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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Given a directed graph with non-negative edge weights, suppose we have computed the shortest paths from a given source to all the other vertices. If we modify the graph in such a way that the weights of all the edges
are doubled, then, the shortest paths remain the same and only the total weights of the paths change.
O True
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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.
Let G be a directed graph with positive and negative weights. You are given that the shortest paths from
source node s to every other vertex is at most k edges long. Give a O(k|E|) algorithm that finds all shortest
paths from s.
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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- Suppose that we are given a weighted, directed graph G=(V,E), in which edges that leave the source vertex s may have negative weights, all other edge weights are nonnegative, and there are no negative-weight cycles. We also assume that the graph has no self-loop. In this question we will argue that Dijkstra's algorithm correctly finds shortest paths from s in this graph (Hint: remember the argument using positive weights in the proof of the Dijkstra's algorithm). To do that you will solve the following questions: a. Regarding the proof of the Dijkstra's algorithm for computing the single source shortest path. Indicate which part of this proof relies on the assumption that the graph does not have negative weight cycles. b. Explain why the original proof still holds true in the particular case described above.arrow_forwardGiven any directed graph G, with weighted edges and marked vertices s and t, there is a known linear time algorithm to compute the longest path from s to t in G.arrow_forwardWrite a pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.arrow_forward
- Give a linear time algorithm via pseudo code that takes as input a directed acyclic graph G (V, E) and two vertices u and v, that returns the number of simple paths from u to v in G. Your algorithm needs only to count the simple paths, not list them. Explain why your code runs in linear time.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_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_forward
- Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1, v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)arrow_forwardWe are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…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 an undirected graph G = (V, E), a vertex cover is a subset of V so that every edge in E has at least one endpoint in the vertex cover. The problem of finding a minimum vertex cover is to find a vertex cover of the smallest possible size. Formulate this problem as an integer linear programming problem.arrow_forwardGiven a directed graph with positive edge lengths (weights) and two distinct vertices u and v in the graph, the “all-pairs u-constrained v-avoiding shortest path problem” is the problem of computing for each pair of vertices i and j the length of the shortest path from i to j that goes through the vertex u and avoids vertex v. If no such path exists, the answer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v as input parameters and computes values L(i,j) that represent the length of u-constrained v-avoiding shortest path from i to j for all 1 ≤ i, j ≤ |V |, i ̸= u, j ̸= u, i ̸= v, j ̸= v. Prove your algorithm correct. Your algorithm must have running time in O(|V |2). Detailed pseudocode is required.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
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