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Unit ii divide and conquer -4
The document discusses algorithms for computing convex hulls, including Graham's scan and quickhull algorithms. Graham's scan finds the convex hull of a set of points by maintaining a stack of candidate points and removing points that are not vertices of the convex hull. It runs in O(n log n) time. Quickhull is a divide and conquer algorithm that recursively partitions points and computes farthest points to partition the space until the convex hull is completed. Both algorithms are efficient ways to compute convex hulls in two dimensions.
Unit ii divide and conquer -3
The document discusses topological sorting of directed acyclic graphs (DAGs). It defines topological sorting as an ordering of the vertices of a DAG such that for every edge from vertex u to vertex v, u comes before v in the ordering. It presents two algorithms for topological sorting - one based on depth-first search (DFS) and one based on source removal. The DFS-based algorithm runs in O(V+E) time and correctly produces a topological ordering by listing vertices in decreasing order of their finishing times from DFS.
Unit ii divide and conquer -2
This document discusses algorithms for finding the maximum and minimum elements in an array, including the straightforward method that requires 2n-2 comparisons and the divide-and-conquer method that requires fewer comparisons. It also covers graph traversal algorithms like breadth-first search (BFS) and depth-first search (DFS), and discusses applications like finding articulation points in a graph. Examples of applying BFS, DFS, and the algorithm for finding articulation points using DFS are provided.
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Unit ii divide and conquer -4
The document discusses algorithms for computing convex hulls, including Graham's scan and quickhull algorithms. Graham's scan finds the convex hull of a set of points by maintaining a stack of candidate points and removing points that are not vertices of the convex hull. It runs in O(n log n) time. Quickhull is a divide and conquer algorithm that recursively partitions points and computes farthest points to partition the space until the convex hull is completed. Both algorithms are efficient ways to compute convex hulls in two dimensions.
Unit ii divide and conquer -3
The document discusses topological sorting of directed acyclic graphs (DAGs). It defines topological sorting as an ordering of the vertices of a DAG such that for every edge from vertex u to vertex v, u comes before v in the ordering. It presents two algorithms for topological sorting - one based on depth-first search (DFS) and one based on source removal. The DFS-based algorithm runs in O(V+E) time and correctly produces a topological ordering by listing vertices in decreasing order of their finishing times from DFS.
Unit ii divide and conquer -2
This document discusses algorithms for finding the maximum and minimum elements in an array, including the straightforward method that requires 2n-2 comparisons and the divide-and-conquer method that requires fewer comparisons. It also covers graph traversal algorithms like breadth-first search (BFS) and depth-first search (DFS), and discusses applications like finding articulation points in a graph. Examples of applying BFS, DFS, and the algorithm for finding articulation points using DFS are provided.
Unit ii divide and conquer -1
The document discusses divide and conquer algorithms and provides examples of binary search and quicksort. It explains that divide and conquer works by dividing problems into smaller subproblems, solving the subproblems recursively, and then combining the solutions. Binary search uses divide and conquer to search a sorted list by repeatedly dividing the search space in half. Quicksort uses divide and conquer to partition an array around a pivot element, sorting smaller subarrays recursively until the entire array is sorted. The time complexity of quicksort is O(n log n) on average and in the best case but can be O(n^2) in the worst case.
Bs,qs,divide and conquer 1
The document discusses the divide-and-conquer algorithmic paradigm. It provides examples of binary search and quicksort, which both use the divide-and-conquer approach. It explains the three steps of divide-and-conquer as dividing the problem into subproblems, conquering the subproblems by solving them recursively, and combining the solutions to solve the original problem. Details are given on implementing binary search and quicksort, including pseudocode and analysis of their time complexities in best, worst and average cases.
Recursive algorithms
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.
Recurrence relation solutions
The document discusses recurrence relations and algorithms for solving recurrence relations. It begins by defining what a recurrence relation is and provides some examples of natural functions that can be expressed as recurrences. It then discusses different methods for solving recurrence relations, including iteration methods like backward substitution, substitution methods, and recursion tree methods. Specific examples are provided to demonstrate how to apply these different solving methods to common recurrence relations.
Asymptotic notations
The document discusses asymptotic notations used to analyze algorithms. It defines best case, average case, and worst case time complexities, and commonly used notations like Big-O, Big-Omega, and Big-Theta. Examples of common time complexities are given like constant, logarithmic, linear, quadratic, and exponential. Examples are provided for analyzing the time complexity of nested loops, matrix multiplication, and bubble sort algorithms. General rules for algorithm analysis are outlined regarding ignoring lower order terms and constants, summing segment times, and selecting the worst case complexity.
P,NP Hard, NP-Complete problems
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Turing machine types
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Introduction to algorithms
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Disjoint sets union, find
The document discusses disjoint sets and operations on disjoint sets such as union and find. Disjoint sets are sets that do not have any common elements. The union of two disjoint sets combines all the elements of both sets. The find operation takes an element as input and returns the set that contains that element. Disjoint sets can be represented using a tree structure. Algorithms for union and find operations are presented, including weighted union and collapsing find techniques that improve the efficiency.
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哪里办理(csu毕业证书)查尔斯特大学毕业证硕士学历原版一模一样
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本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
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The document discusses divide and conquer algorithms and provides examples of binary search and quicksort. It explains that divide and conquer works by dividing problems into smaller subproblems, solving the subproblems recursively, and then combining the solutions. Binary search uses divide and conquer to search a sorted list by repeatedly dividing the search space in half. Quicksort uses divide and conquer to partition an array around a pivot element, sorting smaller subarrays recursively until the entire array is sorted. The time complexity of quicksort is O(n log n) on average and in the best case but can be O(n^2) in the worst case.
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The document discusses the divide-and-conquer algorithmic paradigm. It provides examples of binary search and quicksort, which both use the divide-and-conquer approach. It explains the three steps of divide-and-conquer as dividing the problem into subproblems, conquering the subproblems by solving them recursively, and combining the solutions to solve the original problem. Details are given on implementing binary search and quicksort, including pseudocode and analysis of their time complexities in best, worst and average cases.
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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.
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Introduction to algorithms
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