A L G O R I T H M S
I N T R O D U C T I O N T O
T H I R D E D I T I O N
T H O M A S H.
C H A R L E S E.
R O N A L D L.
C L...
Introduction to Algorithms
Third Edition
Thomas H. Cormen
Charles E. Leiserson
Ronald L. Rivest
Clifford Stein
Introduction to Algorithms
Third Edition
The MIT Pre...
c 2009 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form or by...
Contents
Preface xiii
I Foundations
Introduction 3
1 The Role of Algorithms in Computing 5
1.1 Algorithms 5
1.2 Algorithms...
vi Contents
II Sorting and Order Statistics
Introduction 147
6 Heapsort 151
6.1 Heaps 151
6.2 Maintaining the heap propert...
Contents vii
12 Binary Search Trees 286
12.1 What is a binary search tree? 286
12.2 Querying a binary search tree 289
12.3...
viii Contents
V Advanced Data Structures
Introduction 481
18 B-Trees 484
18.1 Definition of B-trees 488
18.2 Basic operatio...
Contents ix
24 Single-Source Shortest Paths 643
24.1 The Bellman-Ford algorithm 651
24.2 Single-source shortest paths in d...
x Contents
30 Polynomials and the FFT 898
30.1 Representing polynomials 900
30.2 The DFT and FFT 906
30.3 Efficient FFT imp...
Contents xi
VIII Appendix: Mathematical Background
Introduction 1143
A Summations 1145
A.1 Summation formulas and properti...
Preface
Before there were computers, there were algorithms. But now that there are com-
puters, there are even more algori...
xiv Preface
You should find it easy to organize your course around just the chapters you
need. We have made chapters relati...
Preface xv
What are the prerequisites for reading this book?
You should have some programming experience. In particular, y...
xvi Preface
of algorithms, however. Though it may be hard to believe for a book of this size,
space constraints prevented ...
Preface xvii
We revised our treatment of dynamic programming and greedy algorithms. Dy-
namic programming now leads off wi...
xviii Preface
Web site
You can use our Web site, http://mitpress.mit.edu/algorithms/, to obtain supple-
mentary informatio...
Preface xix
Science and Artificial Intelligence Laboratory, and the Columbia University De-
partment of Industrial Engineer...
Introduction to Algorithms
Third Edition
I Foundations
Introduction
This part will start you thinking about designing and analyzing algorithms. It is
intended to be a gentle int...
4 Part I Foundations
Chapter 4 delves further into the divide-and-conquer method introduced in
Chapter 2. It provides addi...
1 The Role of Algorithms in Computing
What are algorithms? Why is the study of algorithms worthwhile? What is the role
of ...
6 Chapter 1 The Role of Algorithms in Computing
Because many programs use it as an intermediate step, sorting is a fundame...
1.1 Algorithms 7
Chapter 24), and using a search engine to quickly find pages on which particular
information resides (rela...
8 Chapter 1 The Role of Algorithms in Computing
respectively. Selecting all possible subsequences of X and Y and matching
...
1.1 Algorithms 9
Not every problem solved by algorithms has an easily identified set of candidate
solutions. For example, s...
10 Chapter 1 The Role of Algorithms in Computing
that an efficient algorithm for one cannot exist. In other words, no one k...
1.2 Algorithms as a technology 11
Exercises
1.1-1
Give a real-world example that requires sorting or a real-world example ...
12 Chapter 1 The Role of Algorithms in Computing
Efficiency
Different algorithms devised to solve the same problem often di...
1.2 Algorithms as a technology 13
50 107
lg 107
instructions
107 instructions/second
1163 seconds (less than 20 minutes) :...
14 Chapter 1 The Role of Algorithms in Computing
of algorithms. Algorithms are at the core of most technologies used in co...
Notes for Chapter 1 15
1 1 1 1 1 1 1
second minute hour day month year century
lg n
p
n
n
n lg n
n2
n3
2n
nŠ
Chapter notes...
2 Getting Started
This chapter will familiarize you with the framework we shall use throughout the
book to think about the...
2.1 Insertion sort 17
2
♣
♣
♣ 2
♣
4
♣♣ ♣
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♣
5
♣♣ ♣
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7
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10
♣
Figure 2....
18 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
...
2.1 Insertion sort 19
Initialization: It is true prior to the first iteration of the loop.
Maintenance: If it is true befor...
20 Chapter 2 Getting Started
we prefer not to get bogged down in such formalism, and so we rely on our
informal analysis t...
2.1 Insertion sort 21
counter in each iteration, and we use the keyword downto when a for loop
decrements its loop counter...
22 Chapter 2 Getting Started
a pointer to the array is passed, rather than the entire array, and changes to
individual arr...
2.2 Analyzing algorithms 23
an .n C 1/-element array C. State the problem formally and write pseudocode for
adding the two...
24 Chapter 2 Getting Started
Real computers contain instructions not listed above, and such instructions rep-
resent a gra...
2.2 Analyzing algorithms 25
The best notion for input size depends on the problem being studied. For many
problems, such a...
26 Chapter 2 Getting Started
INSERTION-SORT.A/ cost times
1 for j D 2 to A:length c1 n
2 key D AŒj  c2 n 1
3 // Insert AŒ...
2.2 Analyzing algorithms 27
nX
jD2
j D
n.n C 1/
2
1
and
nX
jD2
.j 1/ D
n.n 1/
2
(see Appendix A for a review of how to sol...
28 Chapter 2 Getting Started
The “average case” is often roughly as bad as the worst case. Suppose that we
randomly choose...
2.3 Designing algorithms 29
order of growth. But for large enough inputs, a ‚.n2
/ algorithm, for example, will
run more q...
30 Chapter 2 Getting Started
2.3.1 The divide-and-conquer approach
Many useful algorithms are recursive in structure: to s...
2.3 Designing algorithms 31
the output pile. We repeat this step until one input pile is empty, at which time
we just take...
32 Chapter 2 Getting Started
A
L R
1 2 3 4 1 2 3 4
i j
k
(a)
2 4 5 7 1 2 3 6
A
L R
1 2 3 4 1 2 3 4
i j
k
(b)
2 4 5 7
1
2 3...
2.3 Designing algorithms 33
A
L R
1 2 3 4 1 2 3 4
i j
k
(e)
2 4 5 7
1
2 3 61
1 2 3 62 2 3 A
L R
1 2 3 4 1 2 3 4
i j
k
(f)
...
34 Chapter 2 Getting Started
To see that the MERGE procedure runs in ‚.n/ time, where n D r p C 1,
observe that each of li...
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