Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 25.1, Problem 5E
Program Plan Intro
To show the single source shortest path
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Develop an algorithm that breaks the linearithmic runningtime barrier for the single-source shortest-paths problem in general edge-weighted digraphs for the special case where the weights are integers known to be bounded inabsolute value by a constant
Design a linear-time algorithm for solving the single source shortest path problem on a given graph G, such that weights associated with edges are drawn from [1/2, 1/3, 1/5]
Consider the given figure: and find the shortest path from source s to all othervertices with the Dijkstra Algorithm. Also discuss about its complexity andbenefits and limitations.
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
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- Implement a Java example to solve Dijkstra's Shortest Path Algorithm using Adjacency Matrix.arrow_forwardSuppose you are given a connected weighted undirected graph, G, with n vertices and m edges, such that the weight of each edge in G is an integer in the interval [1, c], for a fixed constant c > 0. Show how to solve the single-source shortest-paths problem, for any given vertex v, in G, in time O(n + m). Please don't copy and paste the other answers to this question. I'm posting it because the other answers either don't answer the question or have a complicated way of solving the problem.arrow_forwardCreate a technique that, in the unusual case when the weights are integers and it is known that their absolute value is constrained by a constant, solves the single-source shortest-paths problem in general edge-weighted digraphs without exceeding the linearithmic running time limit.arrow_forward
- Prove that The number of augmenting paths needed in the shortest-augmenting-path implementation of the Ford-Fulkerson maxflow algorithm for a flow network with V vertices and E edges is at most EV /2.arrow_forwardWrite an algoruthm for Constructing All-Pairs Shortest Paths?arrow_forwardCreate a method that, in the exceptional situation when the weights are integers and are known to have an absolute value restricted by a constant, overcomes the linearithmic running time limit for the single-source shortest-paths issue in general edge-weighted digraphs.arrow_forward
- You are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain Englisharrow_forwardAssume that we are given an undirected graph G=(V,E). Consider that Dijkstra's algorithm found a shortest path in G, called SP, between two nodes A and X of V. Is it true or false that if we reverse the nodes on SP, we get a shortest path from X to A? Prove or disprove.arrow_forwardYou are given an adjacency list representation of a weighted directed graph G = (V, E) with n vertices, m edges, and no negative cycles. Furthermore, you are given two designated vertices s, t ∈ V and a subset P ⊆ E of the edges are labeled as “premium” edges. Describe a O(mn)-time algorithm to find the shortest path from s to t that uses at most one premium edge (along with any number of non-premium edges).arrow_forward
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