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 24.1, Problem 3E
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To suggest a simple change to the BELLMAN-FORD
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You 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).
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.
Given a directed graph with positive edge lengths and two distinct vertices uand v in the graph, the “all-pairs uv-constrained shortest path problem” is the problemof computing for each pair of vertices i and j the length of the shortest path from i toj that goes through the vertex u or through the vertex v. If no such path exists, theanswer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v asinput parameters and computes values L(i, j) that represent the length of uv-constrainedshortest path from i to j for all 1 ≤ i, j ≤ |V|, i ! = u, j ! = u, i != v, j ! = v. Provide clearpseudocode solution. Prove your algorithm correct. Your algorithm must have runningtime in O(|V| ^2).
Chapter 24 Solutions
Introduction to Algorithms
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- Provide an eficient algorithm that given a directed graph G with n vertices and m edges as input, finds the outdegree of each vertex in G. Note that outdegree of a vertex u is the number of edges directed from u to some other vertex v. Discuss the running-time of your algorithm and Provide an algorithm that given a directed graph G with n vertices and m edges as input, nds the indegree of each vertex in G. Note that indegree of a vertex u is the number of edges directed into u from some other vertex v. Discuss the running-time of your algorithm.arrow_forwardplease answer both of the questions. 7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done.. 8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree. Please give proper explanation and typed answer only.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_forward
- Assume 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_forwardMost graph algorithms that take an n×n adjacency-matrix representation as input require at least time O(n^2), but there are some exceptions. Show how to determine whether a simple directed graph G contains a universal sink, that is, a vertex with in degree n − 1 and out-degree 0, in time O(n) given an n × n adjacency matrix for G. (A vertex v has indegree k if there are precisely k edges of the form (u, v), and has outdegree k if there are precisely k edges of the form (v, u).)arrow_forwardConsider a directed graph G with a starting vertex s, a destination t, and nonnegative edge lengths. Under what conditions is the shortest s-t path guaranteed to be unique? a) When all edge lengths are distinct positive integers. b) When all edge lengths are distinct powers of 2. c) When all edge lengths are distinct positive integers and the graph G contains no directed cycles. d) None of the other options are correct.arrow_forward
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