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 10E
Program Plan Intro
To argue that Dijkstra’s
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Consider 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.
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.
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 T
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_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_forwardDraw a (simple) directed weighted graph G with 8 vertices and 18 edges, such that G contains a minimum-weight cycle with at least 4 edges. Show that the Bellman-Ford algorithm will find this cycle.arrow_forward
- Given 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_forwardConsider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.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
- True or False?Given an undirect and connected graph in which all edges have the same weight, we can design an algorithm to compute an MST in worst running time O(|V|)?arrow_forwardConsider 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)arrow_forwardProve Proposition : For any vertex v reachable from s, BFS computes a shortest path from s to v (no path from s to v has fewer edges).arrow_forward
- LONG-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_forwardCreate 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 = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setarrow_forward
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