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 24.3, Problem 8E
Program Plan Intro
To modify the Dijkstra’s
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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.
Consider a directed graph G=(V,E) with n vertices, m edges, a starting vertex s∈V, real-valued edge lengths, and no negative cycles. Suppose you know that every shortest path in G from s to another vertex has at most k edges. How quickly can you solve the single-source shortest path problem? (Choose the strongest statement that is guaranteed to be true.) a) O(m+n) b) O(kn) c) O( km) d) O(mn)
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.
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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- Given a directed graph G=(V,E) with positive weights in the vertex and two subsets S and T of V, propose an algorithm with worst case time complexity O(|E| * log |V|) to find the minimum path of some vertex of S to some vertex of Tarrow_forwardLONG-Route is the issue of finding whether or not there is a simple path in G from u to v with a length of at least k given the inputs (G, u, v, k), where G is a graph, u and v are vertices, and k is an integer.Demonstrate that LONG-PATH is an NP-complete problem.arrow_forwardGiven a directed graph G=(V, E), design an algorithm to find out whether there is a route between two nodes, say, u and v. Your design should be based on depth first search and should be given in pseudo code.arrow_forward
- Given a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.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_forwardShortest paths. Let G = (V,E) be an acyclic weighted directed graph and let s ∈ V be an arbitrary vertex. Describe an algorithm which in time O(|V | + |E|) finds shortest paths from s to all (reachable from s) vertices in the graph G, represented by an adjacency list.arrow_forward
- Create an algorithm that, given a directed graph (g = (v e)) and a unique vertex (s v), finds the shortest route between each vertex (v v) and s. Your algorithm must complete in less time than o(n + e) if g contains n vertices and e edges.arrow_forwardLet G be a directed graph with positive and negative weights. It is known that the shortest paths from source node s to every other vertex are at most k edges long. Give a O(k|E|) algorithm that finds all shortest paths.arrow_forwardFind the shortest path from S to other nodes, on the given directed acyclic graph.Graph: R → A : 3 S → A : 1 A → C : 6 B → D : 3 C → E : 2R → S : 2 S → B : 2 B → A : 4 C → D : 1 D → E : 1 Answer: Topological Ordering: __________________________ Node Edge Relax? Update Shortest Path from S: Length Path R S A B C D Earrow_forward
- Let G = (V;E) be a graph. Suppose we have the shortest path between two Nodes in V calculated correctly. Is the statement true that the computed shortest path always remains unchanged if we increase the weight of each edge in E by 1?arrow_forwardTrue or false: let G be an arbitrary connected, undirected graph with a distinct cost c(e) on every edge e. suppose e* is the cheapest edge in G; that is, c(e*) <c(e) for every edge e is not equal to e*. Any minimum spanning tree T of G contains the edge e*arrow_forwardWe recollect that Kruskal's Algorithm is used to fi nd the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will fi rst sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.arrow_forward
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