This document discusses the divide-and-conquer algorithm design strategy. It begins by defining divide-and-conquer as dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. Examples covered include sorting algorithms like mergesort and quicksort, tree traversals, binary search, integer multiplication, matrix multiplication using Strassen's algorithm, and closest pair problems. Analysis techniques like recursion trees and the master theorem are introduced.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
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.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
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.
Divide and conquer is a general algorithm design paradigm where a problem is divided into subproblems, the subproblems are solved independently, and the results are combined to solve the original problem. Binary search is a divide and conquer algorithm that searches for a target value in a sorted array by repeatedly dividing the search interval in half. It compares the target to the middle element of the array, and then searches either the upper or lower half depending on whether the target is greater or less than the middle element. Finding the maximum and minimum elements in an array can also be solved using divide and conquer by recursively finding the max and min of halves of the array and combining the results.
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
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.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
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.
Divide and conquer is a general algorithm design paradigm where a problem is divided into subproblems, the subproblems are solved independently, and the results are combined to solve the original problem. Binary search is a divide and conquer algorithm that searches for a target value in a sorted array by repeatedly dividing the search interval in half. It compares the target to the middle element of the array, and then searches either the upper or lower half depending on whether the target is greater or less than the middle element. Finding the maximum and minimum elements in an array can also be solved using divide and conquer by recursively finding the max and min of halves of the array and combining the results.
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
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 the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
The document discusses the divide and conquer algorithm design strategy. It begins by explaining the general concept of divide and conquer, which involves splitting a problem into subproblems, solving the subproblems, and combining the solutions. It then provides pseudocode for a generic divide and conquer algorithm. Finally, it gives examples of divide and conquer algorithms like quicksort, binary search, and matrix multiplication.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses divide and conquer algorithms like binary search, merge sort, and quicksort. It explains that divide and conquer algorithms break problems down into smaller subproblems, solve the subproblems recursively, and combine the solutions. This approach reduces the time complexity from O(n) to O(n log n). Binary search uses divide and conquer to find an item in a sorted array using at most log2(n) comparisons. Merge sort and quicksort use it to sort arrays in O(n log n) time.
Divide and Conquer - Part II - Quickselect and Closest Pair of PointsAmrinder Arora
This document discusses divide and conquer algorithms. It covers the closest pair of points problem, which can be solved in O(n log n) time using a divide and conquer approach. It also discusses selection algorithms like quickselect that can find the median or kth element of an unsorted array in linear time O(n) on average. The document provides pseudocode for these algorithms and analyzes their time complexity using recurrence relations. It also provides an overview of topics like mergesort, quicksort, and solving recurrence relations that were covered in previous lectures.
This document discusses two methods for finding the maximum and minimum values in an array: the naive method and divide and conquer approach. The naive method compares all elements to find the max and min in 2n-2 comparisons. The divide and conquer approach recursively divides the array in half, finds the max and min of each half, and returns the overall max and min, reducing the number of comparisons. Pseudocode is provided for the MAXMIN algorithm that implements this divide and conquer solution.
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 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 discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
The document discusses different decrease-and-conquer algorithms. It explains that decrease-and-conquer works by establishing a relationship between solving a problem for an instance of size n and solving a smaller instance of size n-c, where c is some constant. It then decomposes the problem size recursively until reaching a base case. The document provides examples of decrease by a constant, constant factor, and variable amounts. It also discusses insertion sort, binary search, and Euclid's algorithm as examples of decrease-and-conquer approaches.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
The document discusses different methods for solving recurrence equations:
1. Substitution method involves guessing a solution and using induction to prove it is correct.
2. Iterative method expands the recurrence into a summation and evaluates it.
3. Master method handles recurrences of the form T(n)=aT(n/b)+f(n). It provides 3 cases to bound the solution.
4. Recursion tree method models the recurrence as a tree to track problem sizes and costs at each level. Summing costs across levels provides the total solution.
The document provides an introduction to the analysis of algorithms. It discusses key concepts like the definition of an algorithm, properties of algorithms, common computational problems, and basic issues related to algorithms. It also covers algorithm design strategies, fundamental data structures, and the fundamentals of analyzing algorithm efficiency. Examples of algorithms for computing the greatest common divisor and checking for prime numbers are provided to illustrate algorithm design and analysis.
Algorithm designing using divide and conquer algorithmsSiddhantShelake
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms break problems down into smaller subproblems, solve those, and then combine the solutions. Examples given include merge sort, quick sort, binary search, and the Tower of Hanoi problem. Merge sort works by breaking a list into single elements and then merging the sorted sublists. Quick sort selects a pivot value and partitions the array into elements less than and greater than the pivot. Binary search works on a sorted array by comparing the search value to the middle element. The Tower of Hanoi involves moving disks between rods according to rules of only moving one disk at a time and not placing a larger disk on a smaller one.
Iterative improvement is an algorithm design technique for solving optimization problems. It starts with a feasible solution and repeatedly makes small changes to the current solution to find a solution with a better objective function value until no further improvements can be found. The simplex method, Ford-Fulkerson algorithm, and local search heuristics are examples that use this technique. The maximum flow problem can be solved using the iterative Ford-Fulkerson algorithm, which finds augmenting paths in a flow network to incrementally increase the flow from the source to the sink until no more augmenting paths exist.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses the dynamic programming approach to solving the Fibonacci numbers problem and the rod cutting problem. It explains that dynamic programming formulations first express the problem recursively but then optimize it by storing results of subproblems to avoid recomputing them. This is done either through a top-down recursive approach with memoization or a bottom-up approach by filling a table with solutions to subproblems of increasing size. The document also introduces the matrix chain multiplication problem and how it can be optimized through dynamic programming by considering overlapping subproblems.
The document discusses the divide and conquer algorithm design technique. It begins by defining divide and conquer as breaking a problem down into smaller subproblems, solving the subproblems, and then combining the solutions to solve the original problem. It then provides examples of applying divide and conquer to problems like matrix multiplication and finding the maximum subarray. The document also discusses analyzing divide and conquer recurrences using methods like recursion trees and the master theorem.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
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 the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
The document discusses the divide and conquer algorithm design strategy. It begins by explaining the general concept of divide and conquer, which involves splitting a problem into subproblems, solving the subproblems, and combining the solutions. It then provides pseudocode for a generic divide and conquer algorithm. Finally, it gives examples of divide and conquer algorithms like quicksort, binary search, and matrix multiplication.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses divide and conquer algorithms like binary search, merge sort, and quicksort. It explains that divide and conquer algorithms break problems down into smaller subproblems, solve the subproblems recursively, and combine the solutions. This approach reduces the time complexity from O(n) to O(n log n). Binary search uses divide and conquer to find an item in a sorted array using at most log2(n) comparisons. Merge sort and quicksort use it to sort arrays in O(n log n) time.
Divide and Conquer - Part II - Quickselect and Closest Pair of PointsAmrinder Arora
This document discusses divide and conquer algorithms. It covers the closest pair of points problem, which can be solved in O(n log n) time using a divide and conquer approach. It also discusses selection algorithms like quickselect that can find the median or kth element of an unsorted array in linear time O(n) on average. The document provides pseudocode for these algorithms and analyzes their time complexity using recurrence relations. It also provides an overview of topics like mergesort, quicksort, and solving recurrence relations that were covered in previous lectures.
This document discusses two methods for finding the maximum and minimum values in an array: the naive method and divide and conquer approach. The naive method compares all elements to find the max and min in 2n-2 comparisons. The divide and conquer approach recursively divides the array in half, finds the max and min of each half, and returns the overall max and min, reducing the number of comparisons. Pseudocode is provided for the MAXMIN algorithm that implements this divide and conquer solution.
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 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 discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
The document discusses different decrease-and-conquer algorithms. It explains that decrease-and-conquer works by establishing a relationship between solving a problem for an instance of size n and solving a smaller instance of size n-c, where c is some constant. It then decomposes the problem size recursively until reaching a base case. The document provides examples of decrease by a constant, constant factor, and variable amounts. It also discusses insertion sort, binary search, and Euclid's algorithm as examples of decrease-and-conquer approaches.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
The document discusses different methods for solving recurrence equations:
1. Substitution method involves guessing a solution and using induction to prove it is correct.
2. Iterative method expands the recurrence into a summation and evaluates it.
3. Master method handles recurrences of the form T(n)=aT(n/b)+f(n). It provides 3 cases to bound the solution.
4. Recursion tree method models the recurrence as a tree to track problem sizes and costs at each level. Summing costs across levels provides the total solution.
The document provides an introduction to the analysis of algorithms. It discusses key concepts like the definition of an algorithm, properties of algorithms, common computational problems, and basic issues related to algorithms. It also covers algorithm design strategies, fundamental data structures, and the fundamentals of analyzing algorithm efficiency. Examples of algorithms for computing the greatest common divisor and checking for prime numbers are provided to illustrate algorithm design and analysis.
Algorithm designing using divide and conquer algorithmsSiddhantShelake
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms break problems down into smaller subproblems, solve those, and then combine the solutions. Examples given include merge sort, quick sort, binary search, and the Tower of Hanoi problem. Merge sort works by breaking a list into single elements and then merging the sorted sublists. Quick sort selects a pivot value and partitions the array into elements less than and greater than the pivot. Binary search works on a sorted array by comparing the search value to the middle element. The Tower of Hanoi involves moving disks between rods according to rules of only moving one disk at a time and not placing a larger disk on a smaller one.
Iterative improvement is an algorithm design technique for solving optimization problems. It starts with a feasible solution and repeatedly makes small changes to the current solution to find a solution with a better objective function value until no further improvements can be found. The simplex method, Ford-Fulkerson algorithm, and local search heuristics are examples that use this technique. The maximum flow problem can be solved using the iterative Ford-Fulkerson algorithm, which finds augmenting paths in a flow network to incrementally increase the flow from the source to the sink until no more augmenting paths exist.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses the dynamic programming approach to solving the Fibonacci numbers problem and the rod cutting problem. It explains that dynamic programming formulations first express the problem recursively but then optimize it by storing results of subproblems to avoid recomputing them. This is done either through a top-down recursive approach with memoization or a bottom-up approach by filling a table with solutions to subproblems of increasing size. The document also introduces the matrix chain multiplication problem and how it can be optimized through dynamic programming by considering overlapping subproblems.
The document discusses the divide and conquer algorithm design technique. It begins by defining divide and conquer as breaking a problem down into smaller subproblems, solving the subproblems, and then combining the solutions to solve the original problem. It then provides examples of applying divide and conquer to problems like matrix multiplication and finding the maximum subarray. The document also discusses analyzing divide and conquer recurrences using methods like recursion trees and the master theorem.
The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
1) Strassen's algorithm improves on the traditional O(n^3) matrix multiplication algorithm by using divide and conquer to achieve O(n^lg7) time complexity.
2) Finding the median can be done in expected O(n) time using quickselect, or deterministically in O(n) time by choosing the median of medians as the pivot.
Cs6402 design and analysis of algorithms may june 2016 answer keyappasami
The document discusses algorithms and complexity analysis. It provides Euclid's algorithm for computing greatest common divisor, compares the orders of growth of n(n-1)/2 and n^2, and describes the general strategy of divide and conquer methods. It also defines problems like the closest pair problem, single source shortest path problem, and assignment problem. Finally, it discusses topics like state space trees, the extreme point theorem, and lower bounds.
The document discusses various topics in advanced algebra including inequalities, arithmetic progressions, geometric progressions, harmonic progressions, permutations, combinations, matrices, determinants, and solving systems of linear equations using matrices. Key properties and formulas are provided for each topic. Examples are included to demonstrate solving problems related to each concept.
Here are the key steps:
1. Guess the solution: T(n) = O(n log n)
2. Set the induction goal: T(n) ≤ c n log n for some c > 0 and n ≥ n0
3. Apply the induction hypothesis: T(n/2) ≤ c (n/2) log(n/2)
4. Substitute into the recurrence: T(n) = 2T(n/2) + n ≤ 2c(n/2)log(n/2) + n = cn log n
5. Simplify and show it meets the induction goal.
Therefore, by mathematical induction, the solution T(n) =
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach and defining the sorting problem. It then describes the three steps of merge sort as divide, conquer, and combine. It provides pseudocode for the merge sort and merge algorithms. Finally, it analyzes the running time of merge sort, showing that it runs in O(n log n) time using the recursion tree method.
Permutations and Combinations IIT JEE+Olympiad Lecture 4Parth Nandedkar
Continues from PnC lecture 3. The series Follows the JEE Advanced syllabus, but this lecture goes beyond into Mathematical Olympiad territory. Covers the following more advanced topics:
Simple idea of Inclusion Exclusion principle,
Explanation through Venn Diagrams,
Application of I-E Principle,
Counting Derangements using I-E Principle,
Partitioning Indistinguishable Objects(+Comparison with Distinguishable Objects) ,
Problem Session
The document discusses brute force algorithms. It provides examples of problems that can be solved using brute force, including sorting algorithms like selection sort and bubble sort. It then summarizes two geometric problems - the closest pair problem and the convex hull problem - and provides pseudocode for brute force algorithms to solve each problem. The time complexity of these brute force algorithms is O(n^3).
The document discusses matrix multiplication using the divide and conquer algorithm design paradigm. It explains that divide and conquer involves dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions to solve the original problem. As an example, it describes how Strassen's algorithm improves on the basic O(n^3) matrix multiplication algorithm by dividing matrices into smaller submatrices and multiplying them recursively in O(n^2.807) time using fewer operations. The document provides pseudocode for Strassen's algorithm and formulas for multiplying submatrices.
This document provides information about obtaining fully solved assignments. It instructs students to send their semester and specialization name to the email address "help.mbaassignments@gmail.com" or call the provided phone number. The document notes that email is preferred, and calls should only be made in emergency situations. It also provides the phone number for calling as 08263069601.
This document describes the divide-and-conquer algorithm design strategy and provides examples of its use. It discusses:
1) The divide-and-conquer strategy involves dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions.
2) Examples where divide-and-conquer is applied include sorting algorithms like mergesort and quicksort, binary tree traversals, binary search, and multiplying large integers.
3) Analysis techniques like recurrence relations and the master theorem are introduced for analyzing divide-and-conquer algorithms' run times. Specific algorithms like mergesort, quicksort, and binary search are analyzed in more detail.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
The document describes using a genetic algorithm to solve resource-constrained project scheduling problems. It introduces genetic algorithms and describes how they can be applied to optimization problems. It then formulates a sample resource-constrained project scheduling problem as a linear programming problem to find basic feasible solutions and extreme points. The conclusion states that future work will use genetic algorithm operators like selection, reproduction and evaluation on the feasible solutions to find an optimal schedule.
This document discusses determinants and their properties. It begins by introducing determinants as functions that associate a unique number (real or complex) to square matrices. It then provides examples of calculating determinants of matrices of order 1, 2 and 3 by expanding along rows or columns. The key properties discussed are that the determinant remains unchanged under row/column interchange and that the determinant of a scalar multiple of a matrix is the scalar raised to the power of the matrix order times the determinant of the original matrix. Examples are provided to illustrate these concepts.
The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. It provides examples of dynamic programming approaches to computing Fibonacci numbers, binomial coefficients, the knapsack problem, and other problems. It also discusses greedy algorithms and provides examples of their application to problems like the change-making problem, minimum spanning trees, and single-source shortest paths.
The document summarizes the Levenshtein distance algorithm and tree edit distance algorithm. It discusses how Levenshtein distance finds the minimum number of single-character edits (insertions, deletions or substitutions) required to change one word into the other. It then explains how tree edit distance extends this to find the minimum cost sequence of node edit operations (insert, delete, relabel) to transform one tree into another using Tai mappings and the Zhang-Shasha algorithm.
The document provides an overview of indefinite integration or anti-differentiation. It discusses standard integrals, rules of integration, techniques like substitution, integration by parts, partial fractions, and integrals of various functions. Some example integrals are also presented along with their solutions. Reduction formulas for integrals of trigonometric functions like secx and cosecx are outlined as well.
This document provides an outline for the development of a public voice system. It discusses analyzing the existing system, proposing improvements, and planning the design and implementation of the new system. Key sections include analyzing requirements and constraints, creating data flow diagrams and entity relationship diagrams to model the system, planning the database and interface design, and testing and deploying the new system. The goal is to develop a practical public voice system that empowers users to view proposals and provide feedback.
This document outlines the requirements for a store inventory control system. It will automate tasks like tracking inventory levels, calculating reorder amounts, and placing orders. The system must be easy for staff to use, accurate, and reliable. It will allow admin to add, update, and delete stock items. The system aims to streamline the inventory management process and reduce errors compared to manual methods. Non-functional requirements include usability, reliability, performance, supportability and packaging. The detailed design goals are for low response time, high robustness, reliability, and extensibility with low fault tolerance and adaptability.
The document discusses a pill camera, which is a capsule device used to take pictures of the inside of the digestive system. It contains a camera, lights, transmitter and batteries and transmits 50,000 images as it passes through the digestive tract. The images help doctors diagnose conditions of the small intestine. It provides a safer alternative to endoscopy for examining the small intestine. The capsule works by being swallowed and passing naturally through the digestive system, transmitting images to an external receiver worn by the patient.
This document discusses high performance computing with accelerators. It introduces accelerators like GPUs, FPGAs, and IBM's Cell processor that are used in supercomputers and computer clusters to accelerate computations. GPUs are well-suited for graphics and floating-point work, FPGAs for embedded and low-bit applications, and ClearSpeed's accelerators for matrix operations. While accelerators provide improved performance and efficiency over CPUs, programming them can be more difficult due to their specialized architectures.
The document discusses infix, postfix, and prefix notations for mathematical expressions. It provides examples to illustrate the differences between the notations. The key points are:
- Infix notation writes operators between operands, like "X + Y". Postfix writes operators after operands, like "XY+". Prefix writes operators before operands, like "+ XY".
- Expressions can be converted between the notations algorithmically by fully parenthesizing the expression and moving operators to the position of matching parentheses.
- Converting an expression to postfix moves operators to the position of the right parenthesis, while prefix moves operators to the left parenthesis position.
pill camera/ capsule endoscopy used inn medical field to find out the abnormalities of small intestine. This replaced the convectional endoscopy method.
UI5con 2024 - Keynote: Latest News about UI5 and it’s EcosystemPeter Muessig
Learn about the latest innovations in and around OpenUI5/SAPUI5: UI5 Tooling, UI5 linter, UI5 Web Components, Web Components Integration, UI5 2.x, UI5 GenAI.
Recording:
https://www.youtube.com/live/MSdGLG2zLy8?si=INxBHTqkwHhxV5Ta&t=0
Microservice Teams - How the cloud changes the way we workSven Peters
A lot of technical challenges and complexity come with building a cloud-native and distributed architecture. The way we develop backend software has fundamentally changed in the last ten years. Managing a microservices architecture demands a lot of us to ensure observability and operational resiliency. But did you also change the way you run your development teams?
Sven will talk about Atlassian’s journey from a monolith to a multi-tenanted architecture and how it affected the way the engineering teams work. You will learn how we shifted to service ownership, moved to more autonomous teams (and its challenges), and established platform and enablement teams.
What is Master Data Management by PiLog Groupaymanquadri279
PiLog Group's Master Data Record Manager (MDRM) is a sophisticated enterprise solution designed to ensure data accuracy, consistency, and governance across various business functions. MDRM integrates advanced data management technologies to cleanse, classify, and standardize master data, thereby enhancing data quality and operational efficiency.
Unveiling the Advantages of Agile Software Development.pdfbrainerhub1
Learn about Agile Software Development's advantages. Simplify your workflow to spur quicker innovation. Jump right in! We have also discussed the advantages.
GraphSummit Paris - The art of the possible with Graph TechnologyNeo4j
Sudhir Hasbe, Chief Product Officer, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
Atelier - Innover avec l’IA Générative et les graphes de connaissancesNeo4j
Atelier - Innover avec l’IA Générative et les graphes de connaissances
Allez au-delà du battage médiatique autour de l’IA et découvrez des techniques pratiques pour utiliser l’IA de manière responsable à travers les données de votre organisation. Explorez comment utiliser les graphes de connaissances pour augmenter la précision, la transparence et la capacité d’explication dans les systèmes d’IA générative. Vous partirez avec une expérience pratique combinant les relations entre les données et les LLM pour apporter du contexte spécifique à votre domaine et améliorer votre raisonnement.
Amenez votre ordinateur portable et nous vous guiderons sur la mise en place de votre propre pile d’IA générative, en vous fournissant des exemples pratiques et codés pour démarrer en quelques minutes.
SOCRadar's Aviation Industry Q1 Incident Report is out now!
The aviation industry has always been a prime target for cybercriminals due to its critical infrastructure and high stakes. In the first quarter of 2024, the sector faced an alarming surge in cybersecurity threats, revealing its vulnerabilities and the relentless sophistication of cyber attackers.
SOCRadar’s Aviation Industry, Quarterly Incident Report, provides an in-depth analysis of these threats, detected and examined through our extensive monitoring of hacker forums, Telegram channels, and dark web platforms.
Using Query Store in Azure PostgreSQL to Understand Query PerformanceGrant Fritchey
Microsoft has added an excellent new extension in PostgreSQL on their Azure Platform. This session, presented at Posette 2024, covers what Query Store is and the types of information you can get out of it.
AI Fusion Buddy Review: Brand New, Groundbreaking Gemini-Powered AI AppGoogle
AI Fusion Buddy Review: Brand New, Groundbreaking Gemini-Powered AI App
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https://sumonreview.com/ai-fusion-buddy-review
AI Fusion Buddy Review: Key Features
✅Create Stunning AI App Suite Fully Powered By Google's Latest AI technology, Gemini
✅Use Gemini to Build high-converting Converting Sales Video Scripts, ad copies, Trending Articles, blogs, etc.100% unique!
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✅Fully automated AI articles bulk generation!
✅Auto-post or schedule stunning AI content across all your accounts at once—WordPress, Facebook, LinkedIn, Blogger, and more.
✅With one keyword or URL, generate complete websites, landing pages, and more…
✅Automatically create & sell AI content, graphics, websites, landing pages, & all that gets you paid non-stop 24*7.
✅Pre-built High-Converting 100+ website Templates and 2000+ graphic templates logos, banners, and thumbnail images in Trending Niches.
✅Say goodbye to wasting time logging into multiple Chat GPT & AI Apps once & for all!
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See My Other Reviews Article:
(1) AI Genie Review: https://sumonreview.com/ai-genie-review
(2) SocioWave Review: https://sumonreview.com/sociowave-review
(3) AI Partner & Profit Review: https://sumonreview.com/ai-partner-profit-review
(4) AI Ebook Suite Review: https://sumonreview.com/ai-ebook-suite-review
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Measures in SQL (SIGMOD 2024, Santiago, Chile)Julian Hyde
SQL has attained widespread adoption, but Business Intelligence tools still use their own higher level languages based upon a multidimensional paradigm. Composable calculations are what is missing from SQL, and we propose a new kind of column, called a measure, that attaches a calculation to a table. Like regular tables, tables with measures are composable and closed when used in queries.
SQL-with-measures has the power, conciseness and reusability of multidimensional languages but retains SQL semantics. Measure invocations can be expanded in place to simple, clear SQL.
To define the evaluation semantics for measures, we introduce context-sensitive expressions (a way to evaluate multidimensional expressions that is consistent with existing SQL semantics), a concept called evaluation context, and several operations for setting and modifying the evaluation context.
A talk at SIGMOD, June 9–15, 2024, Santiago, Chile
Authors: Julian Hyde (Google) and John Fremlin (Google)
https://doi.org/10.1145/3626246.3653374
E-Invoicing Implementation: A Step-by-Step Guide for Saudi Arabian CompaniesQuickdice ERP
Explore the seamless transition to e-invoicing with this comprehensive guide tailored for Saudi Arabian businesses. Navigate the process effortlessly with step-by-step instructions designed to streamline implementation and enhance efficiency.
Hand Rolled Applicative User ValidationCode KataPhilip Schwarz
Could you use a simple piece of Scala validation code (granted, a very simplistic one too!) that you can rewrite, now and again, to refresh your basic understanding of Applicative operators <*>, <*, *>?
The goal is not to write perfect code showcasing validation, but rather, to provide a small, rough-and ready exercise to reinforce your muscle-memory.
Despite its grandiose-sounding title, this deck consists of just three slides showing the Scala 3 code to be rewritten whenever the details of the operators begin to fade away.
The code is my rough and ready translation of a Haskell user-validation program found in a book called Finding Success (and Failure) in Haskell - Fall in love with applicative functors.
Introducing Crescat - Event Management Software for Venues, Festivals and Eve...Crescat
Crescat is industry-trusted event management software, built by event professionals for event professionals. Founded in 2017, we have three key products tailored for the live event industry.
Crescat Event for concert promoters and event agencies. Crescat Venue for music venues, conference centers, wedding venues, concert halls and more. And Crescat Festival for festivals, conferences and complex events.
With a wide range of popular features such as event scheduling, shift management, volunteer and crew coordination, artist booking and much more, Crescat is designed for customisation and ease-of-use.
Over 125,000 events have been planned in Crescat and with hundreds of customers of all shapes and sizes, from boutique event agencies through to international concert promoters, Crescat is rigged for success. What's more, we highly value feedback from our users and we are constantly improving our software with updates, new features and improvements.
If you plan events, run a venue or produce festivals and you're looking for ways to make your life easier, then we have a solution for you. Try our software for free or schedule a no-obligation demo with one of our product specialists today at crescat.io
What is Augmented Reality Image Trackingpavan998932
Augmented Reality (AR) Image Tracking is a technology that enables AR applications to recognize and track images in the real world, overlaying digital content onto them. This enhances the user's interaction with their environment by providing additional information and interactive elements directly tied to physical images.
Transform Your Communication with Cloud-Based IVR SolutionsTheSMSPoint
Discover the power of Cloud-Based IVR Solutions to streamline communication processes. Embrace scalability and cost-efficiency while enhancing customer experiences with features like automated call routing and voice recognition. Accessible from anywhere, these solutions integrate seamlessly with existing systems, providing real-time analytics for continuous improvement. Revolutionize your communication strategy today with Cloud-Based IVR Solutions. Learn more at: https://thesmspoint.com/channel/cloud-telephony