The document discusses three sorting algorithms: bubble sort, selection sort, and insertion sort. Bubble sort works by repeatedly swapping adjacent elements that are in the wrong order. Selection sort finds the minimum element and swaps it into the sorted portion of the array. Insertion sort inserts elements into the sorted portion of the array, swapping as needed to put the element in the correct position. Both selection sort and insertion sort have a time complexity of O(n^2) in the worst case.
The document discusses asymptotic notations that are used to describe the time complexity of algorithms. It introduces big O notation, which describes asymptotic upper bounds, big Omega notation for lower bounds, and big Theta notation for tight bounds. Common time complexities are described such as O(1) for constant time, O(log N) for logarithmic time, and O(N^2) for quadratic time. The notations allow analyzing how efficiently algorithms use resources like time and space as the input size increases.
This document discusses time and space complexity analysis of algorithms. It analyzes the time complexity of bubble sort, which is O(n^2) as each pass through the array requires n-1 comparisons and there are n passes needed. Space complexity is typically a secondary concern to time complexity. Time complexity analysis allows comparison of algorithms to determine efficiency and whether an algorithm will complete in a reasonable time for a given input size. NP-complete problems cannot be solved in polynomial time but can be verified in polynomial time.
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
- Quicksort is a simple and fast sorting algorithm that can sort arrays "in place" without using extra space.
- It works by recursively partitioning the array around a pivot value, sorting the left and right subarrays, and then combining them.
- On average, it has a runtime of O(n log n) but in the worst case of an already sorted array it can have a quadratic runtime of O(n^2) like bubble sort. However, its randomized choice of pivots means it rarely encounters worst-case inputs in practice.
This document discusses top-down parsing and different types of top-down parsers, including recursive descent parsers, predictive parsers, and LL(1) grammars. It explains how to build predictive parsers without recursion by using a parsing table constructed from the FIRST and FOLLOW sets of grammar symbols. The key steps are: 1) computing FIRST and FOLLOW, 2) filling the predictive parsing table based on FIRST/FOLLOW, 3) using the table to parse inputs in a non-recursive manner by maintaining the parser's own stack. An example is provided to illustrate constructing the FIRST/FOLLOW sets and parsing table for a sample grammar.
Worst-case analysis is sometimes overly pessimistic.
Amortized analysis of an algorithm involves computing the maximum total number of all operations on the various data structures.
Amortized cost applies to each operation, even when there are several types of operations in the sequence.
In amortized analysis, time required to perform a sequence of data structure operations is averaged over all the successive operations performed. That is, a large cost of one operation is spread out over many operations (amortized), where the others are less expensive.
Therefore, amortized anaysis can be used to show that the average cost of an operation is small, if one averages over a sequence of operations, even though one of the single operations might be very expensive.
The document discusses three sorting algorithms: bubble sort, selection sort, and insertion sort. Bubble sort works by repeatedly swapping adjacent elements that are in the wrong order. Selection sort finds the minimum element and swaps it into the sorted portion of the array. Insertion sort inserts elements into the sorted portion of the array, swapping as needed to put the element in the correct position. Both selection sort and insertion sort have a time complexity of O(n^2) in the worst case.
The document discusses asymptotic notations that are used to describe the time complexity of algorithms. It introduces big O notation, which describes asymptotic upper bounds, big Omega notation for lower bounds, and big Theta notation for tight bounds. Common time complexities are described such as O(1) for constant time, O(log N) for logarithmic time, and O(N^2) for quadratic time. The notations allow analyzing how efficiently algorithms use resources like time and space as the input size increases.
This document discusses time and space complexity analysis of algorithms. It analyzes the time complexity of bubble sort, which is O(n^2) as each pass through the array requires n-1 comparisons and there are n passes needed. Space complexity is typically a secondary concern to time complexity. Time complexity analysis allows comparison of algorithms to determine efficiency and whether an algorithm will complete in a reasonable time for a given input size. NP-complete problems cannot be solved in polynomial time but can be verified in polynomial time.
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
- Quicksort is a simple and fast sorting algorithm that can sort arrays "in place" without using extra space.
- It works by recursively partitioning the array around a pivot value, sorting the left and right subarrays, and then combining them.
- On average, it has a runtime of O(n log n) but in the worst case of an already sorted array it can have a quadratic runtime of O(n^2) like bubble sort. However, its randomized choice of pivots means it rarely encounters worst-case inputs in practice.
This document discusses top-down parsing and different types of top-down parsers, including recursive descent parsers, predictive parsers, and LL(1) grammars. It explains how to build predictive parsers without recursion by using a parsing table constructed from the FIRST and FOLLOW sets of grammar symbols. The key steps are: 1) computing FIRST and FOLLOW, 2) filling the predictive parsing table based on FIRST/FOLLOW, 3) using the table to parse inputs in a non-recursive manner by maintaining the parser's own stack. An example is provided to illustrate constructing the FIRST/FOLLOW sets and parsing table for a sample grammar.
Worst-case analysis is sometimes overly pessimistic.
Amortized analysis of an algorithm involves computing the maximum total number of all operations on the various data structures.
Amortized cost applies to each operation, even when there are several types of operations in the sequence.
In amortized analysis, time required to perform a sequence of data structure operations is averaged over all the successive operations performed. That is, a large cost of one operation is spread out over many operations (amortized), where the others are less expensive.
Therefore, amortized anaysis can be used to show that the average cost of an operation is small, if one averages over a sequence of operations, even though one of the single operations might be very expensive.
Greedy algorithms work by making locally optimal choices at each step to arrive at a global optimal solution. They require that the problem exhibits the greedy choice property and optimal substructure. Examples that can be solved with greedy algorithms include fractional knapsack problem, minimum spanning tree, and activity selection. The fractional knapsack problem is solved greedily by sorting items by value/weight ratio and filling the knapsack completely. The 0/1 knapsack problem differs in that items are indivisible.
1) Stacks are linear data structures that follow the LIFO (last-in, first-out) principle. Elements can only be inserted or removed from one end called the top of the stack.
2) The basic stack operations are push, which adds an element to the top of the stack, and pop, which removes an element from the top.
3) Stacks have many applications including evaluating arithmetic expressions by converting them to postfix notation and implementing the backtracking technique in recursive backtracking problems like tower of Hanoi.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
The document discusses the branch and bound algorithm for solving the 15-puzzle problem. It describes the key components of branch and bound including live nodes, e-nodes, and dead nodes. It also defines the cost function used to evaluate nodes as the sum of the path length and number of misplaced tiles. The algorithm generates all possible child nodes from the current node and prunes the search tree by comparing node costs to avoid exploring subtrees without solutions.
This document provides an overview of algorithms and algorithm analysis. It discusses key concepts like what an algorithm is, different types of algorithms, and the algorithm design and analysis process. Some important problem types covered include sorting, searching, string processing, graph problems, combinatorial problems, geometric problems, and numerical problems. Examples of specific algorithms are given for some of these problem types, like various sorting algorithms, search algorithms, graph traversal algorithms, and algorithms for solving the closest pair and convex hull problems.
The document discusses algorithm analysis and asymptotic notation. It defines algorithm analysis as comparing algorithms based on running time and other factors as problem size increases. Asymptotic notation such as Big-O, Big-Omega, and Big-Theta are introduced to classify algorithms based on how their running times grow relative to input size. Common time complexities like constant, logarithmic, linear, quadratic, and exponential are also covered. The properties and uses of asymptotic notation for equations and inequalities are explained.
The document discusses divide and conquer algorithms. It describes divide and conquer as a design strategy that involves dividing a problem into smaller subproblems, solving the subproblems recursively, and combining the solutions. It provides examples of divide and conquer algorithms like merge sort, quicksort, and binary search. Merge sort works by recursively sorting halves of an array until it is fully sorted. Quicksort selects a pivot element and partitions the array into subarrays of smaller and larger elements, recursively sorting the subarrays. Binary search recursively searches half-intervals of a sorted array to find a target value.
This document summarizes graph coloring using backtracking. It defines graph coloring as minimizing the number of colors used to color a graph. The chromatic number is the fewest colors needed. Graph coloring is NP-complete. The document outlines a backtracking algorithm that tries assigning colors to vertices, checks if the assignment is valid (no adjacent vertices have the same color), and backtracks if not. It provides pseudocode for the algorithm and lists applications like scheduling, Sudoku, and map coloring.
The document discusses Strassen's algorithm for matrix multiplication. It begins by explaining traditional matrix multiplication that has a time complexity of O(n3). It then explains how the divide and conquer strategy can be applied by dividing the matrices into smaller square sub-matrices. Strassen improved upon this by reducing the number of multiplications from 8 to 7 terms, obtaining a time complexity of O(n2.81). His key insight was applying different equations on the sub-matrix multiplication formulas to minimize operations.
Abstract data types (adt) intro to data structure part 2Self-Employed
Abstract Data type (ADT), Related to DATA STRUCTURE and ALGORITHMS STACK QUEUE ARRAY LINKED LIST ALGORITHMS AND INSERTION DELETION MERGE TRAVERSE MODIFY AND OTHER related operation in the algorithms of stack queue array and linked list as an ADT type
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
1) The document discusses complexity analysis of algorithms, which involves determining the time efficiency of algorithms by counting the number of basic operations performed based on input size.
2) It covers motivations for complexity analysis, machine independence, and analyzing best, average, and worst case complexities.
3) Simple rules are provided for determining the complexity of code structures like loops, nested loops, if/else statements, and switch cases based on the number of iterations and branching.
This document discusses asymptotic notations that are used to characterize an algorithm's efficiency. It introduces Big-Oh, Big-Omega, and Theta notations. Big-Oh gives an upper bound on running time. Big-Omega gives a lower bound. Theta gives both upper and lower bounds, representing an algorithm's exact running time. Examples are provided for each notation. Time complexity is also introduced, which describes how the time to solve a problem changes with problem size. Worst-case time analysis provides an upper bound on runtime.
1. An algorithm is a sequence of unambiguous instructions to solve a problem and obtain an output for any valid input in a finite amount of time. Pseudocode is used to describe algorithms using a natural language format.
2. Analyzing algorithm efficiency involves determining the theoretical and empirical time complexity by counting the number of basic operations performed relative to the input size. Common measures are best-case, worst-case, average-case, and amortized analysis.
3. Important problem types for algorithms include sorting, searching, string processing, graphs, combinatorics, geometry, and numerical problems. Fundamental algorithms are analyzed for correctness and time/space complexity.
Merge sort is a divide and conquer algorithm that divides an array into halves, recursively sorts the halves, and then merges the sorted halves back together. The key steps are:
1. Divide the array into equal halves until reaching base cases of arrays with one element.
2. Recursively sort the left and right halves by repeating the divide step.
3. Merge the sorted halves back into a single sorted array by comparing elements pairwise and copying the smaller element into the output array.
Merge sort has several advantages including running in O(n log n) time in all cases, accessing data sequentially with low random access needs, and being suitable for external sorting of large data sets that do not fit in memory
PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
The document provides an overview of recursive and iterative algorithms. It discusses key differences between recursive and iterative algorithms such as definition, application, termination, usage, code size, and time complexity. Examples of recursive algorithms like recursive sum, factorial, binary search, tower of Hanoi, and permutation generator are presented along with pseudocode. Analysis of recursive algorithms like recursive sum, factorial, binary search, Fibonacci number, and tower of Hanoi is demonstrated to determine their time complexities. The document also discusses iterative algorithms, proving an algorithm's correctness, the brute force approach, and store and reuse methods.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
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.
Lecture 5 6_7 - divide and conquer and method of solving recurrencesjayavignesh86
The document discusses divide and conquer algorithms and solving recurrences. It covers asymptotic notations, examples of divide and conquer including finding the largest number in a list, recurrence relations, and methods for solving recurrences including iteration, substitution, and recursion trees. The iteration method involves unfolding the recurrence into a summation. The recursion tree method visually depicts recursive calls in a tree to help solve the recurrence. Divide and conquer algorithms break problems into smaller subproblems, solve the subproblems recursively, and combine the solutions.
Greedy algorithms work by making locally optimal choices at each step to arrive at a global optimal solution. They require that the problem exhibits the greedy choice property and optimal substructure. Examples that can be solved with greedy algorithms include fractional knapsack problem, minimum spanning tree, and activity selection. The fractional knapsack problem is solved greedily by sorting items by value/weight ratio and filling the knapsack completely. The 0/1 knapsack problem differs in that items are indivisible.
1) Stacks are linear data structures that follow the LIFO (last-in, first-out) principle. Elements can only be inserted or removed from one end called the top of the stack.
2) The basic stack operations are push, which adds an element to the top of the stack, and pop, which removes an element from the top.
3) Stacks have many applications including evaluating arithmetic expressions by converting them to postfix notation and implementing the backtracking technique in recursive backtracking problems like tower of Hanoi.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
The document discusses the branch and bound algorithm for solving the 15-puzzle problem. It describes the key components of branch and bound including live nodes, e-nodes, and dead nodes. It also defines the cost function used to evaluate nodes as the sum of the path length and number of misplaced tiles. The algorithm generates all possible child nodes from the current node and prunes the search tree by comparing node costs to avoid exploring subtrees without solutions.
This document provides an overview of algorithms and algorithm analysis. It discusses key concepts like what an algorithm is, different types of algorithms, and the algorithm design and analysis process. Some important problem types covered include sorting, searching, string processing, graph problems, combinatorial problems, geometric problems, and numerical problems. Examples of specific algorithms are given for some of these problem types, like various sorting algorithms, search algorithms, graph traversal algorithms, and algorithms for solving the closest pair and convex hull problems.
The document discusses algorithm analysis and asymptotic notation. It defines algorithm analysis as comparing algorithms based on running time and other factors as problem size increases. Asymptotic notation such as Big-O, Big-Omega, and Big-Theta are introduced to classify algorithms based on how their running times grow relative to input size. Common time complexities like constant, logarithmic, linear, quadratic, and exponential are also covered. The properties and uses of asymptotic notation for equations and inequalities are explained.
The document discusses divide and conquer algorithms. It describes divide and conquer as a design strategy that involves dividing a problem into smaller subproblems, solving the subproblems recursively, and combining the solutions. It provides examples of divide and conquer algorithms like merge sort, quicksort, and binary search. Merge sort works by recursively sorting halves of an array until it is fully sorted. Quicksort selects a pivot element and partitions the array into subarrays of smaller and larger elements, recursively sorting the subarrays. Binary search recursively searches half-intervals of a sorted array to find a target value.
This document summarizes graph coloring using backtracking. It defines graph coloring as minimizing the number of colors used to color a graph. The chromatic number is the fewest colors needed. Graph coloring is NP-complete. The document outlines a backtracking algorithm that tries assigning colors to vertices, checks if the assignment is valid (no adjacent vertices have the same color), and backtracks if not. It provides pseudocode for the algorithm and lists applications like scheduling, Sudoku, and map coloring.
The document discusses Strassen's algorithm for matrix multiplication. It begins by explaining traditional matrix multiplication that has a time complexity of O(n3). It then explains how the divide and conquer strategy can be applied by dividing the matrices into smaller square sub-matrices. Strassen improved upon this by reducing the number of multiplications from 8 to 7 terms, obtaining a time complexity of O(n2.81). His key insight was applying different equations on the sub-matrix multiplication formulas to minimize operations.
Abstract data types (adt) intro to data structure part 2Self-Employed
Abstract Data type (ADT), Related to DATA STRUCTURE and ALGORITHMS STACK QUEUE ARRAY LINKED LIST ALGORITHMS AND INSERTION DELETION MERGE TRAVERSE MODIFY AND OTHER related operation in the algorithms of stack queue array and linked list as an ADT type
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
1) The document discusses complexity analysis of algorithms, which involves determining the time efficiency of algorithms by counting the number of basic operations performed based on input size.
2) It covers motivations for complexity analysis, machine independence, and analyzing best, average, and worst case complexities.
3) Simple rules are provided for determining the complexity of code structures like loops, nested loops, if/else statements, and switch cases based on the number of iterations and branching.
This document discusses asymptotic notations that are used to characterize an algorithm's efficiency. It introduces Big-Oh, Big-Omega, and Theta notations. Big-Oh gives an upper bound on running time. Big-Omega gives a lower bound. Theta gives both upper and lower bounds, representing an algorithm's exact running time. Examples are provided for each notation. Time complexity is also introduced, which describes how the time to solve a problem changes with problem size. Worst-case time analysis provides an upper bound on runtime.
1. An algorithm is a sequence of unambiguous instructions to solve a problem and obtain an output for any valid input in a finite amount of time. Pseudocode is used to describe algorithms using a natural language format.
2. Analyzing algorithm efficiency involves determining the theoretical and empirical time complexity by counting the number of basic operations performed relative to the input size. Common measures are best-case, worst-case, average-case, and amortized analysis.
3. Important problem types for algorithms include sorting, searching, string processing, graphs, combinatorics, geometry, and numerical problems. Fundamental algorithms are analyzed for correctness and time/space complexity.
Merge sort is a divide and conquer algorithm that divides an array into halves, recursively sorts the halves, and then merges the sorted halves back together. The key steps are:
1. Divide the array into equal halves until reaching base cases of arrays with one element.
2. Recursively sort the left and right halves by repeating the divide step.
3. Merge the sorted halves back into a single sorted array by comparing elements pairwise and copying the smaller element into the output array.
Merge sort has several advantages including running in O(n log n) time in all cases, accessing data sequentially with low random access needs, and being suitable for external sorting of large data sets that do not fit in memory
PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
The document provides an overview of recursive and iterative algorithms. It discusses key differences between recursive and iterative algorithms such as definition, application, termination, usage, code size, and time complexity. Examples of recursive algorithms like recursive sum, factorial, binary search, tower of Hanoi, and permutation generator are presented along with pseudocode. Analysis of recursive algorithms like recursive sum, factorial, binary search, Fibonacci number, and tower of Hanoi is demonstrated to determine their time complexities. The document also discusses iterative algorithms, proving an algorithm's correctness, the brute force approach, and store and reuse methods.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
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.
Lecture 5 6_7 - divide and conquer and method of solving recurrencesjayavignesh86
The document discusses divide and conquer algorithms and solving recurrences. It covers asymptotic notations, examples of divide and conquer including finding the largest number in a list, recurrence relations, and methods for solving recurrences including iteration, substitution, and recursion trees. The iteration method involves unfolding the recurrence into a summation. The recursion tree method visually depicts recursive calls in a tree to help solve the recurrence. Divide and conquer algorithms break problems into smaller subproblems, solve the subproblems recursively, and combine the solutions.
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.
Tail recursion is a type of recursion where the recursive call is the last statement executed in the recursive function. This allows compilers to optimize tail recursive functions by reusing the same stack frame rather than creating new frames on each recursion. An example of a tail recursive function prints numbers from 3 to 1 by recursively calling itself and passing the decremented value. While recursion uses more space than a loop, tail recursion can be optimized to have similar space complexity to a loop.
This document discusses analyzing recursive algorithms and forming recurrence relations. It provides examples of writing recurrence relations for recursive functions. The key steps are:
1) Identify the base case(s) where recursive calls stop.
2) Express the work done and size of subproblems at each recursive call.
3) Derive the recurrence relation relating the function at different inputs sizes.
The recurrence relation captures the work at each level of recursion and sums the costs to determine overall runtime. Analyzing recurrences helps understand the asymptotic complexity of recursive algorithms.
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.
35000120060_Nitesh Modi_CSE Presentation on recursion.pptx15AnasKhan
The document summarizes recursion and provides examples. It begins by defining recursion as a function that calls itself directly or indirectly. It then provides examples including calculating factorials and the Towers of Hanoi problem. It discusses the different types of recursion including direct, indirect, tail, and non-tail recursion. Finally, it discusses the N-Queens problem and provides an algorithm to solve it using backtracking and recursion.
The document discusses unconstrained optimization problems and algorithms for solving them. It begins by defining an unconstrained optimization problem as one where the decision vector x can take any value in Rn. The goal is to find the point x that minimizes the objective function f(x) for all x in Rn. It then describes the descent property, quadratic termination property, and global convergence property that desirable algorithms should have. It also discusses line search methods like the Golden Section Rule and Fibonacci Search Method for solving unconstrained optimization problems involving unimodal functions.
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
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.
Recursion is introduced, including definitions and examples of recursion in mathematics and programming. Key aspects of recursion covered include the recursive case, base case, and use of the call stack. The document discusses recursion compared to iteration, advantages and disadvantages of recursion, and techniques like memoization to improve recursive solutions. Examples of recursive algorithms like calculating factorials and Fibonacci numbers are provided to illustrate recursive concepts.
The document discusses recursion, which is a technique that solves a problem by solving smaller instances of the same problem. It provides examples of recursive functions, including factorial, combinations, multiplication, Fibonacci, and greatest common divisor (GCD). Recursive functions contain a base case with a straightforward solution and recursively define the problem in terms of smaller instances until reaching the base case. C uses a call stack to maintain the state of each recursive call.
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.
1. Algorithm analysis helps determine which algorithm is most efficient in terms of time and space consumed by analyzing the rate of growth of the time and space complexity functions as the input size increases.
2. Big-O notation provides an asymptotic upper bound on the growth rate of an algorithm, while Omega and Theta notations provide asymptotic lower and tight bounds respectively.
3. Recurrence relations can be used to analyze the time complexity of recursive algorithms, where the running time T(n) is expressed as a function of smaller inputs.
This document discusses recursion, including:
- Recursive functions call themselves, with a base case to end recursion and a general case that calls a smaller version.
- Recursive solutions are generally less efficient than iterative ones but work for some problems.
- Examples include computing powers and factorials recursively, and solving the Towers of Hanoi puzzle recursively by breaking it into smaller subproblems.
The document discusses algorithm analysis and complexity analysis. It introduces big-O notation to classify algorithms based on how their runtime scales with input size. Common complexity classes include constant, logarithmic, linear, quadratic, and exponential time. The document explains how to determine the time complexity of algorithms by analyzing basic operations and ignoring constant factors. Loops are particularly important, as their runtime is determined by the number of iterations.
Digital Signal Processing (DSP) from basics introduction to medium level book based on Anna University Syllabus! This is just a share of worthfull book!
-Prabhaharan Ellaiyan
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-www.insmartworld.blogspot.in
The document discusses algorithm analysis and complexity. It defines a priori and a posteriori analysis, and explains that algorithm analysis deals with running time. There are two main complexity measures: time complexity, which describes how time scales with input size, and space complexity, which describes how memory usage scales with input size. Time complexity can be best-case, average-case, or worst-case. Asymptotic notation like Big-O, Big-Omega, and Big-Theta are used to describe these complexities. Common loop types like linear, logarithmic, quadratic, and dependent quadratic loops are covered along with their time complexities.
Recursion is a programming technique in which a function calls itself directly or indirectly. Using a recursive algorithm, certain problems can be solved quite easily. Examples of such problems are Towers of Hanoi (TOH), Inorder/Preorder/Postorder Tree Traversals, DFS of Graph, etc.
Recursion is a fundamental programming technique where a method calls itself to solve a problem. This chapter discusses recursive thinking and programming, providing examples of recursion in sorting, graphics, and solving puzzles like the Towers of Hanoi. Some key algorithms covered are merge sort, quick sort, maze traversal, and generating fractals. Recursive solutions can provide elegant code but require careful structuring to avoid infinite recursion.
Similar to Complexity Analysis of Recursive Function (20)
This document defines and provides examples of partial order relations. It discusses the key properties of a partial order being reflexive, antisymmetric, and transitive. Examples are given to show that the relation of greater than or equal to (≥) forms a partial order on integers, while division (|) forms a partial order on positive integers. The document also discusses comparability, total orders, well-ordered sets, and Hasse diagrams which are used to visually represent partial orders.
The primary focus of the PPT is to develop the initial skill of using HTML & CSS programming language to develop a static web page like Portfolio.
This PowerPoint Presentation is of Front End Design.
This PPT will give an entire view on developing the static web page.
This PPT covers the entire topic of Macro Assembler. This Includes the topic such as design of a macro assembler, 3 passes of macro assembler etc.
This is the PPT of System Programming.
This is an PPT about the Icons that are used in Graphical User Interface, the Images that are used for developing a web page & the use of multimedia for various purpose.
This is an PowerPoint Presentation of Front End Design.
This PPT describes about the "Project Tracking" activity & statistical process control at Infosys.
It covers the entire topic such as project tracking, activities tracking, defect tracking, issue tracking, etc.
It covers all main activity of SPC such as SPC analysis, control chart for SPC etc.
This PowerPoint presentation is of "Software Project Management".
This is the PowerPoint presentation on the topic "Peephole Optimization". This presentation covers the entire topic of peephole optimization.
This PowerPoint presentation is of Compiler Design.
This is the PPT of "Routing in Manet". It covers the entire topic of routing protocol.
This PowerPoint presentation is of Data Communication & Computer Network.
The document discusses the design of a two-pass macro preprocessor. In pass one, macro definitions are identified and stored in a macro definition table along with their parameters. A macro name table is also created. In pass two, macro calls are identified and replaced by retrieving the corresponding macro definition and substituting actual parameters for formal parameters using an argument list array. Databases like the macro definition table, macro name table, and argument list array are used to store and retrieve macro information to enable expansion of macro calls. The algorithm scans the input sequentially in each pass to process macro definitions and calls.
This document discusses Vehicular Ad-Hoc Networks (VANETs) which allow vehicles to communicate with each other to share safety and traffic information. It outlines the architecture of VANETs including vehicle-to-vehicle and vehicle-to-infrastructure communication. The document also discusses security issues in VANETs such as bogus information attacks, identity disclosure, and denial-of-service attacks. It proposes the use of authentication, message integrity, privacy, traceability and availability to address these security requirements. The document assumes that roadways are divided into regions managed by trusted roadside infrastructure units.
This document discusses breadth-first search (BFS) and depth-first search (DFS) algorithms for traversing graphs. It provides examples of how BFS uses a queue to search all neighbors at the current level before moving to the next level, while DFS uses a stack and explores each branch as far as possible before backtracking. The document compares key differences between BFS and DFS such as their time and space complexities, usefulness for finding shortest paths, and whether queues or stacks are used. Application areas for each algorithm are also mentioned.
Secant method in Numerical & Statistical MethodMeghaj Mallick
This is an PPT of a Mathematical Paper i.e Numerical & Statistical Method. It contsin the following topic such as "Secant method in Numerical & Statistical Method ".
This document discusses communication and barriers to effective communication. It defines communication as the exchange of information, ideas, thoughts and feelings between individuals through speech, writing and behavior. It then outlines some common barriers to communication, including badly expressed messages, loss in transmission, semantic problems, over or under communication, prejudices on the sender's part, and poor attention, inattentive listening, evaluation, interests/attitudes and refutation on the receiver's part. The document suggests identifying and addressing such barriers to improve communication.
This document provides an introduction to hashing and hash tables. It defines hashing as a data structure that uses a hash function to map values to keys for fast retrieval. It gives an example of mapping list values to array indices using modulo. The document discusses hash tables and their operations of search, insert and delete in O(1) time. It describes collisions that occur during hash function mapping and resolution techniques like separate chaining and linear probing.
This presentation by Professor Giuseppe Colangelo, Jean Monnet Professor of European Innovation Policy, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
This presentation by OECD, OECD Secretariat, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
XP 2024 presentation: A New Look to Leadershipsamililja
Presentation slides from XP2024 conference, Bolzano IT. The slides describe a new view to leadership and combines it with anthro-complexity (aka cynefin).
This presentation by Yong Lim, Professor of Economic Law at Seoul National University School of Law, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
This presentation by Nathaniel Lane, Associate Professor in Economics at Oxford University, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
This presentation was uploaded with the author’s consent.
This presentation by Juraj Čorba, Chair of OECD Working Party on Artificial Intelligence Governance (AIGO), was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
This presentation by Tim Capel, Director of the UK Information Commissioner’s Office Legal Service, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
Why Psychological Safety Matters for Software Teams - ACE 2024 - Ben Linders.pdfBen Linders
Psychological safety in teams is important; team members must feel safe and able to communicate and collaborate effectively to deliver value. It’s also necessary to build long-lasting teams since things will happen and relationships will be strained.
But, how safe is a team? How can we determine if there are any factors that make the team unsafe or have an impact on the team’s culture?
In this mini-workshop, we’ll play games for psychological safety and team culture utilizing a deck of coaching cards, The Psychological Safety Cards. We will learn how to use gamification to gain a better understanding of what’s going on in teams. Individuals share what they have learned from working in teams, what has impacted the team’s safety and culture, and what has led to positive change.
Different game formats will be played in groups in parallel. Examples are an ice-breaker to get people talking about psychological safety, a constellation where people take positions about aspects of psychological safety in their team or organization, and collaborative card games where people work together to create an environment that fosters psychological safety.
The importance of sustainable and efficient computational practices in artificial intelligence (AI) and deep learning has become increasingly critical. This webinar focuses on the intersection of sustainability and AI, highlighting the significance of energy-efficient deep learning, innovative randomization techniques in neural networks, the potential of reservoir computing, and the cutting-edge realm of neuromorphic computing. This webinar aims to connect theoretical knowledge with practical applications and provide insights into how these innovative approaches can lead to more robust, efficient, and environmentally conscious AI systems.
Webinar Speaker: Prof. Claudio Gallicchio, Assistant Professor, University of Pisa
Claudio Gallicchio is an Assistant Professor at the Department of Computer Science of the University of Pisa, Italy. His research involves merging concepts from Deep Learning, Dynamical Systems, and Randomized Neural Systems, and he has co-authored over 100 scientific publications on the subject. He is the founder of the IEEE CIS Task Force on Reservoir Computing, and the co-founder and chair of the IEEE Task Force on Randomization-based Neural Networks and Learning Systems. He is an associate editor of IEEE Transactions on Neural Networks and Learning Systems (TNNLS).
Suzanne Lagerweij - Influence Without Power - Why Empathy is Your Best Friend...Suzanne Lagerweij
This is a workshop about communication and collaboration. We will experience how we can analyze the reasons for resistance to change (exercise 1) and practice how to improve our conversation style and be more in control and effective in the way we communicate (exercise 2).
This session will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
Abstract:
Let’s talk about powerful conversations! We all know how to lead a constructive conversation, right? Then why is it so difficult to have those conversations with people at work, especially those in powerful positions that show resistance to change?
Learning to control and direct conversations takes understanding and practice.
We can combine our innate empathy with our analytical skills to gain a deeper understanding of complex situations at work. Join this session to learn how to prepare for difficult conversations and how to improve our agile conversations in order to be more influential without power. We will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
In the session you will experience how preparing and reflecting on your conversation can help you be more influential at work. You will learn how to communicate more effectively with the people needed to achieve positive change. You will leave with a self-revised version of a difficult conversation and a practical model to use when you get back to work.
Come learn more on how to become a real influencer!
This presentation by OECD, OECD Secretariat, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
This presentation by OECD, OECD Secretariat, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
This presentation was uploaded with the author’s consent.
1. ‘COMPLEXITY ANALYSIS OF RECURSIVE FUNCTION’
PRESENTED BY: PRASHI JAIN
ROLL NO. : MCA/25022/18
2. RECURSIVE ALGORITHM
A recursive function is a function that is defined in terms of itself. Similarly, an algorithm is said to
be recursive if the same algorithm is invoked in the body.
Direct recursion occurs when an algorithm calls itself .
Indirect recursion occurs when a function calls another function , eventually resulting in the original
method being called again.
A
A
A
B
3. PROPERTIES
• A recursive function can go infinite like a loop. To avoid infinite running of
recursive function, there are two properties that a recursive function must have −
Base criteria − There must be at least one base criteria or condition, such that, when this
condition is met the function stops calling itself recursively.
Progressive approach − The recursive calls should progress in such a way that each time
a recursive call is made it comes closer to the base criteria.
4. ANALYSIS OF RECURSION
• One may argue why to use recursion, as the same task can be done with
iteration. The first reason is, recursion makes a program more readable and
because of latest enhanced CPU systems, recursion is more efficient than
iterations.
• There are mainly two ways for analysis of any algorithm :-
1- Time Complexity
2- Space Complexity
5. TIME COMPLEXITY
Time complexity is the amount of time it takes to run an algorithm.
In case of iterations, we take number of iterations to count the time complexity.
Likewise, in case of recursion, assuming everything is constant, we try to figure out
the number of times a recursive call is being made.
A call made to a function is Ο(1), hence the (n) number of times a recursive call is
made makes the recursive function Ο(n).
6. SPACE COMPLEXITY:
Space complexity is counted as what amount of extra space is required for a module to
execute.
In case of iterations, the compiler hardly requires any extra space. The compiler
keeps updating the values of variables used in the iterations.
In case of recursion, the system needs to store activation record each time a recursive
call is made. Hence, it is considered that space complexity of recursive function may
go higher than that of a function with iteration.
7. FINDING COMPLEXITY USING TFC
• We can find complexity of iteration algorithm using frequency count.
For example:
For(i=1;i<n;i++) n times
{
For(j=1 ; j<n ;j++) (n-1)n times
{
Statement; n2 times
}
}
The total frequency count for the above algorithm is 4n2 -1. Then the complexity of algorithm is O(n2).
8. WHY RECURRENCE RELATION ?
• When we use tfc (total frequency count) method for analysis of any algorithm then it is useful
for many algorithms . But Many algorithms such as Tower of Hanoi, selection sort , binary
search , Fibonacci and many more , where we use a recursive then finding the complexity of
that using tfc is not possible , because you can not exactly find out how many times a statement
will execute in a recursive function.
• For solving the above problem we use recurrence relation to find out the complexity of any
recursive algorithm.
9. WHAT IS A RECURRENCE RELATION?
When analysing a recursive function for its step count ( running time), we often obtain a recursive
formula. These recursive formulas are referred to as recurrence relations which are solved by repeated
substitutions method.
A recurrence relation, T(n), is a recursive function of an integer variable n.
Example:
1 if n=0
T(n)=
1+T(n-1) for n>o
The portion of the definition that does not contain T is called the base case of the recurrence relation.
The portion that contains T is called the recurrent or recursive case.
10. STEPS FOR FINDING COMPLEXITY OF RECURSIVE FUNCTION
forming a recurrence relation
solving the recurrence relation
11. FORMING A RECURRENCE RELATION
Example
function factorial(n)
1-2. if (n=1) then factorial=1;
Or else
3. Factorial =n*factorial(n-1)
And end factorial
The base case is reached when n = = 0. The method performs one comparison. Thus, the number of
operations when n = = 0, T(0), is some constant a.
When n > 0, the method performs two basic operations and then calls itself, using ONE recursive call, with
a parameter n – 1.
• Therefore the recurrence relation is:
T(0) = a for some constant a
T(n) = b + T(n – 1) for some constant b
12. SOLVING RECURRENCE RELATIONS
• Steps:
Expand the recurrence relation.
Express the expansion as a summation by plugging the recurrence back into itself
until you see a pattern.
Evaluate the summation
15. ALGORITHM : TOWER OF HANOI
TOH(n ,x , y , z)
{
If(n>=1)
{
//put (n-1) disk to z by using y
TOH((n-1), x ,z , y)
// move larger disk to right place
Move : x- -> y
//put (n-1) disk to right place
TOH((n-1), z ,y ,x)
}
x y z
16. SOLUTION:
• The recurrence relation for the running time of the method hanoi is:
T(n) = a if n = 1
T(n) = 2T(n - 1) + b if n > 1
17. Expanding:
T(1) = a (1)
T(n) = 2T(n – 1) + b if n > 1 (2)
= 2[2T(n – 2) + b] + b = 22 T(n – 2) + 2b + b by substituting T(n – 1) in (2)
= 22 [2T(n – 3) + b] + 2b + b = 23 T(n – 3) + 22b + 2b + b by substituting T(n-2) in (2)
= 23 [2T(n – 4) + b] + 22b + 2b + b = 24 T(n – 4) + 23 b + 22b + 21b + 20b by substituting
T(n – 3) in (2)
= ……
= 2k T(n – k) + b[2k- 1 + 2k– 2 + . . . 21 + 20]
The base case is reached when n – k = 1 k = n – 1, we then have:
Therefore the complexity of Tower of Hanoi is O(2n).