recruits, flower-bedecked and carrying each his carton of travel rations, marched in column of squads from the railway station to what had once been the Kaiser’s barracks, singing as they went some rousing Polish song of the olden days. At least half of them wore more or less complete German uniforms. Some were so under- sized that a rifle in their hands would have resembled a machine-
gun. But with few exceptions their military bearing testified to previous training under the exacting drill-sergeants of their former rulers. Watching this new addition each day to the hordes in uniform that already crowded the city, one could not but wonder whether the new Poland was not giving refuge, perhaps unconsciously, to the discredited spirit of militarism that had so recently been expelled from its German Fatherland. The “revolution,” or “Putsch,” as the Poles call it, that brought about all this new state of affairs had been brief and to the point. Paderewski, relying, perhaps, on Germany’s promise to help re- establish the ancient Polish Kingdom, had come to Posen for the Christmas holidays. The hotel he occupied had been decorated with the flags of the Allies. It is scarcely surprising that the Germans proceeded to tear them down in spite of the armistice that had recently been concluded. According to several observers, they might even have “got away with” this had they not persisted in their Prussian aggressiveness. On December 27th a Polish youth paused to ask another for a light from his cigarette. Matches had long been precious things in Posen. A German officer pounced upon the pair and demanded to know what conspiracy they were hatching together. The Polish youths quite properly knocked him down. Their companions joined in the fracas. The Polish turnvereins had long had everything prepared for just such an eventuality. Word swept like prairie fire through the city. French and Italian prisoners of war sprang to such arms as they could lay hands on and added their assistance. The soldiers of the garrison, being chiefly Poles or of Polish sympathies, walked out almost in a body and joined the revolt. It raged for twenty-four hours. In the words of the sergeant-major already introduced: “It was a busy day from four in the morning until the following dawn. At least sixty ribs were broken—mostly German ones.” There have been bloodier revolutions, however, for the number killed is set at ten. The Polish leaders were soon masters of the situation. In three days they had established order. Their search for arms was thorough and included Polish as well as German houses. The government they had already established in secret soon tautened the reins that had been struck from the hands of the Germans, and by New Year’s Day Poznan had already settled down to peace and to a contentment it had not known in more than a century.
As far, at least, as outward appearances go, there was nothing particularly oppressive about the new rule. Civilians were not permitted on the streets after midnight, but those with any legitimate excuse for night-hawking were granted special passes. The Poles showed a tendency to meet half-way their next-door neighbor and late oppressor. With the exception of a few “Polen-fresser,” German residents were not driven out, as in Metz and Strassburg. Boche merchants continued to do business at the old stand. Newspapers published in Germany were refused admittance, but that was a fair retaliation for similar action by the new authorities of the late Empire. Even the detested statues were not overthrown until March, when the Germans declined to give the Poles port facilities at Danzig. The language of the schools, as well as of government offices, was changed to Polish; but as soon as Berlin consented to a reciprocal arrangement, German was restored to the curriculum, though it was taught only a few hours a week, as a foreign tongue. In short, the conditions of Bromberg had been nicely reversed in Poznan. It must, to be sure, have been rather a tough life for the town braggart who had always espoused the German cause; but there was apparently nothing to be feared by those who know how to hold their tongues and confine their attention to their own affairs—and the German is a past-master at lying low when it is to his interest to do so. His native tongue was almost never heard on the streets, such arrogance as existed was confined now to the Poles, and the just-let-us-alone-and- we’ll-be-good rôle had been assumed by the Teutons. There were suggestions, however, that the Poles were not yet adepts at governing, nor likely soon to establish a modern Utopia. Already they had succeeded in encumbering themselves with fully as much red-tape as the French. A musician as national leader and rallying-point seemed to be in keeping with the Polish temperament. There was a lack of practical directness in their methods, a tendency toward the erratic, at the expense of orderly progress. One of their foremost business men turned high official, to whom I applied for a signature and the imprint of a government stamp, received me with a protest that he was “too busy to breathe”—and spent two hours reciting Polish poetry to me and demonstrating how he had succeeded in photographing every secret document that had reached Posen during the war without being once suspected by the Germans. “I am not experienced in this business of government,” he
apologized, when I succeeded at last in taking my leave, “but I am ready to sacrifice myself and all I have to the new Poland.” The statement rang true in his case, but there were others whose repetition of it would have raised grave suspicions that they were putting the cart before the horse. The rush for government jobs under the new régime had in it something of the attitude of the faithful henchmen toward the periodical return to power of their beloved Tammany. There were tender reminiscences of the A. E. F. in the flocks of incompetent pretty girls who encumbered government offices, dipping their charming noses into everything except that which concerned them, as there was in the tendency on the part of both sexes to consider government transportation synonymous with opportunity for “joy-riding.” It will be strange if the Polish servant-girls and factory hands who come to us in the future bring with them the accept-anything spirit of the past, at least after the period of orientation to their new environment is over. They are “feeling their oats” at home now and will be apt to set their worth and their rights to full equality correspondingly higher. The Poles, evidently, are not by nature a frolicsome people, but they seemed to have thrown away the “lid” in Poznan and given free play to all the joy within them. Pianos were more in evidence than they had been during all the twenty months I had spent in war-torn Europe. Children appeared to have taken on a new gaiety. Night life was almost Parisian, except in the more reprehensible features of the “City of Light.” It may have been due only to a temporary difference of mood in the two races, but Polish Poznan struck me as a far more livable place than German Berlin. Evidently the people of the provinces were not letting this new attractiveness of the restored city escape them; the newspapers bristled with offers of reward for any one giving information of apartments or houses for rent. Underneath their merriness, however, the religious current of the race still ran strong and swift. The churches discharged multitudes daily at the end of morning mass; no male, be he coachman, policeman, soldier, or newsboy, ever passed the crucifix at the end of the principal bridge without reverently raising his hat. There are Protestant Poles, but they apparently do not live in Poznan. Now and again, too, there were episodes quite the opposite of gay to give the city pause in the midst of its revelry—the drunken sots in uniform, for instance, who
canvassed the shops demanding alms and prophesying the firing- squad for those who declined to contribute. Were they not perhaps the outposts of Bolshevism? But all this was immersed in the general gaiety, tinged with a mild Orientalism that showed itself not only in the architecture, but in such leisurely customs as closing shops and offices from one to three, in defiance of nearly a century and a half of the sterner German influence. It is quite possible that the increased liveliness of the Poznanians was as much due to the fact that they had plenty to eat as to their release from Teutonic bondage. The two things had come together. Being perhaps the richest agricultural district of the late Empire, the province of Posen was quick to recover its alimentary footing, once its frontiers had been closed against the all-devouring German. With the exception of potatoes, of which the supply was well in excess of local needs, the exportation of foodstuffs toward the hungry West had absolutely ceased. The result was more than noticeable in Poznan; it was conspicuous, all but overpowering, particularly to those arriving from famished Germany. Street after street was lined with a constant tantalization to the new-comer from the West, arousing his resentment at the appetite that was so easily satisfied after its constant vociferations in days gone by—and still to come. Butcher shops displayed an abundance of everything from frankfurters to sides of beef. Cheese, butter, eggs by the bushel, candy, sugar, sweetmeats were heaped high behind glass fronts that would have been slight protection for them in Berlin. In what were now known as “restauracya” one might order a breakfast of eggs, bacon, milk, butter, and all the other things the mere mention of which would have turned a German Wirt livid with rage, without so much as exciting a ripple on the waiter’s brow. At the rathskeller of Poznan’s artistic old city hall a “steak and everything,” such a steak as not even a war-profiteer could command anywhere in Germany, cost a mere seven marks, including the inevitable mug of beer and the “10 per cent. for service” that was exacted here also by the Kellners’ union. With the low rate of exchange—for Poznan was still using German money—the price was considerably less than it would have been in New York at the same date. Far from being short of fats, the Poles were overgenerous with their grease and gravies. Bacon could be had in any quantity at six marks a pound; eggs at thirty pfennigs each. Bread, brown but excellent, was unlimited. Food-
tickets, unknown in hotels and restaurants, were theoretically required for a few of the principal articles in the shops, but there was little difficulty in purchasing without them, at least with the payment of a slight “premium.” On market-days the immense square allotted to them was densely crowded from corner to corner by curiously garbed female hawkers and countrymen offering every conceivable product of their farms and gardens. Poznan still consumed a few things that do not appear on the American bill of fare, such as doves, gull eggs, and various species of weeds and grasses; but the fact remains that the well-to-do could get anything their appetites craved, and the poor were immensely better off than in any city of Germany. There was only one shortage that irked the popular soul. Expression of it rang incessantly in my ears—“Please tell America to send us tobacco!” The queues before tobacconists’ shops were as long and as persistent as in Germany. Ragged men of the street eagerly parted with a precious fifty-pfennig “shin-plaster” for a miserable “cigarette” filled for only half its length with an unsuccessful imitation of tobacco. The principal café, having husbanded its supply of the genuine article, placed a thousand of them on sale each evening at eight, “as a special favor to our clients.” By that hour entrance was quite impossible, and though only two were allowed each purchaser, there was nothing but the empty box left five minutes later. Unselfishness is not one of mankind’s chief virtues, particularly in that chaos of conflicting interests known to the world as central Europe. In view of all they had won in so short a time, and amid the German shrieks of protest, it was disconcerting to find that the Poles were far from satisfied with what had been granted them by the Peace Conference. From high government officials to the man in the street they deluged me with their complaints, often naïvely implying that I had personally had some hand in framing the terms of the proposed treaty, or at least the power to have them altered before it was too late. They were dissatisfied with the western frontier that had been set for them, especially in West Prussia; they were particularly disgruntled because they had not been given Danzig outright. A nation of thirty million people should have a harbor of its own. Danzig was essentially Polish in its sympathies, in spite of the deliberate Germanization that had been practised upon it. Strangely enough they accused America of having blocked their aspirations in
that particular. They blamed Wilson personally for having shut them out of Danzig, as well as for the annoying delay in drawing up the treaty. The Germans had “got at him” through the Jews. The latter had far too much power in the American government, as well as in American finances. The impression was wide-spread in Poznan that Mrs. Wilson is Jewish. The Germans and the Jews had always stuck together. Poland had always been far too lenient with the Jews. She had let them in too easily; had granted them citizenship too readily. As they spoke either Yiddish, an offshoot of German, or Russian, they had always lined up with the enemies of Poland. Half the German spies, every one of the Russian spies with whom Polish territory had been flooded during the war, had been Jews. The Poles in America had gathered money for the alleviation of suffering in their home-land, and had given it to Jews, Germans, and Poles, irrespective of race. The Jews in America had collected similar funds and had expended them only among the Jews. From whatever point of view one approached him, the resident of Poznan had nothing good to say of the Chosen People. The story of Posen’s existence under German rule, now happily ended, was largely a repetition of what had already been told me in Bromberg. In some ways this region had been even more harshly treated, if my informants were trustworthy. Polish skilled workmen “clear down to button-makers” had been driven out of the province. Great numbers had been more or less forcibly compelled to migrate into Germany. There were at least four hundred thousand Poles in the mines and factories of Westphalia. Saxony was half Polish; the district between Hamburg and Bremen was almost entirely Slavish in population. The Ansiedler—the German settlers whom the government had brought to Posen—had acquired all the best land. On the other hand, German Catholics were not allowed to establish themselves in the province of Posen, lest they join their coreligionists against the Protestant oppressors. Perhaps the thing that rankled most was the banishment of the Polish language from the schools. One could scarcely speak it with one’s children at home, for fear of their using it before the teacher. Many of the youngsters had never more than half learned it. In twenty years more no one would have dared speak Polish in public. Men had been given three, and even four, months in prison for privately teaching their children Polish history. The schools were hopelessly Prussianized; the German
teachers received a special premium of one thousand marks or more a year over the regular salaries. All railway jobs went to Germans, except those of section men at two marks a day. There had been Polish newspapers and theaters, but they had never been allowed any freedom of thought or action. “The trouble with the German, or at least the Prussian,” one new official put in, “is that it is his nature to get things by force. He was born that way. Why, the Prussians stole even their name; it was originally Barrusen, as the little corner of Russia was called where the robbers first banded together. They marauded their way westward and southward, treading first little people and then little nations under their iron heels. The very word the German uses for “get” or “obtain” tells his history. It is kriegen, to win by war—krieg. You seldom hear him use the gentler bekommen. Everything he possesses he has gekriegt. Then he is such a hypocrite! In 1916, when we Poles first began to suffer seriously from hunger, some German officers came with baskets of fruit and sandwiches, gathered a group of Polish urchins, filled their hands with the food, and had themselves photographed with them, to show the world how generous and kind-hearted they were. But they did not tell the world that the moment the photographs had been taken the food was snatched away from the hungry children again, some of the officers boxing their ears, and sent back to the German barracks. How do you think the Poles who have been crippled for life fighting for the ‘Fatherland’ feel as they hobble about our streets? What would you say to serving five years in the German army only to be interned as a dangerous enemy alien at the end of it, as is the case with thousands of our sons who were not able to get across the frontier in time? No, the Germans in Poznan are not oppressed as our people were under their rule. We are altogether too soft-hearted with them.” The German residents themselves, as was to be expected, took a different view of the situation. When the Polish authorities had decorated my passport with permission to return to Berlin, I took no chances of being held up by the cantankerous dyspeptic at Kreuz and applied for a new visé by the German Volksrat of Posen. It occupied a modest little dwelling-house on the wide, curving avenue no longer recognizable under its former title of “Kaiser Wilhelm Ring.” Barely had I established my identity when the gloomy Germans took me to
their bosom. Had I been fully informed of their side of the situation? Would I not do them the kindness to return at eleven, when they would see to it that men of high standing were there to give me the real facts of the case? My impressions of Posen would be wholly false if I left it after having consorted only with Poles. As a matter of fact I had already “consorted” with no small number of German residents, chiefly of the small-merchant class. Those I had found somewhat mixed in their minds. A few still prophesied a “peasants’ war” in the territory allotted to Poland; a number of them shivered with apprehension of a “general Bolshevist uprising.” But fully as many pooh-poohed both those cheerful bogies. One thing only was certain—that without exception they were doing business as usual and would continue to do so as long as the Poles permitted it. The feeling for the “Fatherland” did not seem strong enough among the overwhelming majority of them to stand the strain of personal sacrifice. When I returned at eleven the Volksrat had been convoked in unofficial special session. A half-dozen of the men who had formerly held high places in the Municipal Council rose ostentatiously to their feet as I was ushered into the chief sanctum, and did not sit down again until I had been comfortably seated. The chief spokesman had long been something corresponding to chairman of the Board of Aldermen. His close-cropped head glistened in the sunshine that entered through the window at his elbow, and his little ferret-like eyes alternately sought to bore their way into my mental processes and to light up with a winsome naïveté which he did not really possess. Most of the words I set down here are his, though some of them were now and then thrown in by his subservient but approving companions. “With us Germans,” he began, “it has become a case of ‘Vogel friss oder starb’—eat crow or die. We are forced, for the time at least, to accept what the Poles see fit to allow us. The German residents of Posen are not exactly oppressed, but our lives are hemmed in by a thousand petty annoyances, some of them highly discouraging. Take, for instance, this matter of the street names. Granted that the Poles had the right to put them up in their own language. It was certainly a sign of fanaticism to tear down the German names. More than a fourth of the residents of Posen cannot read the new street placards.
There is not a Polish map of the city in existence. When the province of Posen came back to us the Polish street names were allowed to remain until 1879—for more than a hundred years. It is a sign of childishness, of retarded mentality, to daub with red paint all the German signs they cannot remove! It isn’t much more than that to have forbidden the use of our tongue in governmental affairs. We Germans used both languages officially clear up to 1876. We even had the old Prussian laws translated into Polish. It is only during the last ten years that nothing but German was permitted in the public schools; and there have always been plenty of Polish private schools. I am still technically a member of the Municipal Council, but I cannot understand a word of the proceedings, because they are in Polish. Our lawyers cannot practise unless they use that language, although the judges, who pretend not to know German, speak it as readily as you or I. Yet these same lawyers cannot get back into Germany. At least give us time to learn Polish before abolishing German! Many a man born here cannot speak it. There are German children of eighteen or twenty, who have never been outside the province, who are now learning Polish—that is, to write and speak it correctly. “Oh yes, to be sure, we can most of us get permission in three or four weeks to leave the province, but only by abandoning most of our possessions and taking an oath never to return. No wonder so many Germans become Poles overnight. You can hardly expect otherwise, when they have lived here all their lives and have all their property and friends and interests here. No, military service is not required of Germans, even if they were born here; but many of our youths have voluntarily become Polish soldiers, for the same reason that their parents have suddenly turned Poles. Naturally, there is fighting along the boundary of the province. The Poles want to fight, so they can have an excuse to keep their men under arms, and what can Germany do but protect herself? Poland is planning to become an aggressive, militaristic nation, as was falsely charged against the Fatherland by her enemies. “The complaints of the Poles at our rule were ridiculous. We paid German teachers a premium because they had harder work in teaching German to Polish children and in seeing that they did not speak the language that was unwisely used at home. Railroad jobs,
except common labor, were given to Germans because they were more efficient and trustworthy. Besides, does not Germany own the railroads? They complain that the best land was taken by German settlers; but the Poles were only too glad to sell to our Ansiedler—at high prices. Now they are attacking us with a fanaticism of the Middle Ages. Eighteen hundred German teachers, men who have been educating the Poles for twenty or twenty-five years, have suddenly been discharged and ordered to vacate government property within four weeks—yet they are not allowed to go back to Germany. The Pole is still part barbarian; he is more heartless than his cousin the Russian. “Seventy per cent. of the taxes in the province of Posen are paid by Germans. Yet no German who was not born here can vote, though Poles who were not can. I know a village where there are seventy Germans and five Poles—and the five Poles run things to suit themselves. Husbands, wives, and sons often have different rights of suffrage. The family of Baron X has lived here for a hundred and fifty years. The baron himself happens to have been born in Berlin, because his mother went there to see a doctor. So he cannot vote, though his Polish coachman, who has not been here ten years, has all the rights of citizenship. The result is that government affairs are getting into a hopeless muddle. An ignorant fellow by the name of Korfanti—a Polish ‘German-eater’—has now the chief voice in the Municipal Council. The Poles boycott German merchants. They deluge the city with placards and appeals not to buy of Germans. For a long time they refused to trade even a miserable little Polish theater for our splendid big Stadttheater. When the director of that finally got permission to take over the wholly inadequate little playhouse for next season he had to advertise in order to find out how many Germans intend to stay in Posen—as you have seen in our German paper. What can the Poles do with our magnificent Stadttheater? They have no classics to give in it, nor people of sufficient culture to make up an audience. We are still allowed to give German opera, because they know they cannot run that themselves, and a few of the more educated Poles like it. But our splendid spoken classics seem to be doomed. “Then there is their ridiculous hatred of the Jews. The race may have its faults, but the five or six thousand Jews of Posen province
play a most important business and financial rôle. They have always understood the advantages of German Kultur far better than the Poles. There is a Jewish Volksrat here that tries to keep independent of both the other elements of the population; but the great majority of the Jews stand with the Germans. They have no use for this new Zionism—except for the other fellow—unless you take seriously the aspirations of a few impractical young idealists”—a statement, by the way, which I heard from Jews of all classes in various parts of Germany. “We Germans lifted the Poles out of their semi-savagery. We brought them Kultur. Do not be deceived by what you see in Posen. It is a magnificent city, is it not?—finer, perhaps, than you Americans found Coblenz? Yet everything that gives it magnificence was built by the Germans—the well-paved streets, the big, wide boulevards, the splendid parks, all the government buildings and the best of the private ones, the street-cars, the electric lights, even the higher state of civilization you find among the masses. There is not a Pole in the province of Posen who cannot read and write. Do not make the mistake of thinking all these things are Polish because the Poles have stolen them. Before you leave, go and compare Posen with the Polish cities outside Germany. That will tell the story. In non-German Poland you will be struck by the appalling lack of schools, roads, doctors, hospitals, education, culture, by the sad condition of the workmen and the peasants—all those things that are included in the German word Kultur. In Galizia, where Austria virtually allowed the Poles to run themselves, the houses are only six feet high, and you could walk all day without finding a man who can read and write, or who can even speak German. Their cities are sunk in a degradation of the Middle Ages. Posen will fall into the same state, if the present Municipal Council continues in power. There are already frontier troubles between German and Russian Poland, and quarrels between the different sections that confirm what we Germans have always known—that the Poles cannot govern themselves. Warsaw does not wish to keep up our splendid system of workmen and old-age insurance because there is none in Russian Poland. Galizia complains that farm land is several times higher in price in the province of Posen, without admitting that it is German railroads and German settlers that have made it so. That advantage will soon disappear. The Poles will make a mess of the whole province and will
have it sunk into the degradation in which we found it by the time a real ruling nation takes charge of it again.” Just how much truth there was mixed in with the considerable amount of patent nonsense in the ex-chairman’s declamation only a long stay in Poznan, or time itself, would show. The fact that the Poles allowed many of these statements, particularly the protests against the sudden change of language, to be published in the local German newspaper speaks at least for their spirit of tolerance. Though the new government was visibly making mistakes, and had not yet settled down to the orderliness that should come from experience, no one but a prejudiced critic could have discovered immediate evidence that it was making any such complete “mess” of matters as the German Volksrat testified. Even if it had been, at least the mass of the population showed itself happy and contented with the change, and contentment, after all, may in time result in more genuine and lasting progress than that which comes from the forcible feeding of German Kultur. I dropped in at the Teatro Apollo one evening, chiefly to find out how it feels to see a play without understanding a word of it. An immense barnlike building, that looked as if it had once been a skating-rink or a dancing-pavilion, was crowded to suffocation with Poles of every class and variety, from servant-girls in their curious leg-of-mutton sleeves to colonels in the latest cut of Polish uniform. The actors—if they could have been dignified with that title—had recently been imported from Warsaw, and the alleged play they perpetrated could scarcely have been equaled by our silliest rough- and-tumble “comedians.” The herd-like roar with which their inane sallies were unfailingly greeted testified that the audience found them entertaining. But it may be that Poznan was in a particularly simple-minded mood during its first months of relief from a century of bitter oppression. I hope so, for I should regret to find that the startling contrast between this Polish audience and the German one at the artistic Stadttheater the following evening fairly represented the difference between the two races. I believe I am not prejudiced by the fact that the Volksrat presented me with a free ticket when I say that the latter performance was one of which any manager might have been justly proud. The audience, too, resembled the other about as a gathering of college professors resembles a collection of factory
hands. There was a well-bred solemnity about it that could not, in this case, have been due merely to hunger, for there was no munching whatever between the acts, none even under cover of the darkened house, except here and there of candy, a luxury so long since forgotten in Berlin that the happy possessor would never have dreamed of giving his attention at the same time to the merely esthetic appeal of the theater. There may have been Poles in the house, but at least the new army was conspicuous by its absence. Not a uniform was to be seen, with the exception of three scattered through the “peanut gallery.” Two crown boxes, destined only for Hohenzollern royalty or its representatives, sat empty, with something of the solemn demeanor of the vacant chair at the head of the table the day after the funeral. Who would occupy them when the Poles had taken over the playhouse? What, moreover, would they do toward maintaining the high standards of the stage before us? For the most indefatigable enemy of the Germans must have admitted that here was something that could ill be spared. If only they had been contented with bringing the masses these genuine benefits, without militarism, with more open competition, without so much appeal to the doctrine of force—but it has ever been Germany’s contention that only by force can the mass of mankind be lifted to higher levels; that only an army can protect the self-appointed missionaries of a loftier civilization. Armed with what those who read Polish assured me was permission to do so, I set out on foot one morning to the eastward. Beyond the last group of guards wearing the silver double-eagle on their threadbare German uniforms, I fell in with three barefooted Polish peasant women. They were barely thirty, yet all three were already well-nigh toothless, and their hardy forms and faces were plainly marked with the signs that testify to grueling labor and the constant bearing of children. The German they spoke was far superior to the dialects of many regions of purely Teutonic population. Their demeanor was cheerful, yet behind it one caught frequent glimpses of that background of patient, unquestioning acceptance of life as it is which distinguishes the country people of Europe. The most energetic of the trio showed a willingness to enter into conversation; the others confined themselves to an occasional nod of
approval, as if the exertion of keeping pace with us left them no strength to expend in mere words. It was plain from the beginning that they were not enthusiastic on the subject then uppermost in the city behind us. They greeted my first reference to it with expressions that might have been called indifferent, had they not been tinged with evidence of a mild resentment. “What does it matter to us people of the fields,” retorted the less taciturn of the group, “whether Poles or Germans sit in the comfort of government offices, so long as they let us alone? Things were all right as they were, before the war came. Why trouble us with all these changes? Now they are breaking our backs with new burdens, as if we had not had enough of them for five years. First they take our men and leave us to do their work. I have not a male relative left, except my husband, and he is so sickly that he is no longer a man. He is paid twelve marks for eight hours’ work; fifteen for ten. But what help is that when he cannot work ten hours, or even eight? They offered him the iron cross. He told them he would rather have something to feed his family with at home. They asked him if he was not already getting forty marks a month for the support of his family. How could I feed four children, even after the other two had died, with forty marks a month? For three winters I had nothing but dried potatoes and salt. I could not have bread for myself because the flour for the children took all the tickets. Now the war is over, yet they are still taking away what we have left. The same soldiers come and drive off our horses—for the silver eagle on their caps has not changed their natures. Pay for them? Ach, what is eight hundred marks for a horse that is worth six thousand? And how can we cultivate our fields without them? Who started the war? Ach, they are all arguing. What does it matter, so long as they stop it? Will the Germans sign? They should, and have done with it. If they don’t, all the men over fifty, including the Germans and even the Jews”—there was a sneer in this last word, even in the country—“will be at it again. We have had enough of it. Yet if the soldiers come and tell my husband to go he must go, sick though he is.” The basket each of the trio carried contained the midday lunch of her husband in the fields. I turned aside to the grassy slope on which two of the couples assembled. The men insisted that I share their meal with them. It was more nourishing than a ten-mark repast in a
Berlin restaurant, but the absence of bread was significant. When I gave the men each a pinch of tobacco crumbs they announced themselves delighted at the exchange, and mumbled halting words about the well-known generosity of Americans. As I turned my kodak upon them they greeted it with a laughing “Oh, là là!” There was no need to ask where they had picked up that expression. It oriented their war experiences as definitely as it will distinguish for years to come the Americans, in whatever garb one finds them, who were members of the A. E. F. in France. The men were less indifferent to the recent change of government than their wives, but even they could not have been called enthusiastic. What struck one most was the wider outlook on life the Germans had been forced to give them in spite of themselves. Had they been left to till their farms, these plodding peasants would probably still have swallowed whole the specious propaganda of their erstwhile rulers. Now, after four years of military service that had carried them through all central Europe, they had developed the habit of forming their own opinions on all questions; they took any unverified statement, from whatever source, with more than a grain of salt. It would be a mistake nowadays to think of the European peasant as the prejudiced conservative, the plaything of deliberate misinformation, which he was five years ago. In the light of his new experiences he is in many cases doing more individual thinking than the average city resident. Yet, I must admit, the conclusions of this well-traveled pair did not boil down into anything very different from the consensus of opinion, even though they reached them by their own peculiar trains of thought. Germany, they were convinced, had the full guilt of the war; not the Kaiser particularly—they call him “Wilhelm” in Posen province now, and even there one detects now and again a tendency toward the old idolatry he seems personally to have enjoyed throughout the whole Empire—but the military crowd, “and the capitalists.” They disclaimed any hatred of the Germans, “until they wanted to rule the earth” and sought to make the peasants the instruments of their ambition. They, too, charged Wilson personally with delaying the conclusion of peace—on the fate of Danzig they seemed to be supremely indifferent.
“It’s all politics, anyway,” concluded one of them. “They are all playing politics. If the Germans don’t sign they will be divided up as Poland was a hundred and forty years ago. But this new government in Posen is no better than the old. What we need is something entirely new—a government of the peasants and of the working- classes.” The women had from the beginning tried to lead their husbands away from “arguing politics,” chiefly with ludicrously heavy attempts at coquetry, and at length they succeeded. I regained the highway. On either hand lay slightly rolling fields of fertile black soil, well cultivated as far as the eye could see, with only a scattering of trees. Miles away an abandoned Zeppelin hangar bulked into the sky. There were more women laborers than men; several gangs of them were working with picks and shovels; another group was slowly but patiently loading bricks. Horses were to be seen here and there, but oxen were in the majority. Farm-houses showed a rough comfort and a tolerable cleanliness, villages a passable neatness that may or may not have been due to German influence. Certainly the architecture, the farming methods, the communal customs, were little different from those of Prussia or the Rhineland. The dinner served me in the chief tavern of a village of some two thousand inhabitants was nothing to complain of, either in variety or price. A general-shop keeper stated that “with the exception of a few semi-luxuries, such as cocoa and toilet soap,” his grocery department could still meet the decreased demands made upon it. In the clothing lines everything was scarce or wholly lacking. Worst of all, there was nothing fit to drink or smoke. The strong spirits that had once been his chief trade had become so weak no one but boys would drink them. If only America would send concentrated alcohol they could doctor the stock of liquor they had on hand so that no one would know the difference. Then if they could only get some American tobacco! Life was not what it used to be, without a real cigarette from one month’s end to the other. The German rule, on the whole, had not been so bad as many of the Allies seemed to believe. They got along, though it was rather pleasant to be relieved of the arrogant fellows, or see them crawl into their shells. No German resident in the village had given any sign of intending to move away. The communal school was still teaching the German language—two or
three hours a week now. No one had noticed any other change of any importance. The French prisoners confined in the province during the war had been brutally treated. There was no doubt about that; he had seen it himself. But on the whole the German authorities had not been much harder on the Polish population than upon their own people, in Prussia and elsewhere. It was all part of the war, and every one in the Empire had to bear his share of the burdens. Happily, it was over now, if only the new Polish government did not grow ambitious for military conquests also, with the millions of soldiers, some of them patriotic to the point of self-sacrifice, under its command. My hope of walking out of Posen province suffered the same fate as my plan of tramping into it from Germany. In the end I was forced to return to Poznan and make my exit by train over the same route by which I had entered. In the third-class compartment I occupied there were five German residents who had renounced forever their right to return, for the privilege of leaving now with the more portable of their possessions. Two of them had been born in the amputated province; the others had lived there most of their lives. All spoke Polish as readily as German. One masterly, yet scholarly youth, who had served through the war as a lieutenant, was a school- teacher by profession, as was the uncle who accompanied him. They had taught six and twenty-six years, respectively, but had been dispossessed of their positions and of their government dwellings by the new rulers. Up to the time we reached the frontier all five of my companions laid careful emphasis on the statement that they were going to seek re-establishment in their civilian professions in what was left of the Fatherland. At Wronki the Polish authorities were far more inquisitive than they had been toward travelers from the other direction. One by one each compartment group was herded together, bag and baggage, and strained through the sieve of a careful search-and-questioning bureau. The soldier who examined my knapsack glared at the half- dozen precious American cigars I had left as if nothing but the presence of his superiors could have prevented him from confiscating them. Only sufficient food for the day’s journey was allowed to pass. In some cases this rule was interpreted rather liberally, but no one got through with more than ten or twelve
pounds to the person. The amount that was confiscated easily sufficed to feed the garrison of Wronki for the twenty-four hours before the next westbound train was due. An old woman, riding fourth class, who resembled one of India’s famine victims, was despoiled of almost the entire contents of her trunk-sized chest— several sacks of flour, a dozen huge loaves of bread, and a generous supply of sausage. The fact that she spoke only Polish did not seem to impress the searchers in her favor, who silenced her wails at last by bundling her bodily back into the coach and tossing her empty coffer after her. When at last we were under way again the Germans in my compartment took to comparing notes. One, a doctor, was bewailing the “plain theft” of a surgical appliance of rubber which the Poles had confiscated in spite of what seemed to be complete proof that it was his private property and not part of the German army supplies. A foxy-faced country youth, who had carefully changed from shoes to high boots just before the arrival at Wronki, changed back again now with the announcement that there were some four thousand marks concealed between the boot soles. The younger schoolmaster threw off the disguise with which he had covered his real thoughts and announced, vociferously: “You drive me out to work for my livelihood! I will work for my Fatherland at the same time. I will go to Bromberg this very evening and join the army again. We shall see whether the Poles can keep Posen.” The two other young men asserted that they, too, had left with exactly that intention. An indignation meeting against the Poles raged for an hour or more. “I could have remained and kept my position,” went on the schoolmaster, “if I had wanted to turn Polack. Both my parents were Polish; I spoke it before I did German; but I shall always remain a true son of the Fatherland, no matter what happens to it.” A few hundred yards from Kreuz station our train halted for more than an hour and gave us the pleasure of watching the Berlin express go on without us. Though it would have been a matter of twenty seconds to have sprinted across the delta between the two lines, armed boy soldiers prevented any one from leaving his compartment. To all appearances it was a case of “pure meanness”
on the part of the German authorities. Our wrath at being forced to wait a half-day for a dawdling local train was soon appeased, however, by the announcement that we were the last travelers who would be allowed to enter Germany from the province of Posen “until the war was over.” The frontier had been closed by orders from Berlin. It is a long way round from Poland to Holland, and amid the turmoil of gloomy men, disheveled women, and squalling children who had been turned back with their goal so near I found cause to be personally thankful, particularly as I succeeded in eluding during all the afternoon the glassy eye of the cantankerous dyspeptic, who buffeted his way now and then through the throng. Some things are still cheap in Germany. A twelve-word telegram from Kreuz to Berlin cost me nine cents—and it was delivered in telegraphic haste. The hungry passengers from farther east with whom I shared a compartment that evening eyed me greedily as I supped on the supplies I had brought from Posen. One man wearing several diamonds leaned toward me as I was cutting my coffee- brown loaf and sighed, reminiscently, “What beautiful white bread!” When I offered to share it with him, however, he refused vigorously, as if his pride would not permit him to accept what his appetite was so loudly demanding. Unable to find a place in the section to which my third-class ticket entitled me, I was riding second-class. The train-guard on his rounds confiscated my ticket and ignored my offer to pay the difference, with a stern, “It is unlawful to ride in a higher class.” On the Friedrichstrasse platform, however, instead of conducting me to his superiors, he sidled up to me in the darkness and murmured, “If you have a five-mark note with you it will be all right.” Germany is changing indeed if her very railway employees are taking on these Latin characteristics.
A XII ON THE ROAD IN BAVARIA n excellent express raced all day southward across a Germany lush-green with May. Cattle were scarcer in the fields, horses so rare a sight as to be almost conspicuous, but the fields themselves seemed as intensively, as thoroughly cultivated as my memory pictured them fifteen and ten years before. Within the train there was no crowding; the wide aisles and corridors were free from soldiers and their packs, for though there were a hundred or more in uniform scattered between the engine and the last car, a furlong behind, seats were still to be had. The question naturally arose, Are the Germans so short of rolling-stock, after complying with the terms of the armistice, as they pretend? A traveler racing across the Empire in this roomy, almost luxurious Schnellzug might easily have concluded that their whining on that score was mere camouflage. There were even curtains at the wide windows, though of rather shoddy stuff, and the window-straps of paper were so nicely disguised as to be almost indistinguishable from real leather. He who took pains, however, to dip a bit more deeply into the question found that even this great trunk line was carrying barely a third of its peace-time traffic. The red figures, indicating expresses, on the huge porcelain time-tables decorating station walls were nearly all pasted over with slips of paper, while the black ones of Personenzüge, the stop-everywhere-a-long-time trains, were more than half canceled. The branch lines had contributed even more to the Allies. Nor did our aristocratic Berlin-München express entirely escape overburdening. At Nürnberg came with sunset such hordes of passengers of all grades that every available foot of the train was as densely packed as a fourth-class coach on market-day. The throng it
disgorged at Munich was sufficient to have peopled a town of very respectable size. I had made the sudden leap to the southern end of the Empire as a starting-point of a tramp across it instead of reversing the process in the hope that here at last I should find “something doing,” some remnants of excitement. Munich had just been snatched from the hands of the Sparticists—or the Bolshevists; the distinction between the two dreaded groups is not very clear in the German mind. Leviné, the half-mad Russian Jew who was reputed the organizing spirit of the revolt, was still dodging from one hiding-place to another somewhere in the vicinity. To read the breathless cables to the foreign press was to fancy Munich under a constant hail of shrapnel and machine-gun bullets. Ours was the second passenger-train that had ventured into the city in weeks. All Bavaria was blazing with huge posters, often blood-red in color, headed by the dread word “Standrecht” in letters to be seen a hundred yards away, proclaiming martial law and threatening sudden and dire fate to any one who strayed from the straight and narrow path of absolute submission to the “government-faithful” troops that were still pouring in from the north. Surely here, if anywhere, was a chance for a wandering American to get into trouble. Like so many dreadful things, however, martial law and beleaguered cities prove more terrible at a distance than on the spot. True, a group of soldiers in full fighting equipment held the station exit; but their only act of belligerency toward the invading throng was to hand each of us a red slip granting permission to walk the streets until two in the morning. A bedraggled hotel directly across the way spared me that necessity. The information its registry-pad required of guests was more exacting than its interior aspect, but neither here nor at the station exit was there any demand for proof of identity. Toward midnight, as I was falling asleep, a score of erratically spaced shots and the brief rat-a-tat of a machine-gun sounded somewhere not far away. Their direction was too uncertain, however, to make it worth while to accept the permission granted by the red slip. In the morning the city was thronged with the business-bent quite as if disorders had never dodged in and out of its wide streets. The main hotels, however, had been partly taken over by the staffs of
the newly arrived troops, and pulsated with field gray. At the doors very young men in iron hats leaned their fixed bayonets in the crook of an elbow while they examined the Ausweis with which each civilian was supposed to prove his identity. I entered several of them in the vain hope that the flash of my American passport would “start something.” The youths in uniform handed it back each time without so much as a flicker of curiosity on their rather dull faces. Inside, another boy volunteer ran his hands hastily over me in quest of concealed weapons; but not even the most obviously harmless Bavarian escaped that attention. The staff evidently had no secrets from the world at large. At any rate, I wandered into a dozen hotel rooms that had been turned into offices and idled about undisturbed while majors gave captains their orders for the day and lieutenants explained to sergeants the latest commands from higher up. What had become of that stern discipline and the far-famed secrecy of the German army? The soldiers of democratic America were automatons in the presence of their officers compared with these free-and-easy youths in gray; over in Posen the Poles were manyfold more exacting. Had I been a spy, there were several opportunities to have pocketed papers strewn about tables and improvised desks. When at last an officer looked up at me inquiringly I explained my presence by asking for written permission to take photographs within the beleaguered city, and it was granted at once without question. Berlin had been sinister of aspect; Munich was bland, a softer, gentler, less verboten land. Its citizens were not merely courteous; they were aggressively good-natured, their cheerfulness bubbled over on all who came in contact with them. It was almost as easy to distinguish a native from the stiff Prussians who had descended upon them as if the two groups had worn distinctive uniforms. Yet Munich had by no means escaped war-time privations. Long lines of hollow-eyed women flowed sluggishly in and out of under-stocked food-shops; still longer ones, chiefly though not entirely male, crept forward to the door of the rare tobacconists prepared to receive them, and emerged clutching two half-length cigarettes each, their faces beaming as if they had suddenly come into an unexpected inheritance. They were good-natured in spite of what must have been the saddest cut of all from the Bavarian point of view—the weakness
and high cost of their beloved beer. In those vast underground Bierhallen for which Munich had been far-famed for centuries, where customers of both sexes and any age that can toddle pick out a stone mug and serve themselves, the price per liter had risen to the breathless height of thirty-four pfennigs. As if this calamity were not of itself enough to disrupt the serenity of the Bavarian temperament, the foaming beverage had sunk to a mere shadow of its former robust strength. In the “cellar” of the beautiful Rathaus a buxom barmaid reminded me that Tuesday and Friday were meatless days in Germany. The fish she served instead brought me the added information that Munich is far from the sea. My fellow-sufferers constituted a truly democratic gathering. The still almost portly mayor chuckled with his cronies at a table barely visible through the smoke-screened forest of massive pillars. Collarless laborers clinked their mugs, quite unawed by the presence of city councilors or “big merchants.” A leather-skinned old peasant sat down opposite me and opened conversation at once, with no suggestion of that aloofness of the north. From the rucksack that had slipped from his shoulders he took a half-loaf of dull-brown peasant bread and a square of boiled smoked pork, ordering nothing but a half-bottle of wine. Beer, he explained, had fallen too low in its estate to be worthy of his patronage, at least city beer. In his village, three hours away, he could still endure it. Ach, how the famous beer of Munich had deteriorated! How far away those happy days seemed! And to think of paying three marks for a half-bottle of wine! Why, in the good old days.... And this dinner of mine—a plate of fish bones, some stewed grass, city bread, and city beer—worthless stuff—potatoes, to be sure, but not enough to keep a man’s legs under him for half the afternoon —and a bill of more than eight marks! I restrained my impulse to tell him of that prize dinner in Berlin. He had not always been a peasant. Twenty years before he had started a factory—roof tiles and bricks. But in 1915 he had gone back to the farm. At least a Bauer got something to eat. The peace terms? What else could Germany do but sign? If the shoe had been on the other foot the war lords in Berlin would have demanded as much or more. If they hadn’t wanted war in the first place! Wilhelm and all his crowd should have quit two or three years ago while the quitting
was good. What did it all matter, anyway, so long as order returned and the peasants could work without being pestered with all this military service, and the taxes, not to mention the “hamsterers,” the pests! American, was I? He had noticed I was not a Bavarian. (So had I, straining my ears to catch the meaning of his atrocious dialect.) He had taken me for a man from the north, a Hamburger perhaps. American? They say that is a rich country. He had read somewhere that even the peasants sometimes had automobiles! How about the beer? Deteriorating there, too, eh? Ach, this war! Going to abolish beer! What an insane idea! What will people live on? They can’t afford wine, and Schnapps is not good for a man in the long run, and too strong for the women and children. Well, he must be getting back to his beet-field. Glad to have met an American. He had often heard of them. Good day and a happy journey. Troops were still pouring into Munich. That afternoon what before the war would have looked to Americans like a large army marched in column of fours along the bank of the swift, pale-blue Isar and swung in through the heart of town. There were infantry, machine- gun, and light-artillery sections, both horse- and motor-drawn, and from end to end they were decorated with flowers, which clung even to the horses’ bridles and peered from the mouths of the cannon. All the aspect of a conquering army was there, an army that had retaken one of its own cities after decades of occupation by the enemy. Greetings showered upon the columns, a trifle stiff and irresponsive with pride, after the manner of popular heroes; but it was chiefly voiceless greetings, the waving of hands and handkerchiefs, in striking contrast to similar scenes among the French. The Boy Scouts of a year or two ago filled a large portion, possibly a majority, of the ranks. The older men scattered among them bore plainly imprinted on their faces the information that they had remained chiefly for lack of ambition or opportunity to re-enter civil life. Their bronzed features were like frames for those of the eager, life-tasting youths they surrounded, not so much in color as in their disillusioned, nothing-new-to-us expressions. All wore on their collars the gold or silver oak-leaves of volunteers for “home and border protection”; an insignia belonging to generals only before the flight of the Kaiser. Rumor had it, however, that there were many still held under the old conscription laws, particularly those of Polish
blood. The same inarticulate voices whispered that, despite the opinion of Allied staffs, Germany still had a million men under arms; on the books they were carried as discharged; in reality they were sustained by the government as “out-of-works” and housed in barracks near enough to arsenals or munition dumps to equip themselves in a twinkling. What percentage of truth the assertion possessed could only have been determined by long and deliberate study, for though Munich, like many another city and even the country districts, seemed to swarm with soldiers, many of them were so only in outward appearance. Discharged men were permitted to use their uniforms until they were worn out; the mere removal of the shoulder-straps made one a civilian—unlike the soldiers resident in the occupied region, where civilian garb of field gray was furnished with the discharged papers—and boys of all ages, in many cases large enough to have the appearance of real soldiers, were as apt to wear the uniform and the red-banded cap without visor as anything else. The Sparticist uprising in Munich, now crushed, evidently made less trouble on the spot, as usual, than in foreign newspapers. All classes of the population—except perhaps that to which the turn of events had brought the wisdom of silence—admitted that it had been a nuisance, but it had left none of them ashen with fear or gaunt with suffering. Indeed, business seemed to have gone on as usual during all but the two or three days of retaking the city. Banks and the larger merchants had been more or less heavily levied upon; lawyers and a few other classes whom the new doctrine ranked as “parasitic” had found it wise to leave their offices closed; but in the main all agreed that the population at large was never troubled in their homes and seldom on the street. The mistreatment of women, with rumors of which foreign newspapers reeked, was asserted to have been rare, and their “nationalization,” which the cables seem to have announced, had not, so far, at least, been contemplated. All in all, the Bavarian capital suffered far less than Winnipeg under a similar uprising of like date. The moving spirit had come from Russia, as already mentioned, with a few local theorists or self-seekers of higher social standing as its chief auxiliaries. The rank and file of the movement were escaped Russian prisoners and Munich’s own out-of-works, together with such disorderly elements as always hover about any upheaval
promising loot or unearned gain. But the city’s chief scare seemed to have been its recapture by government troops under orders from Berlin. Then for some fifty hours the center of town was no proper place for those to dally who had neglected their insurance premiums. A hundred more or less of fashionable shop-fronts bore witness to the ease with which a machine-gunner can make a plate-glass look like a transparent sieve without once cracking it; rival sharpshooters had all but rounded off the corners of a few of the principal buildings. The meek, plaster-faced Protestant church had been the worst sufferer, as so often happens to the innocent bystander. The most fire-eating Münchener admitted that barter and business had lagged in the heart of town during that brief period. But Munich’s red days had already faded to a memory. Even the assassination of hostages, among them some of the city’s most pompous citizens, by the fleeing Sparticists was now mentioned in much the same impersonal tone with which the Swiss might refer to the death of William Tell or an Englishman regret the loss of Kitchener. The blue-and-white flag of Bavaria fluttered again from the staffs that had been briefly usurped by the red banner of revolt; the dark-blue uniform of the once half-autonomous kingdom again asserted its sway over local matters in the new Volksreich Bayern. At the Deutsches Theater a large audience placidly sipping its beer set on little shelves before each seat alternately roared and sniffled at the bare-kneed mountaineers in feathered hats and the buxom Mädels who bounced through a home-made but well-done “custom picture” in the local dialect. It was evident that life in Munich was not likely to afford any more excitement than had the apathetic north. The atmosphere of the place only helped to confirm the ever-hardening conviction that the German, north or south, east or west, had little real sympathy for revolutions compared with the privilege of pursuing his calling steadily and undisturbed. It was high time to take to the road while a faint hope still remained that something might lay in wait for me along the way to put a bit of ginger into a journey that had thus far lamentably failed to fulfil its promise. I breakfasted next morning with the German staff. At least I was the only civilian in the palm-decked dining-room where a score of high ranking wearers of the iron cross munched their black bread and purple Ersatz marmalade with punctilious formality. Away from
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    Thomas H. Cormen CharlesE. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Fourth Edition The MIT Press Cambridge, Massachusetts London, England
  • 10.
    c 2022 Massachusetts Instituteof Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. The MIT Press would like to thank the anonymous peer reviewers who provided comments on drafts of this book. The generous work of academic experts is essential for establishing the authority and quality of our publications. We acknowledge with gratitude the contributions of these otherwise uncredited readers. This book was set in Times Roman and MathTime Professional II by the authors. Names: Cormen, Thomas H., author. jLeiserson, Charles Eric, author. j Rivest, Ronald L., author. jStein, Clifford, author. Title: Introduction to algorithms / Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Description: Fourth edition. jCambridge, Massachusetts : The MIT Press, [2022] jIncludes bibliographical references and index. Identiûers: LCCN 2021037260 jISBN 9780262046305 Subjects: LCSH: Computer programming. jComputer algorithms. Classiûcation: LCC QA76.6 .C662 2022 jDDC 005.13--dc23 LC record available at http://lccn.loc.gov/2021037260 10 9 8 7 6 5 4 3 2 1
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    Contents Preface xiii I Foundations Introduction3 1 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 12 2 Getting Started 17 2.1 Insertion sort 17 2.2 Analyzing algorithms 25 2.3 Designing algorithms 34 3 Characterizing Running Times 49 3.1 O-notation, �-notation, and ‚-notation 50 3.2 Asymptotic notation: formal deûnitions 53 3.3 Standard notations and common functions 63 4 Divide-and-Conquer 76 4.1 Multiplying square matrices 80 4.2 Strassen’s algorithm for matrix multiplication 85 4.3 The substitution method for solving recurrences 90 4.4 The recursion-tree method for solving recurrences 95 4.5 The master method for solving recurrences 101 ? 4.6 Proof of the continuous master theorem 107 ? 4.7 Akra-Bazzi recurrences 115
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    vi Contents 5 ProbabilisticAnalysis and Randomized Algorithms 126 5.1 The hiring problem 126 5.2 Indicator random variables 130 5.3 Randomized algorithms 134 ? 5.4 Probabilistic analysis and further uses of indicator random variables 140 II Sorting and Order Statistics Introduction 157 6 Heapsort 161 6.1 Heaps 161 6.2 Maintaining the heap property 164 6.3 Building a heap 167 6.4 The heapsort algorithm 170 6.5 Priority queues 172 7 Quicksort 182 7.1 Description of quicksort 183 7.2 Performance of quicksort 187 7.3 A randomized version of quicksort 191 7.4 Analysis of quicksort 193 8 Sorting in Linear Time 205 8.1 Lower bounds for sorting 205 8.2 Counting sort 208 8.3 Radix sort 211 8.4 Bucket sort 215 9 Medians and Order Statistics 227 9.1 Minimum and maximum 228 9.2 Selection in expected linear time 230 9.3 Selection in worst-case linear time 236 III Data Structures Introduction 249 10 Elementary Data Structures 252 10.1 Simple array-based data structures: arrays, matrices, stacks, queues 252 10.2 Linked lists 258 10.3 Representing rooted trees 265
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    Contents vii 11 HashTables 272 11.1 Direct-address tables 273 11.2 Hash tables 275 11.3 Hash functions 282 11.4 Open addressing 293 11.5 Practical considerations 301 12 Binary Search Trees 312 12.1 What is a binary search tree? 312 12.2 Querying a binary search tree 316 12.3 Insertion and deletion 321 13 Red-Black Trees 331 13.1 Properties of red-black trees 331 13.2 Rotations 335 13.3 Insertion 338 13.4 Deletion 346 IV Advanced Design and Analysis Techniques Introduction 361 14 Dynamic Programming 362 14.1 Rod cutting 363 14.2 Matrix-chain multiplication 373 14.3 Elements of dynamic programming 382 14.4 Longest common subsequence 393 14.5 Optimal binary search trees 400 15 Greedy Algorithms 417 15.1 An activity-selection problem 418 15.2 Elements of the greedy strategy 426 15.3 Huffman codes 431 15.4 Ofüine caching 440 16 Amortized Analysis 448 16.1 Aggregate analysis 449 16.2 The accounting method 453 16.3 The potential method 456 16.4 Dynamic tables 460
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    viii Contents V AdvancedData Structures Introduction 477 17 Augmenting Data Structures 480 17.1 Dynamic order statistics 480 17.2 How to augment a data structure 486 17.3 Interval trees 489 18 B-Trees 497 18.1 Deûnition of B-trees 501 18.2 Basic operations on B-trees 504 18.3 Deleting a key from a B-tree 513 19 Data Structures for Disjoint Sets 520 19.1 Disjoint-set operations 520 19.2 Linked-list representation of disjoint sets 523 19.3 Disjoint-set forests 527 ? 19.4 Analysis of union by rank with path compression 531 VI Graph Algorithms Introduction 547 20 Elementary Graph Algorithms 549 20.1 Representations of graphs 549 20.2 Breadth-ûrst search 554 20.3 Depth-ûrst search 563 20.4 Topological sort 573 20.5 Strongly connected components 576 21 Minimum Spanning Trees 585 21.1 Growing a minimum spanning tree 586 21.2 The algorithms of Kruskal and Prim 591 22 Single-Source Shortest Paths 604 22.1 The Bellman-Ford algorithm 612 22.2 Single-source shortest paths in directed acyclic graphs 616 22.3 Dijkstra’s algorithm 620 22.4 Difference constraints and shortest paths 626 22.5 Proofs of shortest-paths properties 633
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    Contents ix 23 All-PairsShortest Paths 646 23.1 Shortest paths and matrix multiplication 648 23.2 The Floyd-Warshall algorithm 655 23.3 Johnson’s algorithm for sparse graphs 662 24 Maximum Flow 670 24.1 Flow networks 671 24.2 The Ford-Fulkerson method 676 24.3 Maximum bipartite matching 693 25 Matchings in Bipartite Graphs 704 25.1 Maximum bipartite matching (revisited) 705 25.2 The stable-marriage problem 716 25.3 The Hungarian algorithm for the assignment problem 723 VII Selected Topics Introduction 745 26 Parallel Algorithms 748 26.1 The basics of fork-join parallelism 750 26.2 Parallel matrix multiplication 770 26.3 Parallel merge sort 775 27 Online Algorithms 791 27.1 Waiting for an elevator 792 27.2 Maintaining a search list 795 27.3 Online caching 802 28 Matrix Operations 819 28.1 Solving systems of linear equations 819 28.2 Inverting matrices 833 28.3 Symmetric positive-deûnite matrices and least-squares approximation 838 29 Linear Programming 850 29.1 Linear programming formulations and algorithms 853 29.2 Formulating problems as linear programs 860 29.3 Duality 866 30 Polynomials and the FFT 877 30.1 Representing polynomials 879 30.2 The DFT and FFT 885 30.3 FFT circuits 894
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    x Contents 31 Number-TheoreticAlgorithms 903 31.1 Elementary number-theoretic notions 904 31.2 Greatest common divisor 911 31.3 Modular arithmetic 916 31.4 Solving modular linear equations 924 31.5 The Chinese remainder theorem 928 31.6 Powers of an element 932 31.7 The RSA public-key cryptosystem 936 ? 31.8 Primality testing 942 32 String Matching 957 32.1 The naive string-matching algorithm 960 32.2 The Rabin-Karp algorithm 962 32.3 String matching with ûnite automata 967 ? 32.4 The Knuth-Morris-Pratt algorithm 975 32.5 Sufûx arrays 985 33 Machine-Learning Algorithms 1003 33.1 Clustering 1005 33.2 Multiplicative-weights algorithms 1015 33.3 Gradient descent 1022 34 NP-Completeness 1042 34.1 Polynomial time 1048 34.2 Polynomial-time veriûcation 1056 34.3 NP-completeness and reducibility 1061 34.4 NP-completeness proofs 1072 34.5 NP-complete problems 1080 35 Approximation Algorithms 1104 35.1 The vertex-cover problem 1106 35.2 The traveling-salesperson problem 1109 35.3 The set-covering problem 1115 35.4 Randomization and linear programming 1119 35.5 The subset-sum problem 1124 VIII Appendix: Mathematical Background Introduction 1139 A Summations 1140 A.1 Summation formulas and properties 1140 A.2 Bounding summations 1145
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    Contents xi B Sets,Etc. 1153 B.1 Sets 1153 B.2 Relations 1158 B.3 Functions 1161 B.4 Graphs 1164 B.5 Trees 1169 C Counting and Probability 1178 C.1 Counting 1178 C.2 Probability 1184 C.3 Discrete random variables 1191 C.4 The geometric and binomial distributions 1196 ? C.5 The tails of the binomial distribution 1203 D Matrices 1214 D.1 Matrices and matrix operations 1214 D.2 Basic matrix properties 1219 Bibliography 1227 Index 1251
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    Preface Not so longago, anyone who had heard the word <algorithm= was almost certainly a computer scientist or mathematician. With computers having become prevalent in our modern lives, however, the term is no longer esoteric. If you look around your home, you’ll ûnd algorithms running in the most mundane places: your microwave oven, your washing machine, and, of course, your computer. You ask algorithms to make recommendations to you: what music you might like or what route to take when driving. Our society, for better or for worse, asks algorithms to suggest sentences for convicted criminals. You even rely on algorithms to keep you alive, or at least not to kill you: the control systems in your car or in medical equipment.1 The word <algorithm= appears somewhere in the news seemingly every day. Therefore, it behooves you to understand algorithms not just as a student or practitioner of computer science, but as a citizen of the world. Once you understand algorithms, you can educate others about what algorithms are, how they operate, and what their limitations are. This book provides a comprehensive introduction to the modern study of com- puter algorithms. It presents many algorithms and covers them in considerable depth, yet makes their design accessible to all levels of readers. All the analyses are laid out, some simple, some more involved. We have tried to keep explanations clear without sacriûcing depth of coverage or mathematical rigor. Each chapter presents an algorithm, a design technique, an application area, or a related topic. Algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The book contains 231 ûgures4many with multiple parts4illustrating how the algorithms work. Since we emphasize efficiency as a design criterion, we include careful analyses of the running times of the algorithms. 1 To understand many of the ways in which algorithms inüuence our daily lives, see the book by Fry [162].
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    xiv Preface The textis intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Because it discusses engineering issues in algorithm design, as well as mathematical aspects, it is equally well suited for self-study by technical professionals. In this, the fourth edition, we have once again updated the entire book. The changes cover a broad spectrum, including new chapters and sections, color illus- trations, and what we hope you’ll ûnd to be a more engaging writing style. To the teacher We have designed this book to be both versatile and complete. You should ûnd it useful for a variety of courses, from an undergraduate course in data structures up through a graduate course in algorithms. Because we have provided considerably more material than can ût in a typical one-term course, you can select the material that best supports the course you wish to teach. You should ûnd it easy to organize your course around just the chapters you need. We have made chapters relatively self-contained, so that you need not worry about an unexpected and unnecessary dependence of one chapter on an- other. Whereas in an undergraduate course, you might use only some sections from a chapter, in a graduate course, you might cover the entire chapter. We have included 931 exercises and 162 problems. Each section ends with exer- cises, and each chapter ends with problems. The exercises are generally short ques- tions that test basic mastery of the material. Some are simple self-check thought exercises, but many are substantial and suitable as assigned homework. The prob- lems include more elaborate case studies which often introduce new material. They often consist of several parts that lead the student through the steps required to ar- rive at a solution. As with the third edition of this book, we have made publicly available solutions to some, but by no means all, of the problems and exercises. You can ûnd these so- lutions on our website, http://mitpress.mit.edu/algorithms/. You will want to check this site to see whether it contains the solution to an exercise or problem that you plan to assign. Since the set of solutions that we post might grow over time, we recommend that you check the site each time you teach the course. We have starred (?) the sections and exercises that are more suitable for graduate students than for undergraduates. A starred section is not necessarily more difû- cult than an unstarred one, but it may require an understanding of more advanced mathematics. Likewise, starred exercises may require an advanced background or more than average creativity.
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    Preface xv To thestudent We hope that this textbook provides you with an enjoyable introduction to the ûeld of algorithms. We have attempted to make every algorithm accessible and inter- esting. To help you when you encounter unfamiliar or difûcult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms and sup- porting ûgures to help you visualize what is going on. Since this book is large, your class will probably cover only a portion of its material. Although we hope that you will ûnd this book helpful to you as a course textbook now, we have also tried to make it comprehensive enough to warrant space on your future professional bookshelf. What are the prerequisites for reading this book?  You need some programming experience. In particular, you should understand recursive procedures and simple data structures, such as arrays and linked lists (although Section 10.2 covers linked lists and a variant that you may ûnd new).  You should have some facility with mathematical proofs, and especially proofs by mathematical induction. A few portions of the book rely on some knowledge of elementary calculus. Although this book uses mathematics throughout, Part I and Appendices A–D teach you all the mathematical techniques you will need. Our website, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises. Feel free to check your solutions against ours. We ask, however, that you not send your solutions to us. To the professional The wide range of topics in this book makes it an excellent handbook on algo- rithms. Because each chapter is relatively self-contained, you can focus on the topics most relevant to you. Since most of the algorithms we discuss have great practical utility, we address implementation concerns and other engineering issues. We often provide practical alternatives to the few algorithms that are primarily of theoretical interest. If you wish to implement any of the algorithms, you should ûnd the transla- tion of our pseudocode into your favorite programming language to be a fairly straightforward task. We have designed the pseudocode to present each algorithm clearly and succinctly. Consequently, we do not address error handling and other software-engineering issues that require speciûc assumptions about your program- ming environment. We attempt to present each algorithm simply and directly with- out allowing the idiosyncrasies of a particular programming language to obscure its essence. If you are used to 0-origin arrays, you might ûnd our frequent practice of
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    xvi Preface indexing arraysfrom 1 a minor stumbling block. You can always either subtract 1 from our indices or just overallocate the array and leave position 0 unused. We understand that if you are using this book outside of a course, then you might be unable to check your solutions to problems and exercises against solutions provided by an instructor. Our website, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises so that you can check your work. Please do not send your solutions to us. To our colleagues We have supplied an extensive bibliography and pointers to the current literature. Each chapter ends with a set of chapter notes that give historical details and ref- erences. The chapter notes do not provide a complete reference to the whole ûeld of algorithms, however. Though it may be hard to believe for a book of this size, space constraints prevented us from including many interesting algorithms. Despite myriad requests from students for solutions to problems and exercises, we have adopted the policy of not citing references for them, removing the temp- tation for students to look up a solution rather than to discover it themselves. Changes for the fourth edition As we said about the changes for the second and third editions, depending on how you look at it, the book changed either not much or quite a bit. A quick look at the table of contents shows that most of the third-edition chapters and sections appear in the fourth edition. We removed three chapters and several sections, but we have added three new chapters and several new sections apart from these new chapters. We kept the hybrid organization from the ûrst three editions. Rather than organizing chapters only by problem domains or only according to techniques, this book incorporates elements of both. It contains technique-based chapters on divide-and-conquer, dynamic programming, greedy algorithms, amortized analy- sis, augmenting data structures, NP-completeness, and approximation algorithms. But it also has entire parts on sorting, on data structures for dynamic sets, and on algorithms for graph problems. We ûnd that although you need to know how to ap- ply techniques for designing and analyzing algorithms, problems seldom announce to you which techniques are most amenable to solving them. Some of the changes in the fourth edition apply generally across the book, and some are speciûc to particular chapters or sections. Here is a summary of the most signiûcant general changes:  We added 140 new exercises and 22 new problems. We also improved many of the old exercises and problems, often as the result of reader feedback. (Thanks to all readers who made suggestions.)
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    Preface xvii  Wehave color! With designers from the MIT Press, we selected a limited palette, devised to convey information and to be pleasing to the eye. (We are delighted to display red-black trees in4get this4red and black!) To enhance readability, deûned terms, pseudocode comments, and page numbers in the in- dex are in color.  Pseudocode procedures appear on a tan background to make them easier to spot, and they do not necessarily appear on the page of their ûrst reference. When they don’t, the text directs you to the relevant page. In the same vein, nonlocal references to numbered equations, theorems, lemmas, and corollaries include the page number.  We removed topics that were rarely taught. We dropped in their entirety the chapters on Fibonacci heaps, van Emde Boas trees, and computational geom- etry. In addition, the following material was excised: the maximum-subarray problem, implementing pointers and objects, perfect hashing, randomly built binary search trees, matroids, push-relabel algorithms for maximum üow, the iterative fast Fourier transform method, the details of the simplex algorithm for linear programming, and integer factorization. You can ûnd all the removed material on our website, http://mitpress.mit.edu/algorithms/.  We reviewed the entire book and rewrote sentences, paragraphs, and sections to make the writing clearer, more personal, and gender neutral. For example, the <traveling-salesman problem= in the previous editions is now called the <traveling-salesperson problem.= We believe that it is critically important for engineering and science, including our own ûeld of computer science, to be welcoming to everyone. (The one place that stumped us is in Chapter 13, which requires a term for a parent’s sibling. Because the English language has no such gender-neutral term, we regretfully stuck with <uncle.=)  The chapter notes, bibliography, and index were updated, reüecting the dra- matic growth of the ûeld of algorithms since the third edition.  We corrected errors, posting most corrections on our website of third-edition errata. Those that were reported while we were in full swing preparing this edition were not posted, but were corrected in this edition. (Thanks again to all readers who helped us identify issues.) The speciûc changes for the fourth edition include the following:  We renamed Chapter 3 and added a section giving an overview of asymptotic notation before delving into the formal deûnitions.  Chapter 4 underwent substantial changes to improve its mathematical founda- tion and make it more robust and intuitive. The notion of an algorithmic re- currence was introduced, and the topic of ignoring üoors and ceilings in recur-
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    xviii Preface rences wasaddressed more rigorously. The second case of the master theorem incorporates polylogarithmic factors, and a rigorous proof of a <continuous= version of the master theorem is now provided. We also present the powerful and general Akra-Bazzi method (without proof).  The deterministic order-statistic algorithm in Chapter 9 is slightly different, and the analyses of both the randomized and deterministic order-statistic algorithms have been revamped.  In addition to stacks and queues, Section 10.1 discusses ways to store arrays and matrices.  Chapter 11 on hash tables includes a modern treatment of hash functions. It also emphasizes linear probing as an efûcient method for resolving collisions when the underlying hardware implements caching to favor local searches.  To replace the sections on matroids in Chapter 15, we converted a problem in the third edition about ofüine caching into a full section.  Section 16.4 now contains a more intuitive explanation of the potential func- tions to analyze table doubling and halving.  Chapter 17 on augmenting data structures was relocated from Part III to Part V, reüecting our view that this technique goes beyond basic material.  Chapter 25 is a new chapter about matchings in bipartite graphs. It presents algorithms to ûnd a matching of maximum cardinality, to solve the stable- marriage problem, and to ûnd a maximum-weight matching (known as the <as- signment problem=).  Chapter 26, on task-parallel computing, has been updated with modern termi- nology, including the name of the chapter.  Chapter 27, which covers online algorithms, is another new chapter. In an online algorithm, the input arrives over time, rather than being available in its entirety at the start of the algorithm. The chapter describes several examples of online algorithms, including determining how long to wait for an elevator before taking the stairs, maintaining a linked list via the move-to-front heuristic, and evaluating replacement policies for caches.  In Chapter 29, we removed the detailed presentation of the simplex algorithm, as it was math heavy without really conveying many algorithmic ideas. The chapter now focuses on the key aspect of how to model problems as linear programs, along with the essential duality property of linear programming.  Section 32.5 adds to the chapter on string matching the simple, yet powerful, structure of sufûx arrays.
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    Preface xix  Chapter33, on machine learning, is the third new chapter. It introduces sev- eral basic methods used in machine learning: clustering to group similar items together, weighted-majority algorithms, and gradient descent to ûnd the mini- mizer of a function.  Section 34.5.6 summarizes strategies for polynomial-time reductions to show that problems are NP-hard.  The proof of the approximation algorithm for the set-covering problem in Sec- tion 35.3 has been revised. Website You can use our website, http://mitpress.mit.edu/algorithms/, to obtain supplemen- tary information and to communicate with us. The website links to a list of known errors, material from the third edition that is not included in the fourth edition, solutions to selected exercises and problems, Python implementations of many of the algorithms in this book, a list explaining the corny professor jokes (of course), as well as other content, which we may add to. The website also tells you how to report errors or make suggestions. How we produced this book Like the previous three editions, the fourth edition was produced in L A TEX 2". We used the Times font with mathematics typeset using the MathTime Professional II fonts. As in all previous editions, we compiled the index using Windex, a C pro- gram that we wrote, and produced the bibliography using BIBTEX. The PDF ûles for this book were created on a MacBook Pro running macOS 10.14. Our plea to Apple in the preface of the third edition to update MacDraw Pro for macOS 10 went for naught, and so we continued to draw illustrations on pre-Intel Macs running MacDraw Pro under the Classic environment of older versions of macOS 10. Many of the mathematical expressions appearing in illustrations were laid in with the psfrag package for L A TEX 2". Acknowledgments for the fourth edition We have been working with the MIT Press since we started writing the ûrst edi- tion in 1987, collaborating with several directors, editors, and production staff. Throughout our association with the MIT Press, their support has always been out- standing. Special thanks to our editors Marie Lee, who put up with us for far too long, and Elizabeth Swayze, who pushed us over the ûnish line. Thanks also to Director Amy Brand and to Alex Hoopes.
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    xx Preface As inthe third edition, we were geographically distributed while producing the fourth edition, working in the Dartmouth College Department of Computer Science; the MIT Computer Science and Artiûcial Intelligence Laboratory and the MIT Department of Electrical Engineering and Computer Science; and the Columbia University Department of Industrial Engineering and Operations Re- search, Department of Computer Science, and Data Science Institute. During the COVID-19 pandemic, we worked largely from home. We thank our respective universities and colleagues for providing such supportive and stimulating environ- ments. As we complete this book, those of us who are not retired are eager to return to our respective universities now that the pandemic seems to be abating. Julie Sussman, P.P.A., came to our rescue once again with her technical copy- editing under tremendous time pressure. If not for Julie, this book would be riddled with errors (or, let’s say, many more errors than it has) and would be far less read- able. Julie, we will be forever indebted to you. Errors that remain are the responsi- bility of the authors (and probably were inserted after Julie read the material). Dozens of errors in previous editions were corrected in the process of creating this edition. We thank our readers4too many to list them all4who have reported errors and suggested improvements over the years. We received considerable help in preparing some of the new material in this edition. Neville Campbell (unafûliated), Bill Kuszmaul of MIT, and Chee Yap of NYU provided valuable advice regarding the treatment of recurrences in Chapter 4. Yan Gu of the University of California, Riverside, provided feedback on parallel algorithms in Chapter 26. Rob Shapire of Microsoft Research altered our approach to the material on machine learning with his detailed comments on Chapter 33. Qi Qi of MIT helped with the analysis of the Monty Hall problem (Problem C-1). Molly Seaman and Mary Reilly of the MIT Press helped us select the color palette in the illustrations, and Wojciech Jarosz of Dartmouth College suggested design improvements to our newly colored ûgures. Yichen (Annie) Ke and Linda Xiao, who have since graduated from Dartmouth, aided in colorizing the illus- trations, and Linda also produced many of the Python implementations that are available on the book’s website. Finally, we thank our wives4Wendy Leiserson, Gail Rivest, Rebecca Ivry, and the late Nicole Cormen4and our families. The patience and encouragement of those who love us made this project possible. We affectionately dedicate this book to them. THOMAS H. CORMEN Lebanon, New Hampshire CHARLES E. LEISERSON Cambridge, Massachusetts RONALD L. RIVEST Cambridge, Massachusetts CLIFFORD STEIN New York, New York June, 2021
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    Introduction When you designand analyze algorithms, you need to be able to describe how they operate and how to design them. You also need some mathematical tools to show that your algorithms do the right thing and do it efûciently. This part will get you started. Later parts of this book will build upon this base. Chapter 1 provides an overview of algorithms and their place in modern com- puting systems. This chapter deûnes what an algorithm is and lists some examples. It also makes a case for considering algorithms as a technology, alongside tech- nologies such as fast hardware, graphical user interfaces, object-oriented systems, and networks. In Chapter 2, we see our ûrst algorithms, which solve the problem of sorting a sequence of n numbers. They are written in a pseudocode which, although not directly translatable to any conventional programming language, conveys the struc- ture of the algorithm clearly enough that you should be able to implement it in the language of your choice. The sorting algorithms we examine are insertion sort, which uses an incremental approach, and merge sort, which uses a recursive tech- nique known as <divide-and-conquer.= Although the time each requires increases with the value of n, the rate of increase differs between the two algorithms. We determine these running times in Chapter 2, and we develop a useful <asymptotic= notation to express them. Chapter 3 precisely deûnes asymptotic notation. We’ll use asymptotic notation to bound the growth of functions4most often, functions that describe the running time of algorithms4from above and below. The chapter starts by informally deûn- ing the most commonly used asymptotic notations and giving an example of how to apply them. It then formally deûnes ûve asymptotic notations and presents conven- tions for how to put them together. The rest of Chapter 3 is primarily a presentation of mathematical notation, more to ensure that your use of notation matches that in this book than to teach you new mathematical concepts.
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    4 Part IFoundations Chapter 4 delves further into the divide-and-conquer method introduced in Chapter 2. It provides two additional examples of divide-and-conquer algorithms for multiplying square matrices, including Strassen’s surprising method. Chapter 4 contains methods for solving recurrences, which are useful for describing the run- ning times of recursive algorithms. In the substitution method, you guess an answer and prove it correct. Recursion trees provide one way to generate a guess. Chap- ter 4 also presents the powerful technique of the <master method,= which you can often use to solve recurrences that arise from divide-and-conquer algorithms. Al- though the chapter provides a proof of a foundational theorem on which the master theorem depends, you should feel free to employ the master method without delv- ing into the proof. Chapter 4 concludes with some advanced topics. Chapter 5 introduces probabilistic analysis and randomized algorithms. You typically use probabilistic analysis to determine the running time of an algorithm in cases in which, due to the presence of an inherent probability distribution, the running time may differ on different inputs of the same size. In some cases, you might assume that the inputs conform to a known probability distribution, so that you are averaging the running time over all possible inputs. In other cases, the probability distribution comes not from the inputs but from random choices made during the course of the algorithm. An algorithm whose behavior is determined not only by its input but by the values produced by a random-number generator is a randomized algorithm. You can use randomized algorithms to enforce a probability distribution on the inputs4thereby ensuring that no particular input always causes poor performance4or even to bound the error rate of algorithms that are allowed to produce incorrect results on a limited basis. Appendices A–D contain other mathematical material that you will ûnd helpful as you read this book. You might have seen much of the material in the appendix chapters before having read this book (although the speciûc deûnitions and nota- tional conventions we use may differ in some cases from what you have seen in the past), and so you should think of the appendices as reference material. On the other hand, you probably have not already seen most of the material in Part I. All the chapters in Part I and the appendices are written with a tutorial üavor.
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    1 The Roleof Algorithms in Computing What are algorithms? Why is the study of algorithms worthwhile? What is the role of algorithms relative to other technologies used in computers? This chapter will answer these questions. 1.1 Algorithms Informally, an algorithm is any well-deûned computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output in a ûnite amount of time. An algorithm is thus a sequence of computational steps that transform the input into the output. You can also view an algorithm as a tool for solving a well-speciûed computa- tional problem. The statement of the problem speciûes in general terms the desired input/output relationship for problem instances, typically of arbitrarily large size. The algorithm describes a speciûc computational procedure for achieving that in- put/output relationship for all problem instances. As an example, suppose that you need to sort a sequence of numbers into mono- tonically increasing order. This problem arises frequently in practice and provides fertile ground for introducing many standard design techniques and analysis tools. Here is how we formally deûne the sorting problem: Input: A sequence of n numbers ha1; a2; : : : ; ani. Output: A permutation (reordering) ha0 1; a0 2; : : : ; a0 ni of the input sequence such that a0 1 හ a0 2 හ    හ a0 n. Thus, given the input sequence h31; 41; 59; 26; 41; 58i, a correct sorting algorithm returns as output the sequence h26; 31; 41; 41; 58; 59i. Such an input sequence is
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    6 Chapter 1The Role of Algorithms in Computing called an instance of the sorting problem. In general, an instance of a problem1 consists of the input (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem. Because many programs use it as an intermediate step, sorting is a fundamental operation in computer science. As a result, you have a large number of good sort- ing algorithms at your disposal. Which algorithm is best for a given application depends on4among other factors4the number of items to be sorted, the extent to which the items are already somewhat sorted, possible restrictions on the item values, the architecture of the computer, and the kind of storage devices to be used: main memory, disks, or even4archaically4tapes. An algorithm for a computational problem is correct if, for every problem in- stance provided as input, it halts4ûnishes its computing in ûnite time4and out- puts the correct solution to the problem instance. A correct algorithm solves the given computational problem. An incorrect algorithm might not halt at all on some input instances, or it might halt with an incorrect answer. Contrary to what you might expect, incorrect algorithms can sometimes be useful, if you can control their error rate. We’ll see an example of an algorithm with a controllable error rate in Chapter 31 when we study algorithms for ûnding large prime numbers. Ordi- narily, however, we’ll concern ourselves only with correct algorithms. An algorithm can be speciûed in English, as a computer program, or even as a hardware design. The only requirement is that the speciûcation must provide a precise description of the computational procedure to be followed. What kinds of problems are solved by algorithms? Sorting is by no means the only computational problem for which algorithms have been developed. (You probably suspected as much when you saw the size of this book.) Practical applications of algorithms are ubiquitous and include the follow- ing examples:  The Human Genome Project has made great progress toward the goals of iden- tifying all the roughly 30,000 genes in human DNA, determining the sequences of the roughly 3 billion chemical base pairs that make up human DNA, stor- ing this information in databases, and developing tools for data analysis. Each of these steps requires sophisticated algorithms. Although the solutions to the various problems involved are beyond the scope of this book, many methods to solve these biological problems use ideas presented here, enabling scientists to accomplish tasks while using resources efûciently. Dynamic programming, as 1 Sometimes, when the problem context is known, problem instances are themselves simply called <problems.=
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    1.1 Algorithms 7 inChapter 14, is an important technique for solving several of these biological problems, particularly ones that involve determining similarity between DNA sequences. The savings realized are in time, both human and machine, and in money, as more information can be extracted by laboratory techniques.  The internet enables people all around the world to quickly access and retrieve large amounts of information. With the aid of clever algorithms, sites on the internet are able to manage and manipulate this large volume of data. Exam- ples of problems that make essential use of algorithms include ûnding good routes on which the data travels (techniques for solving such problems appear in Chapter 22), and using a search engine to quickly ûnd pages on which par- ticular information resides (related techniques are in Chapters 11 and 32).  Electronic commerce enables goods and services to be negotiated and ex- changed electronically, and it depends on the privacy of personal informa- tion such as credit card numbers, passwords, and bank statements. The core technologies used in electronic commerce include public-key cryptography and digital signatures (covered in Chapter 31), which are based on numerical algo- rithms and number theory.  Manufacturing and other commercial enterprises often need to allocate scarce resources in the most beneûcial way. An oil company might wish to know where to place its wells in order to maximize its expected proût. A political candidate might want to determine where to spend money buying campaign ad- vertising in order to maximize the chances of winning an election. An airline might wish to assign crews to üights in the least expensive way possible, mak- ing sure that each üight is covered and that government regulations regarding crew scheduling are met. An internet service provider might wish to determine where to place additional resources in order to serve its customers more effec- tively. All of these are examples of problems that can be solved by modeling them as linear programs, which Chapter 29 explores. Although some of the details of these examples are beyond the scope of this book, we do give underlying techniques that apply to these problems and problem areas. We also show how to solve many speciûc problems, including the following:  You have a road map on which the distance between each pair of adjacent in- tersections is marked, and you wish to determine the shortest route from one intersection to another. The number of possible routes can be huge, even if you disallow routes that cross over themselves. How can you choose which of all possible routes is the shortest? You can start by modeling the road map (which is itself a model of the actual roads) as a graph (which we will meet in Part VI and Appendix B). In this graph, you wish to ûnd the shortest path from one vertex to another. Chapter 22 shows how to solve this problem efûciently.
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    8 Chapter 1The Role of Algorithms in Computing  Given a mechanical design in terms of a library of parts, where each part may include instances of other parts, list the parts in order so that each part appears before any part that uses it. If the design comprises n parts, then there are nŠ possible orders, where nŠ denotes the factorial function. Because the factorial function grows faster than even an exponential function, you cannot feasibly generate each possible order and then verify that, within that order, each part appears before the parts using it (unless you have only a few parts). This prob- lem is an instance of topological sorting, and Chapter 20 shows how to solve this problem efûciently.  A doctor needs to determine whether an image represents a cancerous tumor or a benign one. The doctor has available images of many other tumors, some of which are known to be cancerous and some of which are known to be benign. A cancerous tumor is likely to be more similar to other cancerous tumors than to benign tumors, and a benign tumor is more likely to be similar to other be- nign tumors. By using a clustering algorithm, as in Chapter 33, the doctor can identify which outcome is more likely.  You need to compress a large ûle containing text so that it occupies less space. Many ways to do so are known, including <LZW compression,= which looks for repeating character sequences. Chapter 15 studies a different approach, <Huff- man coding,= which encodes characters by bit sequences of various lengths, with characters occurring more frequently encoded by shorter bit sequences. These lists are far from exhaustive (as you again have probably surmised from this book’s heft), but they exhibit two characteristics common to many interesting algorithmic problems: 1. They have many candidate solutions, the overwhelming majority of which do not solve the problem at hand. Finding one that does, or one that is <best,= with- out explicitly examining each possible solution, can present quite a challenge. 2. They have practical applications. Of the problems in the above list, ûnding the shortest path provides the easiest examples. A transportation ûrm, such as a trucking or railroad company, has a ûnancial interest in ûnding shortest paths through a road or rail network because taking shorter paths results in lower labor and fuel costs. Or a routing node on the internet might need to ûnd the shortest path through the network in order to route a message quickly. Or a person wishing to drive from New York to Boston might want to ûnd driving directions using a navigation app. Not every problem solved by algorithms has an easily identiûed set of candi- date solutions. For example, given a set of numerical values representing samples of a signal taken at regular time intervals, the discrete Fourier transform converts
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    1.1 Algorithms 9 thetime domain to the frequency domain. That is, it approximates the signal as a weighted sum of sinusoids, producing the strength of various frequencies which, when summed, approximate the sampled signal. In addition to lying at the heart of signal processing, discrete Fourier transforms have applications in data compres- sion and multiplying large polynomials and integers. Chapter 30 gives an efûcient algorithm, the fast Fourier transform (commonly called the FFT), for this problem. The chapter also sketches out the design of a hardware FFT circuit. Data structures This book also presents several data structures. A data structure is a way to store and organize data in order to facilitate access and modiûcations. Using the appro- priate data structure or structures is an important part of algorithm design. No sin- gle data structure works well for all purposes, and so you should know the strengths and limitations of several of them. Technique Although you can use this book as a <cookbook= for algorithms, you might some- day encounter a problem for which you cannot readily ûnd a published algorithm (many of the exercises and problems in this book, for example). This book will teach you techniques of algorithm design and analysis so that you can develop al- gorithms on your own, show that they give the correct answer, and analyze their ef- ûciency. Different chapters address different aspects of algorithmic problem solv- ing. Some chapters address speciûc problems, such as ûnding medians and order statistics in Chapter 9, computing minimum spanning trees in Chapter 21, and de- termining a maximum üow in a network in Chapter 24. Other chapters introduce techniques, such as divide-and-conquer in Chapters 2 and 4, dynamic programming in Chapter 14, and amortized analysis in Chapter 16. Hard problems Most of this book is about efûcient algorithms. Our usual measure of efûciency is speed: how long does an algorithm take to produce its result? There are some problems, however, for which we know of no algorithm that runs in a reasonable amount of time. Chapter 34 studies an interesting subset of these problems, which are known as NP-complete. Why are NP-complete problems interesting? First, although no efûcient algo- rithm for an NP-complete problem has ever been found, nobody has ever proven that an efûcient algorithm for one cannot exist. In other words, no one knows whether efûcient algorithms exist for NP-complete problems. Second, the set of
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    10 Chapter 1The Role of Algorithms in Computing NP-complete problems has the remarkable property that if an efûcient algorithm exists for any one of them, then efûcient algorithms exist for all of them. This re- lationship among the NP-complete problems makes the lack of efûcient solutions all the more tantalizing. Third, several NP-complete problems are similar, but not identical, to problems for which we do know of efûcient algorithms. Computer scientists are intrigued by how a small change to the problem statement can cause a big change to the efûciency of the best known algorithm. You should know about NP-complete problems because some of them arise sur- prisingly often in real applications. If you are called upon to produce an efûcient algorithm for an NP-complete problem, you are likely to spend a lot of time in a fruitless search. If, instead, you can show that the problem is NP-complete, you can spend your time developing an efûcient approximation algorithm, that is, an algorithm that gives a good, but not necessarily the best possible, solution. As a concrete example, consider a delivery company with a central depot. Each day, it loads up delivery trucks at the depot and sends them around to deliver goods to several addresses. At the end of the day, each truck must end up back at the depot so that it is ready to be loaded for the next day. To reduce costs, the company wants to select an order of delivery stops that yields the lowest overall distance traveled by each truck. This problem is the well-known <traveling-salesperson problem,= and it is NP-complete.2 It has no known efûcient algorithm. Under certain assumptions, however, we know of efûcient algorithms that compute overall distances close to the smallest possible. Chapter 35 discusses such <approximation algorithms.= Alternative computing models For many years, we could count on processor clock speeds increasing at a steady rate. Physical limitations present a fundamental roadblock to ever-increasing clock speeds, however: because power density increases superlinearly with clock speed, chips run the risk of melting once their clock speeds become high enough. In or- der to perform more computations per second, therefore, chips are being designed to contain not just one but several processing <cores.= We can liken these multi- core computers to several sequential computers on a single chip. In other words, they are a type of <parallel computer.= In order to elicit the best performance from multicore computers, we need to design algorithms with parallelism in mind. Chapter 26 presents a model for =task-parallel= algorithms, which take advantage of multiple processing cores. This model has advantages from both theoretical and 2 To be precise, only decision problems4those with a <yes/no= answer4can be NP-complete. The decision version of the traveling salesperson problem asks whether there exists an order of stops whose distance totals at most a given amount.
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    1.1 Algorithms 11 practicalstandpoints, and many modern parallel-programming platforms embrace something similar to this model of parallelism. Most of the examples in this book assume that all of the input data are available when an algorithm begins running. Much of the work in algorithm design makes the same assumption. For many important real-world examples, however, the input actually arrives over time, and the algorithm must decide how to proceed without knowing what data will arrive in the future. In a data center, jobs are constantly arriving and departing, and a scheduling algorithm must decide when and where to run a job, without knowing what jobs will be arriving in the future. Trafûc must be routed in the internet based on the current state, without knowing about where trafûc will arrive in the future. Hospital emergency rooms make triage decisions about which patients to treat ûrst without knowing when other patients will be arriving in the future and what treatments they will need. Algorithms that receive their input over time, rather than having all the input present at the start, are online algorithms, which Chapter 27 examines. Exercises 1.1-1 Describe your own real-world example that requires sorting. Describe one that requires ûnding the shortest distance between two points. 1.1-2 Other than speed, what other measures of efûciency might you need to consider in a real-world setting? 1.1-3 Select a data structure that you have seen, and discuss its strengths and limitations. 1.1-4 How are the shortest-path and traveling-salesperson problems given above similar? How are they different? 1.1-5 Suggest a real-world problem in which only the best solution will do. Then come up with one in which <approximately= the best solution is good enough. 1.1-6 Describe a real-world problem in which sometimes the entire input is available before you need to solve the problem, but other times the input is not entirely available in advance and arrives over time.
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    12 Chapter 1The Role of Algorithms in Computing 1.2 Algorithms as a technology If computers were inûnitely fast and computer memory were free, would you have any reason to study algorithms? The answer is yes, if for no other reason than that you would still like to be certain that your solution method terminates and does so with the correct answer. If computers were inûnitely fast, any correct method for solving a problem would do. You would probably want your implementation to be within the bounds of good software engineering practice (for example, your implementation should be well designed and documented), but you would most often use whichever method was the easiest to implement. Of course, computers may be fast, but they are not inûnitely fast. Computing time is therefore a bounded resource, which makes it precious. Although the saying goes, <Time is money,= time is even more valuable than money: you can get back money after you spend it, but once time is spent, you can never get it back. Memory may be inexpensive, but it is neither inûnite nor free. You should choose algorithms that use the resources of time and space efûciently. Efûciency Different algorithms devised to solve the same problem often differ dramatically in their efûciency. These differences can be much more signiûcant than differences due to hardware and software. As an example, Chapter 2 introduces two algorithms for sorting. The ûrst, known as insertion sort, takes time roughly equal to c1n2 to sort n items, where c1 is a constant that does not depend on n. That is, it takes time roughly proportional to n2. The second, merge sort, takes time roughly equal to c2n lg n, where lg n stands for log2 n and c2 is another constant that also does not depend on n. Inser- tion sort typically has a smaller constant factor than merge sort, so that c1 < c2. We’ll see that the constant factors can have far less of an impact on the running time than the dependence on the input size n. Let’s write insertion sort’s running time as c1n  n and merge sort’s running time as c2n  lg n. Then we see that where insertion sort has a factor of n in its running time, merge sort has a factor of lg n, which is much smaller. For example, when n is 1000, lg n is approximately 10, and when n is 1,000,000, lg n is approximately only 20. Although insertion sort usu- ally runs faster than merge sort for small input sizes, once the input size n becomes large enough, merge sort’s advantage of lg n versus n more than compensates for the difference in constant factors. No matter how much smaller c1 is than c2, there is always a crossover point beyond which merge sort is faster.
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    1.2 Algorithms asa technology 13 For a concrete example, let us pit a faster computer (computer A) running inser- tion sort against a slower computer (computer B) running merge sort. They each must sort an array of 10 million numbers. (Although 10 million numbers might seem like a lot, if the numbers are eight-byte integers, then the input occupies about 80 megabytes, which ûts in the memory of even an inexpensive laptop com- puter many times over.) Suppose that computer A executes 10 billion instructions per second (faster than any single sequential computer at the time of this writing) and computer B executes only 10 million instructions per second (much slower than most contemporary computers), so that computer A is 1000 times faster than computer B in raw computing power. To make the difference even more dramatic, suppose that the world’s craftiest programmer codes insertion sort in machine lan- guage for computer A, and the resulting code requires 2n2 instructions to sort n numbers. Suppose further that just an average programmer implements merge sort, using a high-level language with an inefûcient compiler, with the resulting code taking 50 n lg n instructions. To sort 10 million numbers, computer A takes 2  .107/2 instructions 1010 instructions/second D 20,000 seconds (more than 5:5 hours) ; while computer B takes 50  107 lg 107 instructions 107 instructions/second  1163 seconds (under 20 minutes) : By using an algorithm whose running time grows more slowly, even with a poor compiler, computer B runs more than 17 times faster than computer A! The ad- vantage of merge sort is even more pronounced when sorting 100 million numbers: where insertion sort takes more than 23 days, merge sort takes under four hours. Although 100 million might seem like a large number, there are more than 100 mil- lion web searches every half hour, more than 100 million emails sent every minute, and some of the smallest galaxies (known as ultra-compact dwarf galaxies) con- tain about 100 million stars. In general, as the problem size increases, so does the relative advantage of merge sort. Algorithms and other technologies The example above shows that you should consider algorithms, like computer hard- ware, as a technology. Total system performance depends on choosing efûcient algorithms as much as on choosing fast hardware. Just as rapid advances are being made in other computer technologies, they are being made in algorithms as well. You might wonder whether algorithms are truly that important on contemporary computers in light of other advanced technologies, such as
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    14 Chapter 1The Role of Algorithms in Computing  advanced computer architectures and fabrication technologies,  easy-to-use, intuitive, graphical user interfaces (GUIs),  object-oriented systems,  integrated web technologies,  fast networking, both wired and wireless,  machine learning,  and mobile devices. The answer is yes. Although some applications do not explicitly require algorith- mic content at the application level (such as some simple, web-based applications), many do. For example, consider a web-based service that determines how to travel from one location to another. Its implementation would rely on fast hardware, a graphical user interface, wide-area networking, and also possibly on object ori- entation. It would also require algorithms for operations such as ûnding routes (probably using a shortest-path algorithm), rendering maps, and interpolating ad- dresses. Moreover, even an application that does not require algorithmic content at the application level relies heavily upon algorithms. Does the application rely on fast hardware? The hardware design used algorithms. Does the application rely on graphical user interfaces? The design of any GUI relies on algorithms. Does the application rely on networking? Routing in networks relies heavily on algorithms. Was the application written in a language other than machine code? Then it was processed by a compiler, interpreter, or assembler, all of which make extensive use of algorithms. Algorithms are at the core of most technologies used in contempo- rary computers. Machine learning can be thought of as a method for performing algorithmic tasks without explicitly designing an algorithm, but instead inferring patterns from data and thereby automatically learning a solution. At ûrst glance, machine learning, which automates the process of algorithmic design, may seem to make learning about algorithms obsolete. The opposite is true, however. Machine learning is itself a collection of algorithms, just under a different name. Furthermore, it cur- rently seems that the successes of machine learning are mainly for problems for which we, as humans, do not really understand what the right algorithm is. Promi- nent examples include computer vision and automatic language translation. For algorithmic problems that humans understand well, such as most of the problems in this book, efûcient algorithms designed to solve a speciûc problem are typically more successful than machine-learning approaches. Data science is an interdisciplinary ûeld with the goal of extracting knowledge and insights from structured and unstructured data. Data science uses methods
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    Problems for Chapter1 15 from statistics, computer science, and optimization. The design and analysis of algorithms is fundamental to the ûeld. The core techniques of data science, which overlap signiûcantly with those in machine learning, include many of the algo- rithms in this book. Furthermore, with the ever-increasing capacities of computers, we use them to solve larger problems than ever before. As we saw in the above comparison be- tween insertion sort and merge sort, it is at larger problem sizes that the differences in efûciency between algorithms become particularly prominent. Having a solid base of algorithmic knowledge and technique is one characteristic that deûnes the truly skilled programmer. With modern computing technology, you can accomplish some tasks without knowing much about algorithms, but with a good background in algorithms, you can do much, much more. Exercises 1.2-1 Give an example of an application that requires algorithmic content at the applica- tion level, and discuss the function of the algorithms involved. 1.2-2 Suppose that for inputs of size n on a particular computer, insertion sort runs in 8n2 steps and merge sort runs in 64 n lg n steps. For which values of n does insertion sort beat merge sort? 1.2-3 What is the smallest value of n such that an algorithm whose running time is 100n2 runs faster than an algorithm whose running time is 2n on the same machine? Problems 1-1 Comparison of running times For each function f .n/ and time t in the following table, determine the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes f .n/ microseconds.
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    16 Chapter 1The Role of Algorithms in Computing 1 1 1 1 1 1 1 second minute hour day month year century lg n pn n n lg n n2 n3 2n nŠ Chapter notes There are many excellent texts on the general topic of algorithms, including those by Aho, Hopcroft, and Ullman [5, 6], Dasgupta, Papadimitriou, and Vazirani [107], Edmonds [133], Erickson [135], Goodrich and Tamassia [195, 196], Kleinberg and Tardos [257], Knuth [259, 260, 261, 262, 263], Levitin [298], Louridas [305], Mehlhorn and Sanders [325], Mitzenmacher and Upfal [331], Neapolitan [342], Roughgarden [385, 386, 387, 388], Sanders, Mehlhorn, Dietzfelbinger, and De- mentiev [393], Sedgewick and Wayne [402], Skiena [414], Soltys-Kulinicz [419], Wilf [455], and Williamson and Shmoys [459]. Some of the more practical as- pects of algorithm design are discussed by Bentley [49, 50, 51], Bhargava [54], Kochenderfer and Wheeler [268], and McGeoch [321]. Surveys of the ûeld of al- gorithms can also be found in books by Atallah and Blanton [27, 28] and Mehta and Sahhi [326]. For less technical material, see the books by Christian and Grifûths [92], Cormen [104], Erwig [136], MacCormick [307], and V¨ ocking et al. [448]. Overviews of the algorithms used in computational biology can be found in books by Jones and Pevzner [240], Elloumi and Zomaya [134], and Marchisio [315].
  • 43.
    2 Getting Started Thischapter will familiarize you with the framework we’ll use throughout the book to think about the design and analysis of algorithms. It is self-contained, but it does include several references to material that will be introduced in Chapters 3 and 4. (It also contains several summations, which Appendix A shows how to solve.) We’ll begin by examining the insertion sort algorithm to solve the sorting prob- lem introduced in Chapter 1. We’ll specify algorithms using a pseudocode that should be understandable to you if you have done computer programming. We’ll see why insertion sort correctly sorts and analyze its running time. The analysis introduces a notation that describes how running time increases with the number of items to be sorted. Following a discussion of insertion sort, we’ll use a method called divide-and-conquer to develop a sorting algorithm called merge sort. We’ll end with an analysis of merge sort’s running time. 2.1 Insertion sort Our ûrst algorithm, insertion sort, solves the sorting problem introduced in Chap- ter 1: Input: A sequence of n numbers ha1; a2; : : : ; ani. Output: A permutation (reordering) ha 0 1; a 0 2; : : : ; a 0 ni of the input sequence such that a 0 1 හ a 0 2 හ    හ a 0 n. The numbers to be sorted are also known as the keys. Although the problem is con- ceptually about sorting a sequence, the input comes in the form of an array with n elements. When we want to sort numbers, it’s often because they are the keys associated with other data, which we call satellite data. Together, a key and satel- lite data form a record. For example, consider a spreadsheet containing student records with many associated pieces of data such as age, grade-point average, and number of courses taken. Any one of these quantities could be a key, but when the
  • 44.
    18 Chapter 2Getting Started spreadsheet sorts, it moves the associated record (the satellite data) with the key. When describing a sorting algorithm, we focus on the keys, but it is important to remember that there usually is associated satellite data. In this book, we’ll typically describe algorithms as procedures written in a pseu- docode that is similar in many respects to C, C++, Java, Python,1 or JavaScript. (Apologies if we’ve omitted your favorite programming language. We can’t list them all.) If you have been introduced to any of these languages, you should have little trouble understanding algorithms <coded= in pseudocode. What separates pseudocode from real code is that in pseudocode, we employ whatever expres- sive method is most clear and concise to specify a given algorithm. Sometimes the clearest method is English, so do not be surprised if you come across an En- glish phrase or sentence embedded within a section that looks more like real code. Another difference between pseudocode and real code is that pseudocode often ig- nores aspects of software engineering4such as data abstraction, modularity, and error handling4in order to convey the essence of the algorithm more concisely. We start with insertion sort, which is an efûcient algorithm for sorting a small number of elements. Insertion sort works the way you might sort a hand of playing cards. Start with an empty left hand and the cards in a pile on the table. Pick up the ûrst card in the pile and hold it with your left hand. Then, with your right hand, remove one card at a time from the pile, and insert it into the correct position in your left hand. As Figure 2.1 illustrates, you ûnd the correct position for a card by comparing it with each of the cards already in your left hand, starting at the right and moving left. As soon as you see a card in your left hand whose value is less than or equal to the card you’re holding in your right hand, insert the card that you’re holding in your right hand just to the right of this card in your left hand. If all the cards in your left hand have values greater than the card in your right hand, then place this card as the leftmost card in your left hand. At all times, the cards held in your left hand are sorted, and these cards were originally the top cards of the pile on the table. The pseudocode for insertion sort is given as the procedure INSERTION-SORT on the facing page. It takes two parameters: an array A containing the values to be sorted and the number n of values of sort. The values occupy positions AŒ1� through AŒn� of the array, which we denote by AŒ1 W n�. When the INSERTION- SORT procedure is ûnished, array AŒ1 W n� contains the original values, but in sorted order. 1 If you’re familiar with only Python, you can think of arrays as similar to Python lists.
  • 45.
    2.1 Insertion sort19 2 ♥ ♥ ♥ 2 ♥ 4 ♥♥ ♥ ♥ ♥ 4 ♥ 5 ♥♥ ♥ ♥ ♥ 5 ♥ ♥ 7 ♥ ♥ ♥ ♥ ♥ ♥ ♥ 7 ♥ ♥ 10 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ 1 0 ♥ Figure 2.1 Sorting a hand of cards using insertion sort. INSERTION-SORT.A; n/ 1 for i D 2 to n 2 key D AŒi� 3 / / Insert AŒi� into the sorted subarray AŒ1 Wi  1�. 4 j D i  1 5 while j > 0 and AŒj � > key 6 AŒj C 1� D AŒj � 7 j D j  1 8 AŒj C 1� D key Loop invariants and the correctness of insertion sort Figure 2.2 shows how this algorithm works for an array A that starts out with the sequence h5; 2; 4; 6; 1; 3i. The index i indicates the <current card= being inserted into the hand. At the beginning of each iteration of the for loop, which is indexed by i, the subarray (a contiguous portion of the array) consisting of elements AŒ1 Wi 1� (that is, AŒ1� through AŒi 1�) constitutes the currently sorted hand, and the remaining subarray AŒi C 1 Wn� (elements AŒi C 1� through AŒn�) corresponds to the pile of cards still on the table. In fact, elements AŒ1 Wi  1� are the elements originally in positions 1 through i  1, but now in sorted order. We state these properties of AŒ1 Wi  1� formally as a loop invariant:
  • 46.
    20 Chapter 2Getting Started 1 2 3 4 5 6 5 2 4 6 1 3 (a) 1 2 3 4 5 6 2 5 4 6 1 3 (b) 1 2 3 4 5 6 2 4 5 6 1 3 (c) 1 2 3 4 5 6 2 4 5 6 1 3 (d) 1 2 3 4 5 6 2 4 5 6 1 3 (e) 1 2 3 4 5 6 2 4 5 6 1 3 (f) Figure 2.2 The operation of INSERTION-SORT.A; n/, where A initially contains the sequence h5; 2; 4; 6; 1; 3i and n D 6. Array indices appear above the rectangles, and values stored in the array positions appear within the rectangles. (a)–(e) The iterations of the for loop of lines 138. In each iteration, the blue rectangle holds the key taken from AŒi�, which is compared with the values in tan rectangles to its left in the test of line 5. Orange arrows show array values moved one position to the right in line 6, and blue arrows indicate where the key moves to in line 8. (f) The ûnal sorted array. At the start of each iteration of the for loop of lines 138, the subarray AŒ1 Wi  1� consists of the elements originally in AŒ1 Wi  1�, but in sorted order. Loop invariants help us understand why an algorithm is correct. When you’re using a loop invariant, you need to show three things: Initialization: It is true prior to the ûrst iteration of the loop. Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration. Termination: The loop terminates, and when it terminates, the invariant4usually along with the reason that the loop terminated4gives us a useful property that helps show that the algorithm is correct. When the ûrst two properties hold, the loop invariant is true prior to every iteration of the loop. (Of course, you are free to use established facts other than the loop invariant itself to prove that the loop invariant remains true before each iteration.) A loop-invariant proof is a form of mathematical induction, where to prove that a property holds, you prove a base case and an inductive step. Here, showing that the invariant holds before the ûrst iteration corresponds to the base case, and showing that the invariant holds from iteration to iteration corresponds to the inductive step. The third property is perhaps the most important one, since you are using the loop invariant to show correctness. Typically, you use the loop invariant along with the condition that caused the loop to terminate. Mathematical induction typically applies the inductive step inûnitely, but in a loop invariant the <induction= stops when the loop terminates.
  • 47.
    2.1 Insertion sort21 Let’s see how these properties hold for insertion sort. Initialization: We start by showing that the loop invariant holds before the ûrst loop iteration, when i D 2.2 The subarray AŒ1 Wi  1� consists of just the single element AŒ1�, which is in fact the original element in AŒ1�. Moreover, this subarray is sorted (after all, how could a subarray with just one value not be sorted?), which shows that the loop invariant holds prior to the ûrst iteration of the loop. Maintenance: Next, we tackle the second property: showing that each iteration maintains the loop invariant. Informally, the body of the for loop works by moving the values in AŒi  1�, AŒi  2�, AŒi  3�, and so on by one position to the right until it ûnds the proper position for AŒi� (lines 437), at which point it inserts the value of AŒi� (line 8). The subarray AŒ1 Wi� then consists of the elements originally in AŒ1 Wi�, but in sorted order. Incrementing i (increasing its value by 1) for the next iteration of the for loop then preserves the loop invariant. A more formal treatment of the second property would require us to state and show a loop invariant for the while loop of lines 537. Let’s not get bogged down in such formalism just yet. Instead, we’ll rely on our informal analysis to show that the second property holds for the outer loop. Termination: Finally, we examine loop termination. The loop variable i starts at 2 and increases by 1 in each iteration. Once i’s value exceeds n in line 1, the loop terminates. That is, the loop terminates once i equals n C 1. Substituting n C1 for i in the wording of the loop invariant yields that the subarray AŒ1 Wn� consists of the elements originally in AŒ1 Wn�, but in sorted order. Hence, the algorithm is correct. This method of loop invariants is used to show correctness in various places throughout this book. Pseudocode conventions We use the following conventions in our pseudocode.  Indentation indicates block structure. For example, the body of the for loop that begins on line 1 consists of lines 238, and the body of the while loop that 2 When the loop is a for loop, the loop-invariant check just prior to the ûrst iteration occurs immedi- ately after the initial assignment to the loop-counter variable and just before the ûrst test in the loop header. In the case of INSERTION-SORT, this time is after assigning 2 to the variable i but before the ûrst test of whether i හ n.
  • 48.
    22 Chapter 2Getting Started begins on line 5 contains lines 637 but not line 8. Our indentation style applies to if-else statements3 as well. Using indentation instead of textual indicators of block structure, such as begin and end statements or curly braces, reduces clutter while preserving, or even enhancing, clarity.4  The looping constructs while, for, and repeat-until and the if-else conditional construct have interpretations similar to those in C, C++, Java, Python, and JavaScript.5 In this book, the loop counter retains its value after the loop is exited, unlike some situations that arise in C++ and Java. Thus, immediately after a for loop, the loop counter’s value is the value that ûrst exceeded the for loop bound.6 We used this property in our correctness argument for insertion sort. The for loop header in line 1 is for i D 2 to n, and so when this loop terminates, i equals nC1. We use the keyword to when a for loop increments its loop counter in each iteration, and we use the keyword downto when a for loop decrements its loop counter (reduces its value by 1 in each iteration). When the loop counter changes by an amount greater than 1, the amount of change follows the optional keyword by.  The symbol </ /= indicates that the remainder of the line is a comment.  Variables (such as i, j , and key) are local to the given procedure. We won’t use global variables without explicit indication.  We access array elements by specifying the array name followed by the index in square brackets. For example, AŒi� indicates the ith element of the array A. Although many programming languages enforce 0-origin indexing for arrays (0 is the smallest valid index), we choose whichever indexing scheme is clearest for human readers to understand. Because people usually start counting at 1, not 0, most4but not all4of the arrays in this book use 1-origin indexing. To be 3 In an if-else statement, we indent else at the same level as its matching if. The ûrst executable line of an else clause appears on the same line as the keyword else. For multiway tests, we use elseif for tests after the ûrst one. When it is the ûrst line in an else clause, an if statement appears on the line following else so that you do not misconstrue it as elseif. 4 Each pseudocode procedure in this book appears on one page so that you do not need to discern levels of indentation in pseudocode that is split across pages. 5 Most block-structured languages have equivalent constructs, though the exact syntax may differ. Python lacks repeat-until loops, and its for loops operate differently from the for loops in this book. Think of the pseudocode line <for i D 1 to n= as equivalent to <for i in range(1, n+1)= in Python. 6 In Python, the loop counter retains its value after the loop is exited, but the value it retains is the value it had during the ûnal iteration of the for loop, rather than the value that exceeded the loop bound. That is because a Python for loop iterates through a list, which may contain nonnumeric values.
  • 49.
    2.1 Insertion sort23 clear about whether a particular algorithm assumes 0-origin or 1-origin index- ing, we’ll specify the bounds of the arrays explicitly. If you are implementing an algorithm that we specify using 1-origin indexing, but you’re writing in a programming language that enforces 0-origin indexing (such as C, C++, Java, Python, or JavaScript), then give yourself credit for being able to adjust. You can either always subtract 1 from each index or allocate each array with one extra position and just ignore position 0. The notation <W= denotes a subarray. Thus, AŒi Wj � indicates the subarray of A consisting of the elements AŒi�; AŒi C 1�; : : : ; AŒj �.7 We also use this notation to indicate the bounds of an array, as we did earlier when discussing the array AŒ1 Wn�.  We typically organize compound data into objects, which are composed of attributes. We access a particular attribute using the syntax found in many object-oriented programming languages: the object name, followed by a dot, followed by the attribute name. For example, if an object x has attribute f , we denote this attribute by x:f. We treat a variable representing an array or object as a pointer (known as a reference in some programming languages) to the data representing the array or object. For all attributes f of an object x, setting y D x causes y:f to equal x:f. Moreover, if we now set x:f D 3, then afterward not only does x:f equal 3, but y:f equals 3 as well. In other words, x and y point to the same object after the assignment y D x. This way of treating arrays and objects is consistent with most contemporary programming languages. Our attribute notation can <cascade.= For example, suppose that the attribute f is itself a pointer to some type of object that has an attribute g. Then the notation x:f:g is implicitly parenthesized as .x:f/:g. In other words, if we had assigned y D x:f, then x:f:g is the same as y:g. Sometimes a pointer refers to no object at all. In this case, we give it the special value NIL.  We pass parameters to a procedure by value: the called procedure receives its own copy of the parameters, and if it assigns a value to a parameter, the change is not seen by the calling procedure. When objects are passed, the pointer to the data representing the object is copied, but the object’s attributes are not. For example, if x is a parameter of a called procedure, the assignment x D y within 7 If you’re used to programming in Python, bear in mind that in this book, the subarray AŒi Wj� includes the element AŒj�. In Python, the last element of AŒi Wj� is AŒj 1�. Python allows negative indices, which count from the back end of the list. This book does not use negative array indices.
  • 50.
    24 Chapter 2Getting Started the called procedure is not visible to the calling procedure. The assignment x:f D 3, however, is visible if the calling procedure has a pointer to the same object as x. Similarly, arrays are passed by pointer, so that a pointer to the array is passed, rather than the entire array, and changes to individual array elements are visible to the calling procedure. Again, most contemporary programming languages work this way.  A return statement immediately transfers control back to the point of call in the calling procedure. Most return statements also take a value to pass back to the caller. Our pseudocode differs from many programming languages in that we allow multiple values to be returned in a single return statement without having to create objects to package them together.8  The boolean operators <and= and <or= are short circuiting. That is, evaluate the expression <x and y= by ûrst evaluating x. If x evaluates to FALSE, then the entire expression cannot evaluate to TRUE, and therefore y is not evaluated. If, on the other hand, x evaluates to TRUE, y must be evaluated to determine the value of the entire expression. Similarly, in the expression <x or y= the ex- pression y is evaluated only if x evaluates to FALSE. Short-circuiting operators allow us to write boolean expressions such as <x ¤ NIL and x:f D y= without worrying about what happens upon evaluating x:f when x is NIL.  The keyword error indicates that an error occurred because conditions were wrong for the procedure to have been called, and the procedure immediately terminates. The calling procedure is responsible for handling the error, and so we do not specify what action to take. Exercises 2.1-1 Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on an array initially containing the sequence h31; 41; 59; 26; 41; 58i. 2.1-2 Consider the procedure SUM-ARRAY on the facing page. It computes the sum of the n numbers in array AŒ1 Wn�. State a loop invariant for this procedure, and use its initialization, maintenance, and termination properties to show that the SUM- ARRAY procedure returns the sum of the numbers in AŒ1 Wn�. 8 Python’s tuple notation allows return statements to return multiple values without creating objects from a programmer-deûned class.
  • 51.
    2.2 Analyzing algorithms25 SUM-ARRAY.A; n/ 1 sum D 0 2 for i D 1 to n 3 sum D sum C AŒi� 4 return sum 2.1-3 Rewrite the INSERTION-SORT procedure to sort into monotonically decreasing in- stead of monotonically increasing order. 2.1-4 Consider the searching problem: Input: A sequence of n numbers ha1; a2; : : : ; ani stored in array AŒ1 Wn� and a value x. Output: An index i such that x equals AŒi� or the special value NIL if x does not appear in A. Write pseudocode for linear search, which scans through the array from begin- ning to end, looking for x. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulûlls the three necessary properties. 2.1-5 Consider the problem of adding two n-bit binary integers a and b, stored in two n-element arrays AŒ0 Wn  1� and BŒ0 Wn  1�, where each element is either 0 or 1, a D Pn1 iD0 AŒi�  2i, and b D Pn1 iD0 BŒi�  2i . The sum c D a C b of the two integers should be stored in binary form in an .n C 1/-element array C Œ0 Wn�, where c D Pn iD0 CŒi�  2i. Write a procedure ADD-BINARY-INTEGERS that takes as input arrays A and B, along with the length n, and returns array C holding the sum. 2.2 Analyzing algorithms Analyzing an algorithm has come to mean predicting the resources that the algo- rithm requires. You might consider resources such as memory, communication bandwidth, or energy consumption. Most often, however, you’ll want to measure computational time. If you analyze several candidate algorithms for a problem,
  • 52.
    26 Chapter 2Getting Started you can identify the most efûcient one. There might be more than just one viable candidate, but you can often rule out several inferior algorithms in the process. Before you can analyze an algorithm, you need a model of the technology that it runs on, including the resources of that technology and a way to express their costs. Most of this book assumes a generic one-processor, random-access ma- chine (RAM) model of computation as the implementation technology, with the understanding that algorithms are implemented as computer programs. In the RAM model, instructions execute one after another, with no concurrent operations. The RAM model assumes that each instruction takes the same amount of time as any other instruction and that each data access4using the value of a variable or storing into a variable4takes the same amount of time as any other data access. In other words, in the RAM model each instruction or data access takes a constant amount of time4even indexing into an array.9 Strictly speaking, we should precisely deûne the instructions of the RAM model and their costs. To do so, however, would be tedious and yield little insight into al- gorithm design and analysis. Yet we must be careful not to abuse the RAM model. For example, what if a RAM had an instruction that sorts? Then you could sort in just one step. Such a RAM would be unrealistic, since such instructions do not appear in real computers. Our guide, therefore, is how real computers are de- signed. The RAM model contains instructions commonly found in real computers: arithmetic (such as add, subtract, multiply, divide, remainder, üoor, ceiling), data movement (load, store, copy), and control (conditional and unconditional branch, subroutine call and return). The data types in the RAM model are integer, üoating point (for storing real- number approximations), and character. Real computers do not usually have a separate data type for the boolean values TRUE and FALSE. Instead, they often test whether an integer value is 0 (FALSE) or nonzero (TRUE), as in C. Although we typically do not concern ourselves with precision for üoating-point values in this book (many numbers cannot be represented exactly in üoating point), precision is crucial for most applications. We also assume that each word of data has a limit on the number of bits. For example, when working with inputs of size n, we typically 9 We assume that each element of a given array occupies the same number of bytes and that the elements of a given array are stored in contiguous memory locations. For example, if array AŒ1 Wn� starts at memory address 1000 and each element occupies four bytes, then element AŒi� is at address 1000 C4.i 1/. In general, computing the address in memory of a particular array element requires at most one subtraction (no subtraction for a 0-origin array), one multiplication (often implemented as a shift operation if the element size is an exact power of 2), and one addition. Furthermore, for code that iterates through the elements of an array in order, an optimizing compiler can generate the address of each element using just one addition, by adding the element size to the address of the preceding element.
  • 53.
    2.2 Analyzing algorithms27 assume that integers are represented by c log2 n bits for some constant c  1. We require c  1 so that each word can hold the value of n, enabling us to index the individual input elements, and we restrict c to be a constant so that the word size does not grow arbitrarily. (If the word size could grow arbitrarily, we could store huge amounts of data in one word and operate on it all in constant time4an unrealistic scenario.) Real computers contain instructions not listed above, and such instructions rep- resent a gray area in the RAM model. For example, is exponentiation a constant- time instruction? In the general case, no: to compute xn when x and n are general integers typically takes time logarithmic in n (see equation (31.34) on page 934), and you must worry about whether the result ûts into a computer word. If n is an exact power of 2, however, exponentiation can usually be viewed as a constant-time operation. Many computers have a <shift left= instruction, which in constant time shifts the bits of an integer by n positions to the left. In most computers, shifting the bits of an integer by 1 position to the left is equivalent to multiplying by 2, so that shifting the bits by n positions to the left is equivalent to multiplying by 2n. Therefore, such computers can compute 2n in 1 constant-time instruction by shift- ing the integer 1 by n positions to the left, as long as n is no more than the number of bits in a computer word. We’ll try to avoid such gray areas in the RAM model and treat computing 2n and multiplying by 2n as constant-time operations when the result is small enough to ût in a computer word. The RAM model does not account for the memory hierarchy that is common in contemporary computers. It models neither caches nor virtual memory. Sev- eral other computational models attempt to account for memory-hierarchy effects, which are sometimes signiûcant in real programs on real machines. Section 11.5 and a handful of problems in this book examine memory-hierarchy effects, but for the most part, the analyses in this book do not consider them. Models that include the memory hierarchy are quite a bit more complex than the RAM model, and so they can be difûcult to work with. Moreover, RAM-model analyses are usually excellent predictors of performance on actual machines. Although it is often straightforward to analyze an algorithm in the RAM model, sometimes it can be quite a challenge. You might need to employ mathematical tools such as combinatorics, probability theory, algebraic dexterity, and the ability to identify the most signiûcant terms in a formula. Because an algorithm might behave differently for each possible input, we need a means for summarizing that behavior in simple, easily understood formulas. Analysis of insertion sort How long does the INSERTION-SORT procedure take? One way to tell would be for you to run it on your computer and time how long it takes to run. Of course, you’d
  • 54.
    28 Chapter 2Getting Started ûrst have to implement it in a real programming language, since you cannot run our pseudocode directly. What would such a timing test tell you? You would ûnd out how long insertion sort takes to run on your particular computer, on that particular input, under the particular implementation that you created, with the particular compiler or interpreter that you ran, with the particular libraries that you linked in, and with the particular background tasks that were running on your computer concurrently with your timing test (such as checking for incoming information over a network). If you run insertion sort again on your computer with the same input, you might even get a different timing result. From running just one implementation of insertion sort on just one computer and on just one input, what would you be able to determine about insertion sort’s running time if you were to give it a different input, if you were to run it on a different computer, or if you were to implement it in a different programming language? Not much. We need a way to predict, given a new input, how long insertion sort will take. Instead of timing a run, or even several runs, of insertion sort, we can determine how long it takes by analyzing the algorithm itself. We’ll examine how many times it executes each line of pseudocode and how long each line of pseudocode takes to run. We’ll ûrst come up with a precise but complicated formula for the running time. Then, we’ll distill the important part of the formula using a convenient no- tation that can help us compare the running times of different algorithms for the same problem. How do we analyze insertion sort? First, let’s acknowledge that the running time depends on the input. You shouldn’t be terribly surprised that sorting a thousand numbers takes longer than sorting three numbers. Moreover, insertion sort can take different amounts of time to sort two input arrays of the same size, depending on how nearly sorted they already are. Even though the running time can depend on many features of the input, we’ll focus on the one that has been shown to have the greatest effect, namely the size of the input, and describe the running time of a program as a function of the size of its input. To do so, we need to deûne the terms <running time= and <input size= more carefully. We also need to be clear about whether we are discussing the running time for an input that elicits the worst-case behavior, the best-case behavior, or some other case. The best notion for input size depends on the problem being studied. For many problems, such as sorting or computing discrete Fourier transforms, the most nat- ural measure is the number of items in the input4for example, the number n of items being sorted. For many other problems, such as multiplying two integers, the best measure of input size is the total number of bits needed to represent the input in ordinary binary notation. Sometimes it is more appropriate to describe the size of the input with more than just one number. For example, if the input to an algorithm is a graph, we usually characterize the input size by both the number
  • 55.
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  • 56.
    cheaper than inCoblenz, none of them as high as four marks, even with war tax, poor tax, and “wardrobe.” The house was crowded with the serious-minded of all classes, Poles as well as Germans; the actors were of higher histrionic ability than the average American town of the size of Bromberg sees once a year. Yet equally splendid performances were offered here at these slight prices all the year round. As I strolled hotelward with that pleasant sensation of satisfaction that comes from an evening of genuine entertainment, I could not but wonder whether this, and those other undeniable advantages of German Kultur, whatever sins might justly be charged against it, would be kept up after the Poles had taken Bromberg into their own keeping. As to the walking trip through these eastern provinces which I had planned, fate was once more against me. I might, to be sure, have set out on foot toward the region already amputated from the Empire, but in the course of an hour I should have had the privilege of walking back again. The German-Polish front was just six kilometers from Bromberg, and a wandering stranger would have had exactly the same chance of crossing its succession of trenches as of entering Germany from France a year before. The one and only way of reaching the province of Posen was by train from the village of Kreuz, back along the railway by which I had come. The place had all the appearance of an international frontier, a frontier hastily erected and not yet in efficient running order. Arrangements for examining travelers and baggage consisted only of an improvised fence along the station platform, strewn pellmell with a heterogeneous throng bound in both directions, and their multifarious coffers and bundles. The soldiers who patrolled the line of demarkation with fixed bayonets were callow, thin-faced youths, or men past middle age who had plainly reached the stage of uselessness as combat troops. All wore on their collars the silver oak- leaves of the recently formed “frontier guard.” Their manner toward the harassed travelers was either brutal or cringingly friendly. The Germans in civilian garb who examined passports and baggage were cantankerous and gruff, as if they resented the existence of a frontier where the Fatherland had never admitted that a frontier existed. They vented their wrath especially against men of military age who wished to enter Polish territory—and their interpretation of their
  • 57.
    duties in thatrespect was by no means charitable. Among others, a wretched little dwarf past fifty, whom a glance sufficed to recognize as useless from a military point of view, even had his papers not been stamped with the official Untauglich, was wantonly turned back. Many a family was left only the choice of abandoning the attempt to reach its home or of leaving its adult male members behind. The churls allowed me to pass readily enough, but rescinded their action a moment later. Once beyond the barrier, I had paused to photograph the pandemonium that reigned about it. A lieutenant bellowed and a group of soldiers and officials quickly swarmed about me. Did I not know that photography was forbidden at the front? I protested that the station scenes of Kreuz could scarcely be called military information. What of that? I knew that it was within the zone of the armies, did I not? Rules were rules; it was not the privilege of every Tom, Dick, and Harry to interpret them to his own liking. A lean, hawk-faced civilian, who seemed to be in command, ordered me to open my kodak and confiscated the film it contained. If I set great store by the pictures on it, he would have it developed by the military authorities and let me have those that proved harmless, upon my return. I thanked him for his leniency and strolled toward the compartment I had chosen. Before I had reached it he called me back. “Let me see your papers again,” he demanded, in a far gruffer tone. He glanced casually at them, thrust them into a pocket of his coat, and snapped angrily: “Get your baggage off the train! I am not going to let you through.” It was plain that he was acting from personal rather than official motives. Probably he considered my failure to raise my hat and to smile the sycophant smile with which my fellow-passengers addressed him as an affront to his high Prussian caste. Fortunately he was not alone in command. A more even-tempered official without his dyspeptic leanness beckoned him aside and whispered in his ear. Perhaps he called his attention to the importance of my credentials from Wilhelmstrasse. At any rate, he surrendered my papers after some argument, with an angry shrug of the shoulders, and his less hungry-looking companion brought them back to me. “It has all been arranged,” he smirked. “You may take the train.”
  • 58.
    This was stillmanned by a German crew. For every car that left their territory, however, the Poles required that one of the same class and condition be delivered to them in exchange. Several long freight- trains, loaded from end to end with potatoes, rumbled past us on the parallel track. Two hundred thousand tons of tubers were sent to Germany each month in exchange for coal. It was at that date the only commercial intercourse between the two countries, and explained why potatoes were the one foodstuff of comparative abundance even in Berlin. At Biala the station guards were Polish, but there was little indeed to distinguish them from those of Kreuz and Bromberg. Their uniforms, their rifles, every detail of their equipment, were German, except that some of them wore the square and rather clumsy-looking Polish cap or had decorated their round, red-banded fatigue bonnets with the silver double-eagle of the resurrected empire. Many were without even this insignia of their new allegiance, and only the absence of oak-leaves on their collars showed that they were no longer soldiers of the Fatherland. We halted at Wronki for two hours, which made our departure three hours later, for clocks and watches were turned ahead to correspond with Polish time. Frontier formalities were even more leisurely and disorganized than they had been in Kreuz. The Poles seemed to have something of the amiable but headless temperament of the French. Their officers, too, in their impressive new uniforms with broad red or yellow bands, and their rattling sabers, bore a certain resemblance to children on Christmas morning that did not help to expedite matters under their jurisdiction. They were a bit less “snappy” than the more experienced Germans, somewhat inclined to strut and to flirt, and there were suggestions in their manner that they might not have been horrified at the offer of a tip. When at length my turn had come they found my credentials unsatisfactory. Why had they not been viséed by the Polish consul in Berlin, as well as by the Germans at Frankfurt? I had never dreamed that Berlin boasted a Polish consul. Indeed! Who, then, did I suppose handled the interests of their nation there? However, it was all right. As an American and a fellow-Ally they would let me pass. But I must promise to report at a certain office in Posen within twenty-four hours of my arrival.
  • 59.
    Barefoot boys wereselling huge slabs of bread and generous lengths of sausage through the car windows. All things are relative, and to the travelers from Germany these “ticket-free” viands of doubtful origin seemed a kingly repast. With every mile forward now it was easier to understand why the loss of the province of Posen had been so serious a blow to the hungry Empire. Here were no arid, sandy stretches, but an endless expanse of rich black loam, capable of feeding many times its rather sparse population. If it had been “pumped dry” by the former oppressors, it was already well on the road to recovery. Wheat, corn, and potatoes covered the flat plains to the horizons on either hand. Cattle and sheep were by no means rare; pigs, goats, ducks, and chickens flocked about every village and farm- house, evidently living in democratic equality with the human inhabitants. There were other suggestions that we were approaching the easy-going East. Men in high Russian boots sauntered behind their draft animals with the leisureliness of those who know the world was not built in a day, nor yet in a year. Churches of Oriental aspect, with steep roofs that were still not Gothic, broke the sameness of the prevailing German architecture. There was something softly un-Occidental in the atmosphere of the great city into which we rumbled at sunset, a city which huge new sign-boards on the station platform stridently announced was no longer Posen, but “Poznan.”
  • 60.
    T XI AN AMPUTATED MEMBER (Posenunder the Poles) he same spirit that had led the Poles to impress so forcibly upon the traveler the fact that the city in which he had just arrived was now called Poznan (pronounced Poznánya) had manifested itself in a thousand other changes. In so far as time had permitted, every official signboard had already been rendered into Polish and the detested German ones cast into outer darkness. Only those familiar with the Slavic tongue of the new rulers could have guessed what all those glitteringly new enameled placards that adorned the still Boche-featured station were commanding them to do or not to do. Every street in town had been baptized into the new faith and gaily boasted that fact on every corner. For a time the names had been announced in both languages, as in Metz; but a month or so before my arrival the radicals had prevailed and the older placards had been abolished. True, in most cases the new ones were merely translations of the old. But what did it help the German resident who had neglected to learn Polish to know that the “Alte Markt” was still the “Old Market” so long as he could not recognize it under the new designation of “Stary Rynek”? Imagine, if you can, the sensation of waking up some morning to find that Main Street has become Ulica Glòwna, or to discover that the street-car you had always taken no longer runs to Forest Park but to Ogrott Lass. Nothing but the few things that defied quick change, such as post- boxes or names deeply cut into stone façades, had escaped the all- embracing renovation. Indeed, many of these had been deliberately defaced. The cast-iron “Haltestelle der Strassenbahn” high up on the
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    trolley-supports had beendaubed with red paint, though they were still recognizable to motormen and would-be passengers. Many business houses had followed the official lead, and private signs were more apt than not to have the German words that had once called attention to the excellence of the wares within crudely effaced or changed to the new tongue. Sometimes it was not merely the language that had been altered, but the whole tenor of the proprietor’s allegiance. A popular underground beer-hall in the heart of town was no longer the “Bismarck Tunnel,” but the “Tunel Wilsona.” German trucks thundering by on their iron tires bore the white eagle of Poland instead of the black Prussian bird of prey. German newspapers were still published, but as the streets they mentioned were nowhere to be found in all Poznan, their advertisements and much of their news were rather pointless. It gave me a curiously helpless feeling to find myself for the first time in years unable to guess a word of the language about me. Fortunately all Poznan still spoke German. Only once during my stay there did I find myself hampered by my ignorance of Polish—when a theater- ticket office proved to be in charge of a pair recently arrived from Warsaw. On more than one occasion my advances were received coldly, sometimes with scowls. But a reply was always forthcoming, and whenever I announced myself an American, who spoke the less welcome of the two tongues by necessity rather than by choice, apology and friendly overtures immediately followed. Having effaced the lingual reminders of their late oppressors, the Poznanians had proceeded to pay their respects to the bronze heroes they had left behind. The Germans, as is their custom, had littered the public squares with statues of their chief sword-brandishers, in gigantic size—tender reminders to the conquered people of the blessings that had been forced upon them. The downfall of these had been sudden and unceremonious. Some had descended so hastily that the allegorical figures at their feet had suffered the fate so often overtaking faithful henchmen of the fallen mighty. The stone image of an old woman representing “Sorrow” looked doubly sorrowful with broken nose, legs, and fingers. Kaiser Friedrich, Doctor Bismarck with his panacea of “blood and iron,” the world-famed Wilhelm, had all left behind them imposing pedestals, like university chairs awaiting exponents of newer and more lasting doctrines. Here
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    and there astatue had remained, because it was Polish, but these were few and small and tucked away into the more obscure corners. Next to its change of tongue the most striking feature of the new Poznan was its military aspect. The streets swarmed with soldiers even during the day; in the evening the chief gathering-places became pulsating seas of field gray. For it was still the garb of their former servitude that clothed the vast majority of these warriors of the reborn nation. The silver double-eagle on his service-faded cap was all that was needed to turn a wearer of the German uniform into a soldier of Poland. Many still wore their “Gott mit uns” belt-buckles and their Prussian buttons. A scattering minority, officers for the most part, were conspicuous in the full new Polish uniform—double- breasted, with a forest-green tinge. The high, square cap, distinctive only of the province of Poznan, was more widely in evidence; the less cumbersome headgear of military visitors from Warsaw or Galizia now and then broke the red-banded monotony. But the only universal sign of new fealty was the silver double-eagle. This gleamed everywhere. Men in civilian garb wore it on their hats or in their coat lapels; women adorned their bodices with it; boys and girls proudly displayed it in some conspicuous position. It fluttered on a thousand banners; it bedecked every Polish shop-front; it stared from the covers of newly appeared books, pamphlets, music-sheets in the popular tongue; the very church spires had replaced their crosses with it. One could buy the resurrected insignia, of any size or material, in almost any shop—providing one could produce “legitimation papers” or other proof that it would not be used to disguise a German as a Pole. An over-abundance of swords tended to give the new army a comic-opera aspect, but this detail was offset by the genuine military bearing of all but a few of the multitude in uniform. The great majority, of course, had had German training. Now, however, they put the “pep” of a new game into the old forms of soldierly etiquette. Their two-finger salute was rendered with the precision of ambitious recruits and at the same time with the exactitude of “old-timers.” They sprang unfailingly to attention at sight of a superior officer and stood like automatons until he turned away. Yet there seemed to be an un-German comradeship between the rank and file and the commissioned personnel, a democracy of endeavor, a feeling that
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    they were allembarked together on the same big new adventure. There were, to be sure, some officers and a few men whose sidewalk manners suggested that they had learned Prussian ways a bit too thoroughly, but they were lost in a mass that had something of the easy-going temperament of the East or the South. All classes of the Polish population were represented in the new army from the bulking countryman who ran after me to say that the photograph I had just taken of him would not be a success because he had not been looking at the lens during the operation to the major who granted me special permission to use my kodak in spite of military rules. This officer had been late in reaching his office, and I passed the time in his anteroom in conversation with his sergeant- major. When he entered at last the entire office force sprang to its feet with what in an older army would have been an exaggeration of discipline. The sergeant-major, his middle finger glued to the seams of his trousers, explained my presence and request. The major asked several questions in Polish, which the sergeant repeated to me in German, relaying my replies back to the major in his native tongue. When the latter had nodded his approval and disappeared, and the office force had relaxed into mere human beings, I expressed my surprise that an officer of such high rank knew no German. “Knows no German!” cried the sergeant-major, bursting into laughter. “The major was for nine years a captain in the German army. He is a graduate of the War College in Berlin and was a member of Hindenburg’s staff. But he never lets a word of the accursed tongue pass his lips if he can possibly avoid it.” The new Polish government had established a conscription act as drastic as if it had been taken bodily from the old German statute- books. All males between the ages of seventeen and forty-five were liable to service. Those between eighteen and thirty had already been called to the colors, though thus far German residents had been tacitly exempted. Every afternoon of my stay in Poznan a hundred or two of recruits, flower-bedecked and carrying each his carton of travel rations, marched in column of squads from the railway station to what had once been the Kaiser’s barracks, singing as they went some rousing Polish song of the olden days. At least half of them wore more or less complete German uniforms. Some were so under- sized that a rifle in their hands would have resembled a machine-
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    gun. But withfew exceptions their military bearing testified to previous training under the exacting drill-sergeants of their former rulers. Watching this new addition each day to the hordes in uniform that already crowded the city, one could not but wonder whether the new Poland was not giving refuge, perhaps unconsciously, to the discredited spirit of militarism that had so recently been expelled from its German Fatherland. The “revolution,” or “Putsch,” as the Poles call it, that brought about all this new state of affairs had been brief and to the point. Paderewski, relying, perhaps, on Germany’s promise to help re- establish the ancient Polish Kingdom, had come to Posen for the Christmas holidays. The hotel he occupied had been decorated with the flags of the Allies. It is scarcely surprising that the Germans proceeded to tear them down in spite of the armistice that had recently been concluded. According to several observers, they might even have “got away with” this had they not persisted in their Prussian aggressiveness. On December 27th a Polish youth paused to ask another for a light from his cigarette. Matches had long been precious things in Posen. A German officer pounced upon the pair and demanded to know what conspiracy they were hatching together. The Polish youths quite properly knocked him down. Their companions joined in the fracas. The Polish turnvereins had long had everything prepared for just such an eventuality. Word swept like prairie fire through the city. French and Italian prisoners of war sprang to such arms as they could lay hands on and added their assistance. The soldiers of the garrison, being chiefly Poles or of Polish sympathies, walked out almost in a body and joined the revolt. It raged for twenty-four hours. In the words of the sergeant-major already introduced: “It was a busy day from four in the morning until the following dawn. At least sixty ribs were broken—mostly German ones.” There have been bloodier revolutions, however, for the number killed is set at ten. The Polish leaders were soon masters of the situation. In three days they had established order. Their search for arms was thorough and included Polish as well as German houses. The government they had already established in secret soon tautened the reins that had been struck from the hands of the Germans, and by New Year’s Day Poznan had already settled down to peace and to a contentment it had not known in more than a century.
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    As far, atleast, as outward appearances go, there was nothing particularly oppressive about the new rule. Civilians were not permitted on the streets after midnight, but those with any legitimate excuse for night-hawking were granted special passes. The Poles showed a tendency to meet half-way their next-door neighbor and late oppressor. With the exception of a few “Polen-fresser,” German residents were not driven out, as in Metz and Strassburg. Boche merchants continued to do business at the old stand. Newspapers published in Germany were refused admittance, but that was a fair retaliation for similar action by the new authorities of the late Empire. Even the detested statues were not overthrown until March, when the Germans declined to give the Poles port facilities at Danzig. The language of the schools, as well as of government offices, was changed to Polish; but as soon as Berlin consented to a reciprocal arrangement, German was restored to the curriculum, though it was taught only a few hours a week, as a foreign tongue. In short, the conditions of Bromberg had been nicely reversed in Poznan. It must, to be sure, have been rather a tough life for the town braggart who had always espoused the German cause; but there was apparently nothing to be feared by those who know how to hold their tongues and confine their attention to their own affairs—and the German is a past-master at lying low when it is to his interest to do so. His native tongue was almost never heard on the streets, such arrogance as existed was confined now to the Poles, and the just-let-us-alone-and- we’ll-be-good rôle had been assumed by the Teutons. There were suggestions, however, that the Poles were not yet adepts at governing, nor likely soon to establish a modern Utopia. Already they had succeeded in encumbering themselves with fully as much red-tape as the French. A musician as national leader and rallying-point seemed to be in keeping with the Polish temperament. There was a lack of practical directness in their methods, a tendency toward the erratic, at the expense of orderly progress. One of their foremost business men turned high official, to whom I applied for a signature and the imprint of a government stamp, received me with a protest that he was “too busy to breathe”—and spent two hours reciting Polish poetry to me and demonstrating how he had succeeded in photographing every secret document that had reached Posen during the war without being once suspected by the Germans. “I am not experienced in this business of government,” he
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    apologized, when Isucceeded at last in taking my leave, “but I am ready to sacrifice myself and all I have to the new Poland.” The statement rang true in his case, but there were others whose repetition of it would have raised grave suspicions that they were putting the cart before the horse. The rush for government jobs under the new régime had in it something of the attitude of the faithful henchmen toward the periodical return to power of their beloved Tammany. There were tender reminiscences of the A. E. F. in the flocks of incompetent pretty girls who encumbered government offices, dipping their charming noses into everything except that which concerned them, as there was in the tendency on the part of both sexes to consider government transportation synonymous with opportunity for “joy-riding.” It will be strange if the Polish servant-girls and factory hands who come to us in the future bring with them the accept-anything spirit of the past, at least after the period of orientation to their new environment is over. They are “feeling their oats” at home now and will be apt to set their worth and their rights to full equality correspondingly higher. The Poles, evidently, are not by nature a frolicsome people, but they seemed to have thrown away the “lid” in Poznan and given free play to all the joy within them. Pianos were more in evidence than they had been during all the twenty months I had spent in war-torn Europe. Children appeared to have taken on a new gaiety. Night life was almost Parisian, except in the more reprehensible features of the “City of Light.” It may have been due only to a temporary difference of mood in the two races, but Polish Poznan struck me as a far more livable place than German Berlin. Evidently the people of the provinces were not letting this new attractiveness of the restored city escape them; the newspapers bristled with offers of reward for any one giving information of apartments or houses for rent. Underneath their merriness, however, the religious current of the race still ran strong and swift. The churches discharged multitudes daily at the end of morning mass; no male, be he coachman, policeman, soldier, or newsboy, ever passed the crucifix at the end of the principal bridge without reverently raising his hat. There are Protestant Poles, but they apparently do not live in Poznan. Now and again, too, there were episodes quite the opposite of gay to give the city pause in the midst of its revelry—the drunken sots in uniform, for instance, who
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    canvassed the shopsdemanding alms and prophesying the firing- squad for those who declined to contribute. Were they not perhaps the outposts of Bolshevism? But all this was immersed in the general gaiety, tinged with a mild Orientalism that showed itself not only in the architecture, but in such leisurely customs as closing shops and offices from one to three, in defiance of nearly a century and a half of the sterner German influence. It is quite possible that the increased liveliness of the Poznanians was as much due to the fact that they had plenty to eat as to their release from Teutonic bondage. The two things had come together. Being perhaps the richest agricultural district of the late Empire, the province of Posen was quick to recover its alimentary footing, once its frontiers had been closed against the all-devouring German. With the exception of potatoes, of which the supply was well in excess of local needs, the exportation of foodstuffs toward the hungry West had absolutely ceased. The result was more than noticeable in Poznan; it was conspicuous, all but overpowering, particularly to those arriving from famished Germany. Street after street was lined with a constant tantalization to the new-comer from the West, arousing his resentment at the appetite that was so easily satisfied after its constant vociferations in days gone by—and still to come. Butcher shops displayed an abundance of everything from frankfurters to sides of beef. Cheese, butter, eggs by the bushel, candy, sugar, sweetmeats were heaped high behind glass fronts that would have been slight protection for them in Berlin. In what were now known as “restauracya” one might order a breakfast of eggs, bacon, milk, butter, and all the other things the mere mention of which would have turned a German Wirt livid with rage, without so much as exciting a ripple on the waiter’s brow. At the rathskeller of Poznan’s artistic old city hall a “steak and everything,” such a steak as not even a war-profiteer could command anywhere in Germany, cost a mere seven marks, including the inevitable mug of beer and the “10 per cent. for service” that was exacted here also by the Kellners’ union. With the low rate of exchange—for Poznan was still using German money—the price was considerably less than it would have been in New York at the same date. Far from being short of fats, the Poles were overgenerous with their grease and gravies. Bacon could be had in any quantity at six marks a pound; eggs at thirty pfennigs each. Bread, brown but excellent, was unlimited. Food-
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    tickets, unknown inhotels and restaurants, were theoretically required for a few of the principal articles in the shops, but there was little difficulty in purchasing without them, at least with the payment of a slight “premium.” On market-days the immense square allotted to them was densely crowded from corner to corner by curiously garbed female hawkers and countrymen offering every conceivable product of their farms and gardens. Poznan still consumed a few things that do not appear on the American bill of fare, such as doves, gull eggs, and various species of weeds and grasses; but the fact remains that the well-to-do could get anything their appetites craved, and the poor were immensely better off than in any city of Germany. There was only one shortage that irked the popular soul. Expression of it rang incessantly in my ears—“Please tell America to send us tobacco!” The queues before tobacconists’ shops were as long and as persistent as in Germany. Ragged men of the street eagerly parted with a precious fifty-pfennig “shin-plaster” for a miserable “cigarette” filled for only half its length with an unsuccessful imitation of tobacco. The principal café, having husbanded its supply of the genuine article, placed a thousand of them on sale each evening at eight, “as a special favor to our clients.” By that hour entrance was quite impossible, and though only two were allowed each purchaser, there was nothing but the empty box left five minutes later. Unselfishness is not one of mankind’s chief virtues, particularly in that chaos of conflicting interests known to the world as central Europe. In view of all they had won in so short a time, and amid the German shrieks of protest, it was disconcerting to find that the Poles were far from satisfied with what had been granted them by the Peace Conference. From high government officials to the man in the street they deluged me with their complaints, often naïvely implying that I had personally had some hand in framing the terms of the proposed treaty, or at least the power to have them altered before it was too late. They were dissatisfied with the western frontier that had been set for them, especially in West Prussia; they were particularly disgruntled because they had not been given Danzig outright. A nation of thirty million people should have a harbor of its own. Danzig was essentially Polish in its sympathies, in spite of the deliberate Germanization that had been practised upon it. Strangely enough they accused America of having blocked their aspirations in
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    that particular. Theyblamed Wilson personally for having shut them out of Danzig, as well as for the annoying delay in drawing up the treaty. The Germans had “got at him” through the Jews. The latter had far too much power in the American government, as well as in American finances. The impression was wide-spread in Poznan that Mrs. Wilson is Jewish. The Germans and the Jews had always stuck together. Poland had always been far too lenient with the Jews. She had let them in too easily; had granted them citizenship too readily. As they spoke either Yiddish, an offshoot of German, or Russian, they had always lined up with the enemies of Poland. Half the German spies, every one of the Russian spies with whom Polish territory had been flooded during the war, had been Jews. The Poles in America had gathered money for the alleviation of suffering in their home-land, and had given it to Jews, Germans, and Poles, irrespective of race. The Jews in America had collected similar funds and had expended them only among the Jews. From whatever point of view one approached him, the resident of Poznan had nothing good to say of the Chosen People. The story of Posen’s existence under German rule, now happily ended, was largely a repetition of what had already been told me in Bromberg. In some ways this region had been even more harshly treated, if my informants were trustworthy. Polish skilled workmen “clear down to button-makers” had been driven out of the province. Great numbers had been more or less forcibly compelled to migrate into Germany. There were at least four hundred thousand Poles in the mines and factories of Westphalia. Saxony was half Polish; the district between Hamburg and Bremen was almost entirely Slavish in population. The Ansiedler—the German settlers whom the government had brought to Posen—had acquired all the best land. On the other hand, German Catholics were not allowed to establish themselves in the province of Posen, lest they join their coreligionists against the Protestant oppressors. Perhaps the thing that rankled most was the banishment of the Polish language from the schools. One could scarcely speak it with one’s children at home, for fear of their using it before the teacher. Many of the youngsters had never more than half learned it. In twenty years more no one would have dared speak Polish in public. Men had been given three, and even four, months in prison for privately teaching their children Polish history. The schools were hopelessly Prussianized; the German
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    teachers received aspecial premium of one thousand marks or more a year over the regular salaries. All railway jobs went to Germans, except those of section men at two marks a day. There had been Polish newspapers and theaters, but they had never been allowed any freedom of thought or action. “The trouble with the German, or at least the Prussian,” one new official put in, “is that it is his nature to get things by force. He was born that way. Why, the Prussians stole even their name; it was originally Barrusen, as the little corner of Russia was called where the robbers first banded together. They marauded their way westward and southward, treading first little people and then little nations under their iron heels. The very word the German uses for “get” or “obtain” tells his history. It is kriegen, to win by war—krieg. You seldom hear him use the gentler bekommen. Everything he possesses he has gekriegt. Then he is such a hypocrite! In 1916, when we Poles first began to suffer seriously from hunger, some German officers came with baskets of fruit and sandwiches, gathered a group of Polish urchins, filled their hands with the food, and had themselves photographed with them, to show the world how generous and kind-hearted they were. But they did not tell the world that the moment the photographs had been taken the food was snatched away from the hungry children again, some of the officers boxing their ears, and sent back to the German barracks. How do you think the Poles who have been crippled for life fighting for the ‘Fatherland’ feel as they hobble about our streets? What would you say to serving five years in the German army only to be interned as a dangerous enemy alien at the end of it, as is the case with thousands of our sons who were not able to get across the frontier in time? No, the Germans in Poznan are not oppressed as our people were under their rule. We are altogether too soft-hearted with them.” The German residents themselves, as was to be expected, took a different view of the situation. When the Polish authorities had decorated my passport with permission to return to Berlin, I took no chances of being held up by the cantankerous dyspeptic at Kreuz and applied for a new visé by the German Volksrat of Posen. It occupied a modest little dwelling-house on the wide, curving avenue no longer recognizable under its former title of “Kaiser Wilhelm Ring.” Barely had I established my identity when the gloomy Germans took me to
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    their bosom. HadI been fully informed of their side of the situation? Would I not do them the kindness to return at eleven, when they would see to it that men of high standing were there to give me the real facts of the case? My impressions of Posen would be wholly false if I left it after having consorted only with Poles. As a matter of fact I had already “consorted” with no small number of German residents, chiefly of the small-merchant class. Those I had found somewhat mixed in their minds. A few still prophesied a “peasants’ war” in the territory allotted to Poland; a number of them shivered with apprehension of a “general Bolshevist uprising.” But fully as many pooh-poohed both those cheerful bogies. One thing only was certain—that without exception they were doing business as usual and would continue to do so as long as the Poles permitted it. The feeling for the “Fatherland” did not seem strong enough among the overwhelming majority of them to stand the strain of personal sacrifice. When I returned at eleven the Volksrat had been convoked in unofficial special session. A half-dozen of the men who had formerly held high places in the Municipal Council rose ostentatiously to their feet as I was ushered into the chief sanctum, and did not sit down again until I had been comfortably seated. The chief spokesman had long been something corresponding to chairman of the Board of Aldermen. His close-cropped head glistened in the sunshine that entered through the window at his elbow, and his little ferret-like eyes alternately sought to bore their way into my mental processes and to light up with a winsome naïveté which he did not really possess. Most of the words I set down here are his, though some of them were now and then thrown in by his subservient but approving companions. “With us Germans,” he began, “it has become a case of ‘Vogel friss oder starb’—eat crow or die. We are forced, for the time at least, to accept what the Poles see fit to allow us. The German residents of Posen are not exactly oppressed, but our lives are hemmed in by a thousand petty annoyances, some of them highly discouraging. Take, for instance, this matter of the street names. Granted that the Poles had the right to put them up in their own language. It was certainly a sign of fanaticism to tear down the German names. More than a fourth of the residents of Posen cannot read the new street placards.
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    There is nota Polish map of the city in existence. When the province of Posen came back to us the Polish street names were allowed to remain until 1879—for more than a hundred years. It is a sign of childishness, of retarded mentality, to daub with red paint all the German signs they cannot remove! It isn’t much more than that to have forbidden the use of our tongue in governmental affairs. We Germans used both languages officially clear up to 1876. We even had the old Prussian laws translated into Polish. It is only during the last ten years that nothing but German was permitted in the public schools; and there have always been plenty of Polish private schools. I am still technically a member of the Municipal Council, but I cannot understand a word of the proceedings, because they are in Polish. Our lawyers cannot practise unless they use that language, although the judges, who pretend not to know German, speak it as readily as you or I. Yet these same lawyers cannot get back into Germany. At least give us time to learn Polish before abolishing German! Many a man born here cannot speak it. There are German children of eighteen or twenty, who have never been outside the province, who are now learning Polish—that is, to write and speak it correctly. “Oh yes, to be sure, we can most of us get permission in three or four weeks to leave the province, but only by abandoning most of our possessions and taking an oath never to return. No wonder so many Germans become Poles overnight. You can hardly expect otherwise, when they have lived here all their lives and have all their property and friends and interests here. No, military service is not required of Germans, even if they were born here; but many of our youths have voluntarily become Polish soldiers, for the same reason that their parents have suddenly turned Poles. Naturally, there is fighting along the boundary of the province. The Poles want to fight, so they can have an excuse to keep their men under arms, and what can Germany do but protect herself? Poland is planning to become an aggressive, militaristic nation, as was falsely charged against the Fatherland by her enemies. “The complaints of the Poles at our rule were ridiculous. We paid German teachers a premium because they had harder work in teaching German to Polish children and in seeing that they did not speak the language that was unwisely used at home. Railroad jobs,
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    except common labor,were given to Germans because they were more efficient and trustworthy. Besides, does not Germany own the railroads? They complain that the best land was taken by German settlers; but the Poles were only too glad to sell to our Ansiedler—at high prices. Now they are attacking us with a fanaticism of the Middle Ages. Eighteen hundred German teachers, men who have been educating the Poles for twenty or twenty-five years, have suddenly been discharged and ordered to vacate government property within four weeks—yet they are not allowed to go back to Germany. The Pole is still part barbarian; he is more heartless than his cousin the Russian. “Seventy per cent. of the taxes in the province of Posen are paid by Germans. Yet no German who was not born here can vote, though Poles who were not can. I know a village where there are seventy Germans and five Poles—and the five Poles run things to suit themselves. Husbands, wives, and sons often have different rights of suffrage. The family of Baron X has lived here for a hundred and fifty years. The baron himself happens to have been born in Berlin, because his mother went there to see a doctor. So he cannot vote, though his Polish coachman, who has not been here ten years, has all the rights of citizenship. The result is that government affairs are getting into a hopeless muddle. An ignorant fellow by the name of Korfanti—a Polish ‘German-eater’—has now the chief voice in the Municipal Council. The Poles boycott German merchants. They deluge the city with placards and appeals not to buy of Germans. For a long time they refused to trade even a miserable little Polish theater for our splendid big Stadttheater. When the director of that finally got permission to take over the wholly inadequate little playhouse for next season he had to advertise in order to find out how many Germans intend to stay in Posen—as you have seen in our German paper. What can the Poles do with our magnificent Stadttheater? They have no classics to give in it, nor people of sufficient culture to make up an audience. We are still allowed to give German opera, because they know they cannot run that themselves, and a few of the more educated Poles like it. But our splendid spoken classics seem to be doomed. “Then there is their ridiculous hatred of the Jews. The race may have its faults, but the five or six thousand Jews of Posen province
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    play a mostimportant business and financial rôle. They have always understood the advantages of German Kultur far better than the Poles. There is a Jewish Volksrat here that tries to keep independent of both the other elements of the population; but the great majority of the Jews stand with the Germans. They have no use for this new Zionism—except for the other fellow—unless you take seriously the aspirations of a few impractical young idealists”—a statement, by the way, which I heard from Jews of all classes in various parts of Germany. “We Germans lifted the Poles out of their semi-savagery. We brought them Kultur. Do not be deceived by what you see in Posen. It is a magnificent city, is it not?—finer, perhaps, than you Americans found Coblenz? Yet everything that gives it magnificence was built by the Germans—the well-paved streets, the big, wide boulevards, the splendid parks, all the government buildings and the best of the private ones, the street-cars, the electric lights, even the higher state of civilization you find among the masses. There is not a Pole in the province of Posen who cannot read and write. Do not make the mistake of thinking all these things are Polish because the Poles have stolen them. Before you leave, go and compare Posen with the Polish cities outside Germany. That will tell the story. In non-German Poland you will be struck by the appalling lack of schools, roads, doctors, hospitals, education, culture, by the sad condition of the workmen and the peasants—all those things that are included in the German word Kultur. In Galizia, where Austria virtually allowed the Poles to run themselves, the houses are only six feet high, and you could walk all day without finding a man who can read and write, or who can even speak German. Their cities are sunk in a degradation of the Middle Ages. Posen will fall into the same state, if the present Municipal Council continues in power. There are already frontier troubles between German and Russian Poland, and quarrels between the different sections that confirm what we Germans have always known—that the Poles cannot govern themselves. Warsaw does not wish to keep up our splendid system of workmen and old-age insurance because there is none in Russian Poland. Galizia complains that farm land is several times higher in price in the province of Posen, without admitting that it is German railroads and German settlers that have made it so. That advantage will soon disappear. The Poles will make a mess of the whole province and will
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    have it sunkinto the degradation in which we found it by the time a real ruling nation takes charge of it again.” Just how much truth there was mixed in with the considerable amount of patent nonsense in the ex-chairman’s declamation only a long stay in Poznan, or time itself, would show. The fact that the Poles allowed many of these statements, particularly the protests against the sudden change of language, to be published in the local German newspaper speaks at least for their spirit of tolerance. Though the new government was visibly making mistakes, and had not yet settled down to the orderliness that should come from experience, no one but a prejudiced critic could have discovered immediate evidence that it was making any such complete “mess” of matters as the German Volksrat testified. Even if it had been, at least the mass of the population showed itself happy and contented with the change, and contentment, after all, may in time result in more genuine and lasting progress than that which comes from the forcible feeding of German Kultur. I dropped in at the Teatro Apollo one evening, chiefly to find out how it feels to see a play without understanding a word of it. An immense barnlike building, that looked as if it had once been a skating-rink or a dancing-pavilion, was crowded to suffocation with Poles of every class and variety, from servant-girls in their curious leg-of-mutton sleeves to colonels in the latest cut of Polish uniform. The actors—if they could have been dignified with that title—had recently been imported from Warsaw, and the alleged play they perpetrated could scarcely have been equaled by our silliest rough- and-tumble “comedians.” The herd-like roar with which their inane sallies were unfailingly greeted testified that the audience found them entertaining. But it may be that Poznan was in a particularly simple-minded mood during its first months of relief from a century of bitter oppression. I hope so, for I should regret to find that the startling contrast between this Polish audience and the German one at the artistic Stadttheater the following evening fairly represented the difference between the two races. I believe I am not prejudiced by the fact that the Volksrat presented me with a free ticket when I say that the latter performance was one of which any manager might have been justly proud. The audience, too, resembled the other about as a gathering of college professors resembles a collection of factory
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    hands. There wasa well-bred solemnity about it that could not, in this case, have been due merely to hunger, for there was no munching whatever between the acts, none even under cover of the darkened house, except here and there of candy, a luxury so long since forgotten in Berlin that the happy possessor would never have dreamed of giving his attention at the same time to the merely esthetic appeal of the theater. There may have been Poles in the house, but at least the new army was conspicuous by its absence. Not a uniform was to be seen, with the exception of three scattered through the “peanut gallery.” Two crown boxes, destined only for Hohenzollern royalty or its representatives, sat empty, with something of the solemn demeanor of the vacant chair at the head of the table the day after the funeral. Who would occupy them when the Poles had taken over the playhouse? What, moreover, would they do toward maintaining the high standards of the stage before us? For the most indefatigable enemy of the Germans must have admitted that here was something that could ill be spared. If only they had been contented with bringing the masses these genuine benefits, without militarism, with more open competition, without so much appeal to the doctrine of force—but it has ever been Germany’s contention that only by force can the mass of mankind be lifted to higher levels; that only an army can protect the self-appointed missionaries of a loftier civilization. Armed with what those who read Polish assured me was permission to do so, I set out on foot one morning to the eastward. Beyond the last group of guards wearing the silver double-eagle on their threadbare German uniforms, I fell in with three barefooted Polish peasant women. They were barely thirty, yet all three were already well-nigh toothless, and their hardy forms and faces were plainly marked with the signs that testify to grueling labor and the constant bearing of children. The German they spoke was far superior to the dialects of many regions of purely Teutonic population. Their demeanor was cheerful, yet behind it one caught frequent glimpses of that background of patient, unquestioning acceptance of life as it is which distinguishes the country people of Europe. The most energetic of the trio showed a willingness to enter into conversation; the others confined themselves to an occasional nod of
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    approval, as ifthe exertion of keeping pace with us left them no strength to expend in mere words. It was plain from the beginning that they were not enthusiastic on the subject then uppermost in the city behind us. They greeted my first reference to it with expressions that might have been called indifferent, had they not been tinged with evidence of a mild resentment. “What does it matter to us people of the fields,” retorted the less taciturn of the group, “whether Poles or Germans sit in the comfort of government offices, so long as they let us alone? Things were all right as they were, before the war came. Why trouble us with all these changes? Now they are breaking our backs with new burdens, as if we had not had enough of them for five years. First they take our men and leave us to do their work. I have not a male relative left, except my husband, and he is so sickly that he is no longer a man. He is paid twelve marks for eight hours’ work; fifteen for ten. But what help is that when he cannot work ten hours, or even eight? They offered him the iron cross. He told them he would rather have something to feed his family with at home. They asked him if he was not already getting forty marks a month for the support of his family. How could I feed four children, even after the other two had died, with forty marks a month? For three winters I had nothing but dried potatoes and salt. I could not have bread for myself because the flour for the children took all the tickets. Now the war is over, yet they are still taking away what we have left. The same soldiers come and drive off our horses—for the silver eagle on their caps has not changed their natures. Pay for them? Ach, what is eight hundred marks for a horse that is worth six thousand? And how can we cultivate our fields without them? Who started the war? Ach, they are all arguing. What does it matter, so long as they stop it? Will the Germans sign? They should, and have done with it. If they don’t, all the men over fifty, including the Germans and even the Jews”—there was a sneer in this last word, even in the country—“will be at it again. We have had enough of it. Yet if the soldiers come and tell my husband to go he must go, sick though he is.” The basket each of the trio carried contained the midday lunch of her husband in the fields. I turned aside to the grassy slope on which two of the couples assembled. The men insisted that I share their meal with them. It was more nourishing than a ten-mark repast in a
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    Berlin restaurant, butthe absence of bread was significant. When I gave the men each a pinch of tobacco crumbs they announced themselves delighted at the exchange, and mumbled halting words about the well-known generosity of Americans. As I turned my kodak upon them they greeted it with a laughing “Oh, là là!” There was no need to ask where they had picked up that expression. It oriented their war experiences as definitely as it will distinguish for years to come the Americans, in whatever garb one finds them, who were members of the A. E. F. in France. The men were less indifferent to the recent change of government than their wives, but even they could not have been called enthusiastic. What struck one most was the wider outlook on life the Germans had been forced to give them in spite of themselves. Had they been left to till their farms, these plodding peasants would probably still have swallowed whole the specious propaganda of their erstwhile rulers. Now, after four years of military service that had carried them through all central Europe, they had developed the habit of forming their own opinions on all questions; they took any unverified statement, from whatever source, with more than a grain of salt. It would be a mistake nowadays to think of the European peasant as the prejudiced conservative, the plaything of deliberate misinformation, which he was five years ago. In the light of his new experiences he is in many cases doing more individual thinking than the average city resident. Yet, I must admit, the conclusions of this well-traveled pair did not boil down into anything very different from the consensus of opinion, even though they reached them by their own peculiar trains of thought. Germany, they were convinced, had the full guilt of the war; not the Kaiser particularly—they call him “Wilhelm” in Posen province now, and even there one detects now and again a tendency toward the old idolatry he seems personally to have enjoyed throughout the whole Empire—but the military crowd, “and the capitalists.” They disclaimed any hatred of the Germans, “until they wanted to rule the earth” and sought to make the peasants the instruments of their ambition. They, too, charged Wilson personally with delaying the conclusion of peace—on the fate of Danzig they seemed to be supremely indifferent.
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    “It’s all politics,anyway,” concluded one of them. “They are all playing politics. If the Germans don’t sign they will be divided up as Poland was a hundred and forty years ago. But this new government in Posen is no better than the old. What we need is something entirely new—a government of the peasants and of the working- classes.” The women had from the beginning tried to lead their husbands away from “arguing politics,” chiefly with ludicrously heavy attempts at coquetry, and at length they succeeded. I regained the highway. On either hand lay slightly rolling fields of fertile black soil, well cultivated as far as the eye could see, with only a scattering of trees. Miles away an abandoned Zeppelin hangar bulked into the sky. There were more women laborers than men; several gangs of them were working with picks and shovels; another group was slowly but patiently loading bricks. Horses were to be seen here and there, but oxen were in the majority. Farm-houses showed a rough comfort and a tolerable cleanliness, villages a passable neatness that may or may not have been due to German influence. Certainly the architecture, the farming methods, the communal customs, were little different from those of Prussia or the Rhineland. The dinner served me in the chief tavern of a village of some two thousand inhabitants was nothing to complain of, either in variety or price. A general-shop keeper stated that “with the exception of a few semi-luxuries, such as cocoa and toilet soap,” his grocery department could still meet the decreased demands made upon it. In the clothing lines everything was scarce or wholly lacking. Worst of all, there was nothing fit to drink or smoke. The strong spirits that had once been his chief trade had become so weak no one but boys would drink them. If only America would send concentrated alcohol they could doctor the stock of liquor they had on hand so that no one would know the difference. Then if they could only get some American tobacco! Life was not what it used to be, without a real cigarette from one month’s end to the other. The German rule, on the whole, had not been so bad as many of the Allies seemed to believe. They got along, though it was rather pleasant to be relieved of the arrogant fellows, or see them crawl into their shells. No German resident in the village had given any sign of intending to move away. The communal school was still teaching the German language—two or
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    three hours aweek now. No one had noticed any other change of any importance. The French prisoners confined in the province during the war had been brutally treated. There was no doubt about that; he had seen it himself. But on the whole the German authorities had not been much harder on the Polish population than upon their own people, in Prussia and elsewhere. It was all part of the war, and every one in the Empire had to bear his share of the burdens. Happily, it was over now, if only the new Polish government did not grow ambitious for military conquests also, with the millions of soldiers, some of them patriotic to the point of self-sacrifice, under its command. My hope of walking out of Posen province suffered the same fate as my plan of tramping into it from Germany. In the end I was forced to return to Poznan and make my exit by train over the same route by which I had entered. In the third-class compartment I occupied there were five German residents who had renounced forever their right to return, for the privilege of leaving now with the more portable of their possessions. Two of them had been born in the amputated province; the others had lived there most of their lives. All spoke Polish as readily as German. One masterly, yet scholarly youth, who had served through the war as a lieutenant, was a school- teacher by profession, as was the uncle who accompanied him. They had taught six and twenty-six years, respectively, but had been dispossessed of their positions and of their government dwellings by the new rulers. Up to the time we reached the frontier all five of my companions laid careful emphasis on the statement that they were going to seek re-establishment in their civilian professions in what was left of the Fatherland. At Wronki the Polish authorities were far more inquisitive than they had been toward travelers from the other direction. One by one each compartment group was herded together, bag and baggage, and strained through the sieve of a careful search-and-questioning bureau. The soldier who examined my knapsack glared at the half- dozen precious American cigars I had left as if nothing but the presence of his superiors could have prevented him from confiscating them. Only sufficient food for the day’s journey was allowed to pass. In some cases this rule was interpreted rather liberally, but no one got through with more than ten or twelve
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    pounds to theperson. The amount that was confiscated easily sufficed to feed the garrison of Wronki for the twenty-four hours before the next westbound train was due. An old woman, riding fourth class, who resembled one of India’s famine victims, was despoiled of almost the entire contents of her trunk-sized chest— several sacks of flour, a dozen huge loaves of bread, and a generous supply of sausage. The fact that she spoke only Polish did not seem to impress the searchers in her favor, who silenced her wails at last by bundling her bodily back into the coach and tossing her empty coffer after her. When at last we were under way again the Germans in my compartment took to comparing notes. One, a doctor, was bewailing the “plain theft” of a surgical appliance of rubber which the Poles had confiscated in spite of what seemed to be complete proof that it was his private property and not part of the German army supplies. A foxy-faced country youth, who had carefully changed from shoes to high boots just before the arrival at Wronki, changed back again now with the announcement that there were some four thousand marks concealed between the boot soles. The younger schoolmaster threw off the disguise with which he had covered his real thoughts and announced, vociferously: “You drive me out to work for my livelihood! I will work for my Fatherland at the same time. I will go to Bromberg this very evening and join the army again. We shall see whether the Poles can keep Posen.” The two other young men asserted that they, too, had left with exactly that intention. An indignation meeting against the Poles raged for an hour or more. “I could have remained and kept my position,” went on the schoolmaster, “if I had wanted to turn Polack. Both my parents were Polish; I spoke it before I did German; but I shall always remain a true son of the Fatherland, no matter what happens to it.” A few hundred yards from Kreuz station our train halted for more than an hour and gave us the pleasure of watching the Berlin express go on without us. Though it would have been a matter of twenty seconds to have sprinted across the delta between the two lines, armed boy soldiers prevented any one from leaving his compartment. To all appearances it was a case of “pure meanness”
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    on the partof the German authorities. Our wrath at being forced to wait a half-day for a dawdling local train was soon appeased, however, by the announcement that we were the last travelers who would be allowed to enter Germany from the province of Posen “until the war was over.” The frontier had been closed by orders from Berlin. It is a long way round from Poland to Holland, and amid the turmoil of gloomy men, disheveled women, and squalling children who had been turned back with their goal so near I found cause to be personally thankful, particularly as I succeeded in eluding during all the afternoon the glassy eye of the cantankerous dyspeptic, who buffeted his way now and then through the throng. Some things are still cheap in Germany. A twelve-word telegram from Kreuz to Berlin cost me nine cents—and it was delivered in telegraphic haste. The hungry passengers from farther east with whom I shared a compartment that evening eyed me greedily as I supped on the supplies I had brought from Posen. One man wearing several diamonds leaned toward me as I was cutting my coffee- brown loaf and sighed, reminiscently, “What beautiful white bread!” When I offered to share it with him, however, he refused vigorously, as if his pride would not permit him to accept what his appetite was so loudly demanding. Unable to find a place in the section to which my third-class ticket entitled me, I was riding second-class. The train-guard on his rounds confiscated my ticket and ignored my offer to pay the difference, with a stern, “It is unlawful to ride in a higher class.” On the Friedrichstrasse platform, however, instead of conducting me to his superiors, he sidled up to me in the darkness and murmured, “If you have a five-mark note with you it will be all right.” Germany is changing indeed if her very railway employees are taking on these Latin characteristics.
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    A XII ON THE ROADIN BAVARIA n excellent express raced all day southward across a Germany lush-green with May. Cattle were scarcer in the fields, horses so rare a sight as to be almost conspicuous, but the fields themselves seemed as intensively, as thoroughly cultivated as my memory pictured them fifteen and ten years before. Within the train there was no crowding; the wide aisles and corridors were free from soldiers and their packs, for though there were a hundred or more in uniform scattered between the engine and the last car, a furlong behind, seats were still to be had. The question naturally arose, Are the Germans so short of rolling-stock, after complying with the terms of the armistice, as they pretend? A traveler racing across the Empire in this roomy, almost luxurious Schnellzug might easily have concluded that their whining on that score was mere camouflage. There were even curtains at the wide windows, though of rather shoddy stuff, and the window-straps of paper were so nicely disguised as to be almost indistinguishable from real leather. He who took pains, however, to dip a bit more deeply into the question found that even this great trunk line was carrying barely a third of its peace-time traffic. The red figures, indicating expresses, on the huge porcelain time-tables decorating station walls were nearly all pasted over with slips of paper, while the black ones of Personenzüge, the stop-everywhere-a-long-time trains, were more than half canceled. The branch lines had contributed even more to the Allies. Nor did our aristocratic Berlin-München express entirely escape overburdening. At Nürnberg came with sunset such hordes of passengers of all grades that every available foot of the train was as densely packed as a fourth-class coach on market-day. The throng it
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    disgorged at Munichwas sufficient to have peopled a town of very respectable size. I had made the sudden leap to the southern end of the Empire as a starting-point of a tramp across it instead of reversing the process in the hope that here at last I should find “something doing,” some remnants of excitement. Munich had just been snatched from the hands of the Sparticists—or the Bolshevists; the distinction between the two dreaded groups is not very clear in the German mind. Leviné, the half-mad Russian Jew who was reputed the organizing spirit of the revolt, was still dodging from one hiding-place to another somewhere in the vicinity. To read the breathless cables to the foreign press was to fancy Munich under a constant hail of shrapnel and machine-gun bullets. Ours was the second passenger-train that had ventured into the city in weeks. All Bavaria was blazing with huge posters, often blood-red in color, headed by the dread word “Standrecht” in letters to be seen a hundred yards away, proclaiming martial law and threatening sudden and dire fate to any one who strayed from the straight and narrow path of absolute submission to the “government-faithful” troops that were still pouring in from the north. Surely here, if anywhere, was a chance for a wandering American to get into trouble. Like so many dreadful things, however, martial law and beleaguered cities prove more terrible at a distance than on the spot. True, a group of soldiers in full fighting equipment held the station exit; but their only act of belligerency toward the invading throng was to hand each of us a red slip granting permission to walk the streets until two in the morning. A bedraggled hotel directly across the way spared me that necessity. The information its registry-pad required of guests was more exacting than its interior aspect, but neither here nor at the station exit was there any demand for proof of identity. Toward midnight, as I was falling asleep, a score of erratically spaced shots and the brief rat-a-tat of a machine-gun sounded somewhere not far away. Their direction was too uncertain, however, to make it worth while to accept the permission granted by the red slip. In the morning the city was thronged with the business-bent quite as if disorders had never dodged in and out of its wide streets. The main hotels, however, had been partly taken over by the staffs of
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    the newly arrivedtroops, and pulsated with field gray. At the doors very young men in iron hats leaned their fixed bayonets in the crook of an elbow while they examined the Ausweis with which each civilian was supposed to prove his identity. I entered several of them in the vain hope that the flash of my American passport would “start something.” The youths in uniform handed it back each time without so much as a flicker of curiosity on their rather dull faces. Inside, another boy volunteer ran his hands hastily over me in quest of concealed weapons; but not even the most obviously harmless Bavarian escaped that attention. The staff evidently had no secrets from the world at large. At any rate, I wandered into a dozen hotel rooms that had been turned into offices and idled about undisturbed while majors gave captains their orders for the day and lieutenants explained to sergeants the latest commands from higher up. What had become of that stern discipline and the far-famed secrecy of the German army? The soldiers of democratic America were automatons in the presence of their officers compared with these free-and-easy youths in gray; over in Posen the Poles were manyfold more exacting. Had I been a spy, there were several opportunities to have pocketed papers strewn about tables and improvised desks. When at last an officer looked up at me inquiringly I explained my presence by asking for written permission to take photographs within the beleaguered city, and it was granted at once without question. Berlin had been sinister of aspect; Munich was bland, a softer, gentler, less verboten land. Its citizens were not merely courteous; they were aggressively good-natured, their cheerfulness bubbled over on all who came in contact with them. It was almost as easy to distinguish a native from the stiff Prussians who had descended upon them as if the two groups had worn distinctive uniforms. Yet Munich had by no means escaped war-time privations. Long lines of hollow-eyed women flowed sluggishly in and out of under-stocked food-shops; still longer ones, chiefly though not entirely male, crept forward to the door of the rare tobacconists prepared to receive them, and emerged clutching two half-length cigarettes each, their faces beaming as if they had suddenly come into an unexpected inheritance. They were good-natured in spite of what must have been the saddest cut of all from the Bavarian point of view—the weakness
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    and high costof their beloved beer. In those vast underground Bierhallen for which Munich had been far-famed for centuries, where customers of both sexes and any age that can toddle pick out a stone mug and serve themselves, the price per liter had risen to the breathless height of thirty-four pfennigs. As if this calamity were not of itself enough to disrupt the serenity of the Bavarian temperament, the foaming beverage had sunk to a mere shadow of its former robust strength. In the “cellar” of the beautiful Rathaus a buxom barmaid reminded me that Tuesday and Friday were meatless days in Germany. The fish she served instead brought me the added information that Munich is far from the sea. My fellow-sufferers constituted a truly democratic gathering. The still almost portly mayor chuckled with his cronies at a table barely visible through the smoke-screened forest of massive pillars. Collarless laborers clinked their mugs, quite unawed by the presence of city councilors or “big merchants.” A leather-skinned old peasant sat down opposite me and opened conversation at once, with no suggestion of that aloofness of the north. From the rucksack that had slipped from his shoulders he took a half-loaf of dull-brown peasant bread and a square of boiled smoked pork, ordering nothing but a half-bottle of wine. Beer, he explained, had fallen too low in its estate to be worthy of his patronage, at least city beer. In his village, three hours away, he could still endure it. Ach, how the famous beer of Munich had deteriorated! How far away those happy days seemed! And to think of paying three marks for a half-bottle of wine! Why, in the good old days.... And this dinner of mine—a plate of fish bones, some stewed grass, city bread, and city beer—worthless stuff—potatoes, to be sure, but not enough to keep a man’s legs under him for half the afternoon —and a bill of more than eight marks! I restrained my impulse to tell him of that prize dinner in Berlin. He had not always been a peasant. Twenty years before he had started a factory—roof tiles and bricks. But in 1915 he had gone back to the farm. At least a Bauer got something to eat. The peace terms? What else could Germany do but sign? If the shoe had been on the other foot the war lords in Berlin would have demanded as much or more. If they hadn’t wanted war in the first place! Wilhelm and all his crowd should have quit two or three years ago while the quitting
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    was good. Whatdid it all matter, anyway, so long as order returned and the peasants could work without being pestered with all this military service, and the taxes, not to mention the “hamsterers,” the pests! American, was I? He had noticed I was not a Bavarian. (So had I, straining my ears to catch the meaning of his atrocious dialect.) He had taken me for a man from the north, a Hamburger perhaps. American? They say that is a rich country. He had read somewhere that even the peasants sometimes had automobiles! How about the beer? Deteriorating there, too, eh? Ach, this war! Going to abolish beer! What an insane idea! What will people live on? They can’t afford wine, and Schnapps is not good for a man in the long run, and too strong for the women and children. Well, he must be getting back to his beet-field. Glad to have met an American. He had often heard of them. Good day and a happy journey. Troops were still pouring into Munich. That afternoon what before the war would have looked to Americans like a large army marched in column of fours along the bank of the swift, pale-blue Isar and swung in through the heart of town. There were infantry, machine- gun, and light-artillery sections, both horse- and motor-drawn, and from end to end they were decorated with flowers, which clung even to the horses’ bridles and peered from the mouths of the cannon. All the aspect of a conquering army was there, an army that had retaken one of its own cities after decades of occupation by the enemy. Greetings showered upon the columns, a trifle stiff and irresponsive with pride, after the manner of popular heroes; but it was chiefly voiceless greetings, the waving of hands and handkerchiefs, in striking contrast to similar scenes among the French. The Boy Scouts of a year or two ago filled a large portion, possibly a majority, of the ranks. The older men scattered among them bore plainly imprinted on their faces the information that they had remained chiefly for lack of ambition or opportunity to re-enter civil life. Their bronzed features were like frames for those of the eager, life-tasting youths they surrounded, not so much in color as in their disillusioned, nothing-new-to-us expressions. All wore on their collars the gold or silver oak-leaves of volunteers for “home and border protection”; an insignia belonging to generals only before the flight of the Kaiser. Rumor had it, however, that there were many still held under the old conscription laws, particularly those of Polish
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    blood. The sameinarticulate voices whispered that, despite the opinion of Allied staffs, Germany still had a million men under arms; on the books they were carried as discharged; in reality they were sustained by the government as “out-of-works” and housed in barracks near enough to arsenals or munition dumps to equip themselves in a twinkling. What percentage of truth the assertion possessed could only have been determined by long and deliberate study, for though Munich, like many another city and even the country districts, seemed to swarm with soldiers, many of them were so only in outward appearance. Discharged men were permitted to use their uniforms until they were worn out; the mere removal of the shoulder-straps made one a civilian—unlike the soldiers resident in the occupied region, where civilian garb of field gray was furnished with the discharged papers—and boys of all ages, in many cases large enough to have the appearance of real soldiers, were as apt to wear the uniform and the red-banded cap without visor as anything else. The Sparticist uprising in Munich, now crushed, evidently made less trouble on the spot, as usual, than in foreign newspapers. All classes of the population—except perhaps that to which the turn of events had brought the wisdom of silence—admitted that it had been a nuisance, but it had left none of them ashen with fear or gaunt with suffering. Indeed, business seemed to have gone on as usual during all but the two or three days of retaking the city. Banks and the larger merchants had been more or less heavily levied upon; lawyers and a few other classes whom the new doctrine ranked as “parasitic” had found it wise to leave their offices closed; but in the main all agreed that the population at large was never troubled in their homes and seldom on the street. The mistreatment of women, with rumors of which foreign newspapers reeked, was asserted to have been rare, and their “nationalization,” which the cables seem to have announced, had not, so far, at least, been contemplated. All in all, the Bavarian capital suffered far less than Winnipeg under a similar uprising of like date. The moving spirit had come from Russia, as already mentioned, with a few local theorists or self-seekers of higher social standing as its chief auxiliaries. The rank and file of the movement were escaped Russian prisoners and Munich’s own out-of-works, together with such disorderly elements as always hover about any upheaval
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    promising loot orunearned gain. But the city’s chief scare seemed to have been its recapture by government troops under orders from Berlin. Then for some fifty hours the center of town was no proper place for those to dally who had neglected their insurance premiums. A hundred more or less of fashionable shop-fronts bore witness to the ease with which a machine-gunner can make a plate-glass look like a transparent sieve without once cracking it; rival sharpshooters had all but rounded off the corners of a few of the principal buildings. The meek, plaster-faced Protestant church had been the worst sufferer, as so often happens to the innocent bystander. The most fire-eating Münchener admitted that barter and business had lagged in the heart of town during that brief period. But Munich’s red days had already faded to a memory. Even the assassination of hostages, among them some of the city’s most pompous citizens, by the fleeing Sparticists was now mentioned in much the same impersonal tone with which the Swiss might refer to the death of William Tell or an Englishman regret the loss of Kitchener. The blue-and-white flag of Bavaria fluttered again from the staffs that had been briefly usurped by the red banner of revolt; the dark-blue uniform of the once half-autonomous kingdom again asserted its sway over local matters in the new Volksreich Bayern. At the Deutsches Theater a large audience placidly sipping its beer set on little shelves before each seat alternately roared and sniffled at the bare-kneed mountaineers in feathered hats and the buxom Mädels who bounced through a home-made but well-done “custom picture” in the local dialect. It was evident that life in Munich was not likely to afford any more excitement than had the apathetic north. The atmosphere of the place only helped to confirm the ever-hardening conviction that the German, north or south, east or west, had little real sympathy for revolutions compared with the privilege of pursuing his calling steadily and undisturbed. It was high time to take to the road while a faint hope still remained that something might lay in wait for me along the way to put a bit of ginger into a journey that had thus far lamentably failed to fulfil its promise. I breakfasted next morning with the German staff. At least I was the only civilian in the palm-decked dining-room where a score of high ranking wearers of the iron cross munched their black bread and purple Ersatz marmalade with punctilious formality. Away from
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