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 25.3, Problem 6E
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To show that if the weighted directed graph G is strongly connected then modification by professor Michene gives correct result while applying with Johnson’s
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Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
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- We have the following directed graph G, where the number on each edge is the cost of the edge. 1. Step through Dijkstra’s Algorithm on the graph starting from vertex s, and complete the table below to show what the arrays d and p are at each step of the algorithm. For any vertex x, d[x] stores the current shortest distance from s to x, and p[x] stores the current parent vertex of x. 2. After you complete the table, provide the returned shortest path from s to t and the cost of the path.arrow_forwardhow could someone solce this one? The Chinese postman problem: Consider an undirected connected graph and a given starting node. The Chinese postman has to find the shortest route through the graph that starts and ends in the starting node such that all links are passed. The same problem appears for instance for snow cleaning or garbage collection in a city. For a branch-and-bound algorithm, find a possible lower bound function. (Remark: If the problem is to pass through all nodes of the graph, it is called the travelling salesman problem - which needs different solution algorithms, but also of the branch-and-bound type).).arrow_forwardRecall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique. Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP. Q4.1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G). Prove that for any subset of vertices…arrow_forward
- please 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_forwardGiven a weighted graph G = (V, E, w) and a source vertex s ∈ V, prove the the following properties regardless of the SSSP algorithm. 1. If π[s] ever changes after the initialization, then G contains a negative cycle through s. 2. Show that π[s] may never change after the initialization, even when G contains a negative cycle through s. we need to prove so do not provide codes.arrow_forwardHey, Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result. How can i do that? Thank you in advance!arrow_forward
- 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_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_forwardYou are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E') with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e in E is modified from w(e) to a new value w̃(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree. Note, you are given the tree T and the edge e = (y, z) whose weight is changed; you are told its old weight w(e) and its new weight w~(e) (which you type in latex by widetilde{w}(e) surrounded by double dollar signs). In each case specify if the tree might change. And if it might change then give an algorithm to find the weight of the possibly new MST (just return the weight or the MST, whatever's easier). You can use the algorithms DFS, Explore, BFS, Dijkstra's, SCC, Topological Sort as…arrow_forward
- . Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.arrow_forwardThe Algorithm of AlgebraThe adjacency matrix A of a graph G = is used by the algebraic BFS algorithm (V, E).First, we create a vector x with all zeros except for the index of the source vertex s that we wish to use as the starting point for the algorithm; next, we create the matrix A = A+ I; last, AT x chooses all nodes that are at a distance of 1 (level 1) from the source vertex. The vertices with the fewest number of hops are obtained by multiplying the vector x by the matrix A2. As a rule, the product Ak x will produce neighbours that are at most k hops distant, and the multiplication should be done in a boolean manner as in the algorithm shown below. Algorithm for Algebraic BFS1: Input : Adjacency matrix An,n of a graph G = (V, E) connected, unweighted graph G and asource vertex s2: Output : N, visited, levels a matrix that shows level i vertices at its column i, the visitedvertices in sequence and their levels3: x[n] ← 04: x[s] ← 15: A ← A + I6: for i = 1 to n do7: N ← AT · x8:…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_forward
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