The document discusses recursion, including recursive definitions, recursive programs/algorithms, and applications of recursion. It covers key concepts such as the anatomy of a recursive call, classifying recursions by the number of recursive calls, tail recursion, and avoiding excessive recursion. Examples of recursively defined sequences, functions, and algorithms are provided to illustrate recursive concepts.
This document provides an overview of recursion, including recursive algorithms, functions, data types, and order of execution. Recursion is defined as a function calling itself, either directly or indirectly. Recursive algorithms break down a problem into smaller instances of the same problem until a base case is reached. Examples of recursive functions and data types are provided, such as factorial functions and linked lists. The order of execution for recursive functions is described, with code placed before the recursive call executing once per recursion level.
This presentation describes an approach to analysis of common recursion patterns in purely functional code. The goal is to recognize common patterns and provide the user with hints how to improve them using standard FP techniques (map, fold etc.). This presentation was given as part of the Advanced Functional Programming Seminar at Utrecht University.
The document discusses recursion, including:
1) Recursion is a programming technique where a method calls itself to solve a problem. It involves a base case and recursive calls.
2) Examples of recursive definitions and programs are given, such as computing factorials and sums recursively.
3) Recursion can be direct, where a method calls itself, or indirect through multiple method calls eventually leading back to the original method.
The document discusses recursion, which is when a function calls itself. It provides examples of recursive algorithms, types of recursion including single, multiple, direct, indirect, anonymous, and structural recursion. It also discusses recursive functions like the factorial function, tail recursion, and mutually recursive functions. Additionally, it covers recursive data types, the order of execution in recursive functions, and how recursion uses the call stack in memory.
The document discusses the differences between recursion and iteration. Recursion involves a method calling itself, with each call reducing the problem size until a base case is reached. Iteration uses loops to repeat a process. Examples include calculating factorials, Fibonacci numbers, and binary search recursively and iteratively. Some problems like directory traversal are simpler using recursion due to its divide-and-conquer approach, while iterations may be preferred for easier explanation or to avoid stack overflows with deep recursions.
This document provides an overview of recursion, including recursive algorithms, functions, data types, and order of execution. Recursion is defined as a function calling itself, either directly or indirectly. Recursive algorithms break down a problem into smaller instances of the same problem until a base case is reached. Examples of recursive functions and data types are provided, such as factorial functions and linked lists. The order of execution for recursive functions is described, with code placed before the recursive call executing once per recursion level.
This presentation describes an approach to analysis of common recursion patterns in purely functional code. The goal is to recognize common patterns and provide the user with hints how to improve them using standard FP techniques (map, fold etc.). This presentation was given as part of the Advanced Functional Programming Seminar at Utrecht University.
The document discusses recursion, including:
1) Recursion is a programming technique where a method calls itself to solve a problem. It involves a base case and recursive calls.
2) Examples of recursive definitions and programs are given, such as computing factorials and sums recursively.
3) Recursion can be direct, where a method calls itself, or indirect through multiple method calls eventually leading back to the original method.
The document discusses recursion, which is when a function calls itself. It provides examples of recursive algorithms, types of recursion including single, multiple, direct, indirect, anonymous, and structural recursion. It also discusses recursive functions like the factorial function, tail recursion, and mutually recursive functions. Additionally, it covers recursive data types, the order of execution in recursive functions, and how recursion uses the call stack in memory.
The document discusses the differences between recursion and iteration. Recursion involves a method calling itself, with each call reducing the problem size until a base case is reached. Iteration uses loops to repeat a process. Examples include calculating factorials, Fibonacci numbers, and binary search recursively and iteratively. Some problems like directory traversal are simpler using recursion due to its divide-and-conquer approach, while iterations may be preferred for easier explanation or to avoid stack overflows with deep recursions.
Iteration uses loops to repeatedly execute a block of code a specified number of times or until a condition is met. Recursion provides an alternative approach by having a function call itself repeatedly until a base case is reached. Both approaches can solve problems but recursion uses more memory than iteration due to its use of function calls and stack. Whether to use iteration or recursion depends on the specific problem and considerations of processing speed and memory usage.
The document discusses recursion, including:
1) Recursion involves breaking a problem down into smaller subproblems until a base case is reached, then building up the solution to the overall problem from the solutions to the subproblems.
2) A recursive function is one that calls itself, with each call typically moving closer to a base case where the problem can be solved without recursion.
3) Recursion can be linear, involving one recursive call, or binary, involving two recursive calls to solve similar subproblems.
This document discusses algorithm analysis tools. It explains that algorithm analysis is used to determine which of several algorithms to solve a problem is most efficient. Theoretical analysis counts primitive operations to approximate runtime as a function of input size. Common complexity classes like constant, linear, quadratic, and exponential time are defined based on how quickly runtime grows with size. Big-O notation represents the asymptotic upper bound of a function's growth rate to classify algorithms.
This document discusses analyzing the time efficiency of recursive algorithms. It provides a general 5-step plan: 1) choose a parameter for input size, 2) identify the basic operation, 3) check if operation count varies, 4) set up a recurrence relation, 5) solve the relation to determine growth order. It then gives two examples - computing factorial recursively and solving the Tower of Hanoi puzzle recursively - to demonstrate applying the plan. The document also briefly discusses algorithm visualization using static or dynamic images to convey information about an algorithm's operations and performance.
This document provides an overview of sorting algorithms including bubble sort, insertion sort, shellsort, and others. It discusses why sorting is important, provides pseudocode for common sorting algorithms, and gives examples of how each algorithm works on sample data. The runtime of sorting algorithms like insertion sort and shellsort are analyzed, with insertion sort having quadratic runtime in the worst case and shellsort having unknown but likely better than quadratic runtime.
This document discusses queues and their applications. It begins by defining queues as first-in, first-out (FIFO) data structures and describes how they can be implemented using arrays or linked lists. Common applications of queues include I/O buffering, job scheduling, and round-robin threading. The document also covers double-ended queues (deques), priority queues, and their implementations and applications.
1. The document discusses recursion and provides examples of recursively defined problems and their solutions, including factorials, powers, greatest common divisor, Fibonacci series, Towers of Hanoi, and counting cells in a blob.
2. Recursion involves breaking a problem into smaller subproblems of the same type until reaching a base case that can be solved directly. The solutions to subproblems are then combined to solve the original problem.
3. Examples of recursively defined problems include mathematical formulas, searching and sorting algorithms like binary search, and problems like Towers of Hanoi that are naturally broken down recursively.
This document discusses algorithm analysis and complexity. It introduces algorithm analysis as a way to predict and compare algorithm performance. Different algorithms for computing factorials and finding the maximum subsequence sum are presented, along with their time complexities. The importance of efficient algorithms for problems involving large datasets is discussed.
Heapsort is a sorting algorithm with average-case time complexity of O(n log n). It works by first constructing a max-heap from the input array and then extracting elements from the heap and placing them in sorted order in the array. The key procedures are MAX-HEAPIFY, which maintains the heap property by floating elements down the heap, and BUILD-MAX-HEAP, which constructs a max-heap from an unsorted array in linear time O(n) by calling MAX-HEAPIFY on successive subtrees.
This document provides an introduction and overview of algorithms and analysis of algorithms. It begins with definitions of algorithms and what constitutes an algorithm. It then discusses analyzing algorithms to determine their efficiency, including analyzing worst-case, average-case, and best-case scenarios. It introduces asymptotic notation used to describe algorithm running times at different scales, including big-O notation. It also discusses computational models and analyzing algorithms in terms of counting the basic operations they perform. The document is presented as a lecture on algorithms with definitions, examples, and outlines of key topics to cover.
The document discusses various topics related to analyzing algorithms, including:
i. Analysis of running time and using recurrence equations to predict how long recursive algorithms take on different input sizes.
ii. Iteration, induction, and recursion as fundamental concepts in data structures and algorithms. Recursive programs can sometimes be simpler than iterative programs.
iii. Proving properties of programs formally or informally, such as proving statements are true for each iteration of a loop or recursive call of a function. This is often done using induction.
The document discusses recursion through examples such as calculating factorials and multiplying numbers recursively. It defines recursion as solving a problem by solving smaller instances of the same problem. Recursive algorithms break down a problem into smaller sub-problems until reaching a base case, and use the solutions to the sub-problems to solve the original problem.
This document discusses algorithms and data structures. It covers the requirements of algorithms including inputs, outputs, unambiguousness, generality, correctness, being finite, and efficiency. It also discusses representing algorithms through pseudo-code and flowcharts. Some key aspects of pseudo-code are described such as its purpose to define algorithms simply and its use of limited vocabulary and English-like notation. Common control structures like sequence, selection, and repetition are explained. Examples of bubble sort, selection sort, and insertion sort algorithms are also mentioned.
The document discusses order statistics and medians. It defines order statistics as the ith smallest element in a data set and notes that the median is the middle element when the data set size is odd or the average of the two middle elements when the size is even. It then describes algorithms for finding the minimum, maximum, second smallest element, and any order statistic in expected linear time using a randomized selection algorithm. Finally, it provides an overview of generic programming in C++ using templates for functions and classes.
Introduction to datastructure and algorithmPratik Mota
Introduction to data structure and algorithm
-Basics of Data Structure and Algorithm
-Practical Examples of where Data Structure Algorithms is used
-Asymptotic Notations [ O(n), o(n), θ(n), Ω(n), ω(n) ]
-Calculation of Time and Space Complexity
-GNU gprof basic
1. Data structures organize data in memory for efficient access and processing. They represent relationships between data values through placement and linking of the values.
2. Algorithms are finite sets of instructions that take inputs, produce outputs, and terminate after a finite number of unambiguous steps. Common data structures and algorithms are analyzed based on their time and space complexity.
3. Data structures can be linear, with sequential elements, or non-linear, with branching elements. Abstract data types define operations on values independently of implementation through inheritance and polymorphism.
The document introduces stacks and recursion. It defines stacks as LIFO data structures and discusses how they are used in programming, particularly in method calls and the JVM. Recursion is introduced as a problem-solving technique that breaks problems down into smaller instances of the same problem. It notes recursion uses stacks to store information during method calls and provides examples of recursive algorithms and problems.
An algorithm is a finite set of instructions to accomplish a predefined task. Performance of an algorithm is measured by its time and space complexity, with common metrics being big O, big Omega, and big Theta notation. Common data structures include arrays, linked lists, stacks, queues, trees and graphs. Key concepts are asymptotic analysis of algorithms, recursion, and analyzing complexity classes like constant, linear, quadratic and logarithmic time.
This document discusses recursion, complexity analysis, and sorting algorithms. It defines recursion and how recursive methods use stacks and activation records. Complexity analysis using Big-O notation is introduced, with examples showing orders of growth such as constant, linear, quadratic, and exponential time. Sorting algorithms like quicksort and merge sort are described, with quicksort having a more efficient O(n log n) complexity compared to O(n2) for some other sorts.
Recursion vs. Iteration: Code Efficiency & Structurecogaxor346
This presentation delves into the Recursion vs. Iteration debate, highlighting their operational disparities. Recursion, though elegant, demands caution to prevent system crashes due to infinite loops. While it offers compact code, it suffers from slower execution and increased memory usage. Conversely, iteration, driven by repetition structures, boasts faster performance and lesser memory consumption. Examples illustrate the essence and application of recursion, emphasizing the need for terminating conditions. Additionally, the presentation touches upon modular programming principles and introduces lambda functions and the pass statement, elucidating their roles in code structure and execution control.
Iteration uses loops to repeatedly execute a block of code a specified number of times or until a condition is met. Recursion provides an alternative approach by having a function call itself repeatedly until a base case is reached. Both approaches can solve problems but recursion uses more memory than iteration due to its use of function calls and stack. Whether to use iteration or recursion depends on the specific problem and considerations of processing speed and memory usage.
The document discusses recursion, including:
1) Recursion involves breaking a problem down into smaller subproblems until a base case is reached, then building up the solution to the overall problem from the solutions to the subproblems.
2) A recursive function is one that calls itself, with each call typically moving closer to a base case where the problem can be solved without recursion.
3) Recursion can be linear, involving one recursive call, or binary, involving two recursive calls to solve similar subproblems.
This document discusses algorithm analysis tools. It explains that algorithm analysis is used to determine which of several algorithms to solve a problem is most efficient. Theoretical analysis counts primitive operations to approximate runtime as a function of input size. Common complexity classes like constant, linear, quadratic, and exponential time are defined based on how quickly runtime grows with size. Big-O notation represents the asymptotic upper bound of a function's growth rate to classify algorithms.
This document discusses analyzing the time efficiency of recursive algorithms. It provides a general 5-step plan: 1) choose a parameter for input size, 2) identify the basic operation, 3) check if operation count varies, 4) set up a recurrence relation, 5) solve the relation to determine growth order. It then gives two examples - computing factorial recursively and solving the Tower of Hanoi puzzle recursively - to demonstrate applying the plan. The document also briefly discusses algorithm visualization using static or dynamic images to convey information about an algorithm's operations and performance.
This document provides an overview of sorting algorithms including bubble sort, insertion sort, shellsort, and others. It discusses why sorting is important, provides pseudocode for common sorting algorithms, and gives examples of how each algorithm works on sample data. The runtime of sorting algorithms like insertion sort and shellsort are analyzed, with insertion sort having quadratic runtime in the worst case and shellsort having unknown but likely better than quadratic runtime.
This document discusses queues and their applications. It begins by defining queues as first-in, first-out (FIFO) data structures and describes how they can be implemented using arrays or linked lists. Common applications of queues include I/O buffering, job scheduling, and round-robin threading. The document also covers double-ended queues (deques), priority queues, and their implementations and applications.
1. The document discusses recursion and provides examples of recursively defined problems and their solutions, including factorials, powers, greatest common divisor, Fibonacci series, Towers of Hanoi, and counting cells in a blob.
2. Recursion involves breaking a problem into smaller subproblems of the same type until reaching a base case that can be solved directly. The solutions to subproblems are then combined to solve the original problem.
3. Examples of recursively defined problems include mathematical formulas, searching and sorting algorithms like binary search, and problems like Towers of Hanoi that are naturally broken down recursively.
This document discusses algorithm analysis and complexity. It introduces algorithm analysis as a way to predict and compare algorithm performance. Different algorithms for computing factorials and finding the maximum subsequence sum are presented, along with their time complexities. The importance of efficient algorithms for problems involving large datasets is discussed.
Heapsort is a sorting algorithm with average-case time complexity of O(n log n). It works by first constructing a max-heap from the input array and then extracting elements from the heap and placing them in sorted order in the array. The key procedures are MAX-HEAPIFY, which maintains the heap property by floating elements down the heap, and BUILD-MAX-HEAP, which constructs a max-heap from an unsorted array in linear time O(n) by calling MAX-HEAPIFY on successive subtrees.
This document provides an introduction and overview of algorithms and analysis of algorithms. It begins with definitions of algorithms and what constitutes an algorithm. It then discusses analyzing algorithms to determine their efficiency, including analyzing worst-case, average-case, and best-case scenarios. It introduces asymptotic notation used to describe algorithm running times at different scales, including big-O notation. It also discusses computational models and analyzing algorithms in terms of counting the basic operations they perform. The document is presented as a lecture on algorithms with definitions, examples, and outlines of key topics to cover.
The document discusses various topics related to analyzing algorithms, including:
i. Analysis of running time and using recurrence equations to predict how long recursive algorithms take on different input sizes.
ii. Iteration, induction, and recursion as fundamental concepts in data structures and algorithms. Recursive programs can sometimes be simpler than iterative programs.
iii. Proving properties of programs formally or informally, such as proving statements are true for each iteration of a loop or recursive call of a function. This is often done using induction.
The document discusses recursion through examples such as calculating factorials and multiplying numbers recursively. It defines recursion as solving a problem by solving smaller instances of the same problem. Recursive algorithms break down a problem into smaller sub-problems until reaching a base case, and use the solutions to the sub-problems to solve the original problem.
This document discusses algorithms and data structures. It covers the requirements of algorithms including inputs, outputs, unambiguousness, generality, correctness, being finite, and efficiency. It also discusses representing algorithms through pseudo-code and flowcharts. Some key aspects of pseudo-code are described such as its purpose to define algorithms simply and its use of limited vocabulary and English-like notation. Common control structures like sequence, selection, and repetition are explained. Examples of bubble sort, selection sort, and insertion sort algorithms are also mentioned.
The document discusses order statistics and medians. It defines order statistics as the ith smallest element in a data set and notes that the median is the middle element when the data set size is odd or the average of the two middle elements when the size is even. It then describes algorithms for finding the minimum, maximum, second smallest element, and any order statistic in expected linear time using a randomized selection algorithm. Finally, it provides an overview of generic programming in C++ using templates for functions and classes.
Introduction to datastructure and algorithmPratik Mota
Introduction to data structure and algorithm
-Basics of Data Structure and Algorithm
-Practical Examples of where Data Structure Algorithms is used
-Asymptotic Notations [ O(n), o(n), θ(n), Ω(n), ω(n) ]
-Calculation of Time and Space Complexity
-GNU gprof basic
1. Data structures organize data in memory for efficient access and processing. They represent relationships between data values through placement and linking of the values.
2. Algorithms are finite sets of instructions that take inputs, produce outputs, and terminate after a finite number of unambiguous steps. Common data structures and algorithms are analyzed based on their time and space complexity.
3. Data structures can be linear, with sequential elements, or non-linear, with branching elements. Abstract data types define operations on values independently of implementation through inheritance and polymorphism.
The document introduces stacks and recursion. It defines stacks as LIFO data structures and discusses how they are used in programming, particularly in method calls and the JVM. Recursion is introduced as a problem-solving technique that breaks problems down into smaller instances of the same problem. It notes recursion uses stacks to store information during method calls and provides examples of recursive algorithms and problems.
An algorithm is a finite set of instructions to accomplish a predefined task. Performance of an algorithm is measured by its time and space complexity, with common metrics being big O, big Omega, and big Theta notation. Common data structures include arrays, linked lists, stacks, queues, trees and graphs. Key concepts are asymptotic analysis of algorithms, recursion, and analyzing complexity classes like constant, linear, quadratic and logarithmic time.
This document discusses recursion, complexity analysis, and sorting algorithms. It defines recursion and how recursive methods use stacks and activation records. Complexity analysis using Big-O notation is introduced, with examples showing orders of growth such as constant, linear, quadratic, and exponential time. Sorting algorithms like quicksort and merge sort are described, with quicksort having a more efficient O(n log n) complexity compared to O(n2) for some other sorts.
Recursion vs. Iteration: Code Efficiency & Structurecogaxor346
This presentation delves into the Recursion vs. Iteration debate, highlighting their operational disparities. Recursion, though elegant, demands caution to prevent system crashes due to infinite loops. While it offers compact code, it suffers from slower execution and increased memory usage. Conversely, iteration, driven by repetition structures, boasts faster performance and lesser memory consumption. Examples illustrate the essence and application of recursion, emphasizing the need for terminating conditions. Additionally, the presentation touches upon modular programming principles and introduces lambda functions and the pass statement, elucidating their roles in code structure and execution control.
Iterators in Python allow iteration over iterable objects like lists, tuples, dicts and sets. An iterator is initialized using the iter() method and iterates using the next() method. For loops internally use iter() and next() to iterate. Custom iterators can also be defined by implementing the __iter__() and __next__() methods. Some key differences between recursion and iteration are that recursion involves function calls while iteration uses loops, and recursion has greater overhead. Common algorithms like factorial, Fibonacci sequence and Tower of Hanoi can be implemented using both recursion and iteration.
The document discusses data types, data structures, algorithms, recursion, and asymptotic analysis. It provides definitions and examples of key concepts like abstract data types, data abstraction, algorithms, iterative vs recursive algorithms, complexity analysis using Big-O, Big-Omega and Big-Theta notations. Examples of recursively implementing algorithms to find sum of natural numbers, factorial, GCD, Fibonacci series are presented.
Recursion is a technique in programming where a function calls itself to solve smaller instances of the same problem. It involves two main components: the base case and the recursive case. The base case is the condition that stops the recursive calls, preventing infinite loops. The recursive case is where the function continues to call itself, breaking the problem down into smaller parts. For example, calculating the factorial of a number can be done using recursion, where `n! = n * (n-1)!` with `0! = 1` as the base case.
In data structures, recursion is commonly used for tasks like traversing or searching through trees and linked lists. It simplifies complex problems by dividing them into more manageable subproblems. For instance, in binary trees, recursive methods are used for traversals such as Preorder, Inorder, and Postorder, which visit nodes in a specific order using recursive function calls.
Recursion is the process of defining a problem (or the solution to a problem) in terms of (a simpler version of) itself. For example, we can define the operation "find your way home" as: If you are at home, stop moving. Take one step toward home.
Deep learning (also known as deep structured learning or hierarchical learning) is the application of artificial neural networks (ANNs) to learning tasks that contain more than one hidden layer. Deep learning is part of a broader family of machine learning methods based on learning data representations, as opposed to task-specific algorithms. Learning can be supervised, partially supervised or unsupervised.
This document provides an introduction and overview of the R programming language and statistical software environment. It discusses R's open source development, interoperability with other languages, variety of statistical and numerical methods, and high quality visualization tools. It also introduces key R concepts like vectors, matrices, lists, data frames, functions, operators, subsetting, importing/exporting data, and gives examples of frequently used statistical and graphical functions. Finally, it provides an overview of the Bioconductor project for biological data analysis using R.
Data structure and algorithm using javaNarayan Sau
This presentation created for people who like to go back to basics of data structure and its implementation. This presentation mostly helps B.Tech , Bsc Computer science students as well as all programmer who wants to develop software in core areas.
This document summarizes a massive open online course on Udemy about fundamental data structures and algorithms using the C language. The 15-hour course covers key topics like stacks, queues, linked lists, trees, recursion, and analyzing algorithm efficiency. It aims to help students strengthen programming skills and prepare for technical interviews at top companies. The course consists of 14 sections and includes weekly quizzes on the Udemy platform.
This document provides an overview of core Java concepts including comments, classes, objects, scoping, static methods and fields, arrays, and constructors. It discusses how comments in Java are similar to C++, the fundamentals of classes and objects, variable scoping determined by curly braces, how static methods and fields belong to the class rather than instances, and how arrays are objects allocated dynamically. It also covers how constructors are used to create class instances and automatically invoke superclass constructors.
This document outlines the syllabus for a course on data structures and algorithms using Java. It covers topics such as the role of algorithms and data structures, algorithm design techniques, types of data structures including primitive types, arrays, stacks, queues, linked lists, trees, graphs, and algorithm analysis. Specific algorithms and data structures discussed include sorting, searching, priority queues, stacks, queues, linked lists, trees, graphs, hashing, and complexity theory.
This document discusses deques, priority queues, depth-first search (DFS), and breadth-first search (BFS). It provides code examples for deque and priority queue interfaces. It explains that a deque is a double-ended queue that allows adding/removing from both ends, while a priority queue organizes objects by priority rather than order of arrival. DFS searches deeper first until reaching a goal or dead end, while BFS searches all neighboring nodes first before moving to the next level. The document also discusses converting a general tree to a binary tree for efficient processing and traversal.
This document provides an introduction to data structures and algorithms. It discusses arrays, stacks, queues and their applications. It also covers time and space complexity analysis of algorithms. Arrays are introduced as a linear data structure for storing similar data elements. Array elements can be accessed using an index or subscript. Multi-dimensional arrays can also be implemented for storing data in rows and columns.
The document discusses Java 8 Streams, which provide a way to process data in a functional style. Streams allow operations like filter, map, and reduce to be performed lazily on collections, arrays, or I/O sources. The key aspects of streams are that they are lazy, support both sequential and parallel processing, and represent a sequence of values rather than storing them. The document provides examples of using intermediate operations like filter and map and terminal operations like forEach and collect. It also discusses spliterators, which drive streams and allow parallelization, and functional interfaces which are used with lambda expressions in streams.
- Linear algebra is important for image recognition and other fields like physics, economics, and politics. It allows analyzing relationships between multiple variables without calculus.
- Python is a good platform for linear algebra due to libraries like NumPy that allow fast processing of multi-dimensional data like matrices. It also has simple syntax without semicolons.
- Key concepts discussed include vectors, matrices, linear transformations, abstraction, and how linear algebra solves problems in fields like quantum mechanics. Comprehensions provide a concise way to generate sets, lists, and arrays in Python.
This document provides an overview of various machine learning algorithms and concepts, including supervised learning techniques like linear regression, logistic regression, decision trees, random forests, and support vector machines. It also discusses unsupervised learning methods like principal component analysis and kernel-based PCA. Key aspects of linear regression, logistic regression, and random forests are summarized, such as cost functions, gradient descent, sigmoid functions, and bagging. Kernel methods are also introduced, explaining how the kernel trick can allow solving non-linear problems by mapping data to a higher-dimensional feature space.
This document discusses recursion versus iteration in Lisp and Common Lisp. It notes that the original Lisp language was purely functional and used recursion to solve problems since it lacked local variables and iteration. While recursion is more elegant and leads to less code in some cases, it is also harder to understand, debug, and less efficient due to function call overhead. However, recursion is necessary for some problems like tree and graph traversals. Common Lisp makes recursion easier through its runtime stack and debugger. Several examples of recursive functions are provided, including ones for lists, trees, searching, and factorials. Tail recursion is discussed as a way to make recursion more efficient.
This document provides an overview of data structures and their implementation in C++. It discusses how data structures are used to organize data efficiently to allow for faster programs. Specific data structures covered include arrays, linked lists, stacks, queues, trees and graphs. The document also explains how to select the appropriate data structure based on the operations needed and resource constraints. It emphasizes that each data structure has costs and benefits and no single structure is best for all situations.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
2. Data Structures and Algorithms in Java
Objectives
• Recursive definition
• Recursive program/algorithm
• Recursion application
• Method calls and recursion implementation
• Anatomy of a recursive call
• Classify recursions by number of recursive calls
• Tail recursion
• Non-tail recursion
• Indirect recursion
• Nested recursion
• Excessive recursion
2/25
3. Data Structures and Algorithms in Java
Recursive definition
• Wikipedia: “A recursive definition or inductive definition is one that
defines something in terms of itself (that is, recursively). For it to
work, the definition in any given case must be well-founded,
avoiding an infinite regress”.
• A recursive definition consists of at least 2 parts:
1) A base case (anchor or the ground case) that does not contain a
reference to its own type.
2) An inductive case that does contain a reference to its own type.
For example: Define the set of natural numbers
N
set
the
in
objects
other
no
are
there
N
n
then
N
n
if
N
.
3
)
1
(
,
.
2
0
.
1
3/25
4. Data Structures and Algorithms in Java
Some other examples for recursive definition
1 if n = 0 (anchor)
n*(n-1)! if n>0 (inductive step)
n! =
• The Fibonacci sequence of natural numbers:
n if n<2
fibo(n-1) +fibo(n-2) otherwise
fibo(n) =
• The Factorial of a natural number:
4/25
5. Data Structures and Algorithms in Java 5/27
Recursive program/algorithm - 1
One way to describe repetition within a computer program is the
use of loops, such as Java’s while-loop and for-loop constructs. An
entirely different way to achieve repetition is through a process
known as recursion. Recursion is a technique by which a method
makes one or more calls to itself during execution, or by which a
data structure relies upon smaller instances of the very same type
of structure in its representation. In computing, recursion provides
an elegant and powerful alternative for performing repetitive tasks.
Most modern programming languages support functional recursion
using the identical mechanism that is used to support traditional
forms of method calls. When one invocation of the method makes a
recursive call, that invocation is suspended until the recursive call
completes. Recursion is an important technique in the study of data
structures and algorithms.
6. Data Structures and Algorithms in Java 6/27
Recursive program/algorithm - 2
A recursive program/algorithm is one that calls itself again.
There are three basic rules for developing recursive algorithms.
• Know how to take one step.
• Break each problem down into one step plus a smaller
problem.
• Know how and when to stop.
Example for recursive program:
public static void DecToBin(int n)
{ int q = n/2; // One step
int r = n%2; // One step
if (q > 0)
{DecToBin(q); // smaller problem
}
System.out.print(r); // after all recursive calls have been
// made last remainder printed first
}
7. Data Structures and Algorithms in Java 7/27
Recursion application
• Recursive definitions are used in defining
functions or sequences of elements
• Purpose of recursive definition:
– Generating a new elements
– Testing whether an element belongs to a set (*)
• (*) The problem is solved by reducing it to a
simpler problem
8. Data Structures and Algorithms in Java 8/27
Method calls and recursion implementation - 1
Each time a method is called, an activation record (AR) is allocated for it.
This record usually contains the following information:
• Parameters and local variables used in the called method.
• A dynamic link, which is a pointer to the caller's activation record.
• Return address to resume control by the caller, the address of the
caller’s instruction immediately following the call.
• Return value for a method not declared as void. Because the size of the
activation record may vary from one call to another, the returned value is
placed right above the activation record of the caller.
Each new activation record is placed on the top of the run-time stack
When a method terminates, its activation record is removed from the top
of the run-time stack
Thus, the first AR placed onto the stack is the last one removed
9. Data Structures and Algorithms in Java 9/27
Creating an activation record whenever a method is
called allows the system to handle recursion
properly. Recursion is calling a method that happens
to have the same name as the caller. Therefore, a
recursive call is not literally a method calling it-self,
but rather an instantiation of a method calling
another instantiation of the same original. These
invocation are represented internally by different
activation records and are thus differentiated by the
system.
Method calls and recursion implementation - 2
10. Data Structures and Algorithms in Java 10/27
Anatomy of a recursive Call
N = 4
N <= 1? NO
return (4 * factorial (3)) =
N = 3
N <= 1? NO
return (3 * factorial (2)) =
N = 2
N <= 1? NO
return (2 * factorial (1)) =
N = 1
N <= 1? YES
Return (1)
factorial(4)
factorial(3)
factorial(2)
factorial(1)
2
6
24
1
1
2
6
24
First call
11. Data Structures and Algorithms in Java 11/27
Classification of recursive functions by number
of recursive calls
Considering the maximum number of recursive calls
within the body of a single activation, there are 3 types
of recursions.
• Linear recursion: Only 1 recursive call (to itself) in side
the recursive function (e.g. binary search).
• Binary recursion: There exactly 2 recursive calls (to
itself) in side the recursive function (e.g. Fibonacci
number) .
• Multiple recursion: There are 3 or more recursive calls
(to itself) in side the recursive function.
12. Data Structures and Algorithms in Java 12/27
Tail recursion
• There is only one recursive call at the very end
of a method implementation
Other classification of recursions
class Main
{static void tail(int n)
{if(n >0)
{ System.out.print(n + " ");
tail(n-1);
}
}
public static void main(String [] args)
{tail(10);
System.out.println();
}
}
void nonTail (int i)
{ if (i > 0)
{tail(i-1);
System.out.print (i + "");
tail(i-1);
}
}
13. Data Structures and Algorithms in Java 13/27
Non-tail recursion
• The recursive call is not at the very end of a method
implementation
public class Main
{public static void reverse() throws Exception
{char ch = (char) System.in.read();
if(ch != 'n')
{reverse();
System.out.print(ch);
}
}
public static void main(String [] args) throws Exception
{System.out.println("nEnter a string to be reversed:");
reverse();
System.out.println("n");
}
}
14. Data Structures and Algorithms in Java 14/27
Convert recursion implementation
to iterative implementation using stack
{public static void nonRecursiveReverse() throws Exception
{MyStack t = new MyStack();
char ch;
while(true)
{ch = (char) System.in.read();
if(ch == 'n') break;
t.push(ch);
}
while(!t.isEmpty())
System.out.print(t.pop());
}
15. Data Structures and Algorithms in Java 15/27
Indirect recursion
• If f() calls itself, it is direct recursive
• If f() calls g(), and g() calls f(). It is indirect recursion. The chain of
intermediate calls can be of an arbitrary length, as in:
f() -> f1() -> f2() -> ... -> fn() -> f()
• Example: sin(x) calculation
Recursive call tree for sin(x)
sin(x)
sin(x/3) tan(x/3) tan(x/3)
sin(x/3) cos(x/3) sin(x/3) cos(x)
sin(x/6) sin(x/6)
16. Data Structures and Algorithms in Java 16/27
Nested recursion
• A function is not only defined in terms of itself but also is
used as one of the parameters
• Consider the following examples:
0 if n=0
n if n>4
h(2 +h(2n)) if n<=4
Another example of nested recursive recursion is
Ackermann’s function:
A(0, y) = y + 1
A(x, 0) = A(x - 1, 1)
A(x, y) = A(x - 1, A(x, y - 1))
h(n) =
This function is interesting because of
its remarkably rapid growth.
A(3,1) = 24 - 3
A(4,1) = 265536 - 3
17. Data Structures and Algorithms in Java 17/27
Excessive recursion - 1
• Consider the Fibonacci sequence of numbers
n if n<2
fibo(n-1) +fibo(n-2) otherwise
static long fibo(long n)
{if (n<2)
return n;
else
return(fibo(n-1)+fibo(n-2));
}
This
implementation
looks very natural
but extremely
inefficient!
fibo(n) =
18. Data Structures and Algorithms in Java 18/27
The tree of calls for fibo(4)
18
Fib(4)
Fib(2) Fib(3)
Fib(0) Fib(1)
1
0
Fib(2)
Fib(0) Fib(1)
1
0
Fib(1)
1
Excessive recursion - 2
19. Data Structures and Algorithms in Java 19/27
More Examples – The Tower of Hanoi
19
• Rules:
Only one disk may be moved at a time.
Each move consists of taking the upper disk from one
of the rods and sliding it onto another rod, on top of
the other disks that may already be present on that
rod.
No disk may be placed on top of a smaller disk.
Source: http://en.wikipedia.org/wiki/Hanoi_tower
20. Data Structures and Algorithms in Java 20/27
More Examples – The Tower of Hanoi
Algorithm
20
void moveDisks(int n, char fromTower, char toTower, char auxTower)
{if (n == 1) // Stopping condition
Move disk 1 from the fromTower to the toTower;
else
{moveDisks(n - 1, fromTower, auxTower, toTower);
move disk n from the fromTower to the toTower;
moveDisks(n - 1, auxTower, toTower, fromTower);
}
}
21. Data Structures and Algorithms in Java 21/27
More Examples – Drawing fractals
21
22. Data Structures and Algorithms in Java 22/27
1. Divide an interval side into three even parts
2. Move one-third of side in the direction
specified by angle
Examples of von Koch snowflakes
More Examples – Von Knoch snowflakes
23. Data Structures and Algorithms in Java 23/27
Recursion vs. Iteration
23
• Some recursive algorithms can also be easily
implemented with loops
– When possible, it is usually better to use iteration,
since we don’t have the overhead of the run-time stack
(that we just saw on the previous slide)
• Other recursive algorithms are very difficult to
do any other way
24. Data Structures and Algorithms in Java 24/27
Summary
• Recursive definitions are programming concepts that
define themselves
• Recursive definitions serve two purposes:
– Generating new elements
– Testing whether an element belongs to a set
• Recursive definitions are frequently used to define
functions and sequences of numbers
• Tail recursion is characterized by the use of only one
recursive call at the very end of a method
implementation.
25. Data Structures and Algorithms in Java 25/25
Reading at home
Text book: Data Structures and Algorithms in Java
5 Recursion 189
5.1 Illustrative Examples - 191
5.1.1 The Factorial Function -191
5.1.3 Binary Search - 196
5.1.4 File Systems - 198
5.2 Analyzing Recursive Algorithms - 202
5.3 Further Examples of Recursion - 206
5.3.1 Linear Recursion - 206
5.3.2 Binary Recursion - . 211
5.3.3 Multiple Recursion - 212
5.4 Designing Recursive Algorithms - 214
5.6 Eliminating Tail Recursion -. 219