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 24.2, Problem 2E
Program Plan Intro
To argue that there is no change in the procedure if the line 3 has change for
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Establish how shortest-path computations in edge-weighted digraphs with nonnegative vertices (where the weight of a path is defined as the sum of the vertices' weights) may be handled by generating an edge-weighted digraph with weights only on the edges.
Develop an algorithm for finding an edge whose removal causes maximal increase in the shortest-paths length from one given vertex to another given vertex in a given edge-weighted digraph
Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Consider a graph G that is comprised only of non-negative weight edges such that (u, v) € E, w(u, w) > 0. Is it possible for Bellman-Ford and Dijkstra's algorithm to produce different shortest path trees despite always producing the same shortest-path weights? Justify your answer.arrow_forwardGiven a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple (no repeated vertices).arrow_forwardGet the bitonic shortest route from s to each of the other vertices in a given digraph (if one exists). If a path has an intermediate vertex v and the edges from s to v and from v to t are strictly rising and decreasing, the path is said to be bitonic. The way should be clear-cut.arrow_forward
- When we want to calculate the shortest paths from a vertex using the Bellman-Ford algorithm, it is possible to stop early and not do all |V| - 1 iterations on graphs without a negative cycle. How can we modify the Bellman-Ford Algorithm so that it stops early when all distances are correct?arrow_forwardShow that shortest-paths computations in edge-weighted digraphs with nonnegative weights on vertices (where the weight of a path is defined tobe the sum of the weights of the vertices) can be handled by building an edge-weighteddigraph that has weights on only the edges.arrow_forwardIs it possible to find a minimal spanning tree in O(n) time for a linked, weighted network with n vertices and n edges?arrow_forward
- Let G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.arrow_forwardProve Proposition: The singlesource shortest-paths issue for edge-weighted DAGs can be solved in time proportionate to E + V by relaxing vertices in topological order.arrow_forwardProve by contradiction that BFS computes the shortest path starting from a given source vertex s. Feel free to introduce suitable notation for the proof.arrow_forward
- Show all the steps of Kruskal''s minimum cost spanning tree algorithm for a complete graph of 6 vertices where the weight of the edge between the distinct vertices i and j is |i-j-1|, for 1 <= i, j <= 6.arrow_forwardGiven a weighted graph G = (V, E, w) and a source vertex s ∈ V, prove the the following properties regardless of the SSSP algorithm. 1. If π[s] ever changes after the initialization, then G contains a negative cycle through s. 2. Show that π[s] may never change after the initialization, even when G contains a negative cycle through s. we need to prove so do not provide codes.arrow_forwardGiven a weighted digraph, find a monotonic shortestpath from s to every other vertex. A path is monotonic if the weight of every edge onthe path is either strictly increasing or strictly decreasing. The path should be simple(no repeated vertices). Hint : Relax edges in ascending order and find a best path; thenrelax edges in descending order and find a best path.arrow_forward
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