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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Shortest 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.
Suppose X is a subset of vertices in a G = (V,E). Give a polynomial-time algorithm to test whether there exists a matching in G covering X.
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.
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
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- Let 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_forwardFor any given graph G = (V, E), give a linear time algorithm to determine whether G is a bipartite graph. Note the time complexity of your algorithm is O(n + m), where n = |V|, m = |Earrow_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_forward
- 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_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_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.arrow_forward
- Given a directed graph G(V, E), find an O(|V | + |E|) algorithm that deletes at most half of its edges so that the resulting graph doesn’t contain any directed cycles. Also follow up with proof of correctness.arrow_forwardThe reverse of a directed graph G = (V,E) is another directed graph G^R = (V,E^R) on the same vertex set, but with all edges reversed; that is, E^R = {(v, u) : (u, v) ∈ E}.Give a linear-time algorithm for computing the reverse of a graph in adjacency list format.arrow_forwardLet G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that for all u, v ∈ V , the path between u and v in tree T is uniquearrow_forward
- Given 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_forwardConnecting a Graph, Let G be an undirected graph. Give a linear time algorithm to compute the smallest number of edges that one would need to add to G in order to make it a connected graph??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
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