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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Question
Chapter 26.1, Problem 1E
Program Plan Intro
To show the splitting an edge in a flow network yields an equivalent network.
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Suppose you are given a directed graph G = (V, E) with a positive integer capacity ?? on each edge e, a designated source s ∈ V, and a designated sink t ∈ V. You are also given an integer maximum s-t flow value ?? on each edge e. Now suppose we pick a specific edge e belongs E and increase its capacity by one unit. Show how to find a maximum flow in the resulting capacitated graph in O(m + n), where m is the number of edges in G and n is the number on nodes.
Without intersecting lines, draw the directed graph:
Where:
Let Y = {a, b, c, d, e}
Z = {(a, a), (b, a), (d, b), (d, c), (c, c), (e, a), (a, d), (c, e), (e, b), (b, d)}
True or False
Let G be an arbitrary flow network, with a source s, a sink t, and a positiveinteger capacity ceon every edge e. If f is a maximum s −t flow in G, then f saturates every edge out of s with flow (i.e., for all edges e out of s, we have f (e) = ce).
Chapter 26 Solutions
Introduction to Algorithms
Ch. 26.1 - Prob. 1ECh. 26.1 - Prob. 2ECh. 26.1 - Prob. 3ECh. 26.1 - Prob. 4ECh. 26.1 - Prob. 5ECh. 26.1 - Prob. 6ECh. 26.1 - Prob. 7ECh. 26.2 - Prob. 1ECh. 26.2 - Prob. 2ECh. 26.2 - Prob. 3E
Ch. 26.2 - Prob. 4ECh. 26.2 - Prob. 5ECh. 26.2 - Prob. 6ECh. 26.2 - Prob. 7ECh. 26.2 - Prob. 8ECh. 26.2 - Prob. 9ECh. 26.2 - Prob. 10ECh. 26.2 - Prob. 11ECh. 26.2 - Prob. 12ECh. 26.2 - Prob. 13ECh. 26.3 - Prob. 1ECh. 26.3 - Prob. 2ECh. 26.3 - Prob. 3ECh. 26.3 - Prob. 4ECh. 26.3 - Prob. 5ECh. 26.4 - Prob. 1ECh. 26.4 - Prob. 2ECh. 26.4 - Prob. 3ECh. 26.4 - Prob. 4ECh. 26.4 - Prob. 5ECh. 26.4 - Prob. 6ECh. 26.4 - Prob. 7ECh. 26.4 - Prob. 8ECh. 26.4 - Prob. 9ECh. 26.4 - Prob. 10ECh. 26.5 - Prob. 1ECh. 26.5 - Prob. 2ECh. 26.5 - Prob. 3ECh. 26.5 - Prob. 4ECh. 26.5 - Prob. 5ECh. 26 - Prob. 1PCh. 26 - Prob. 2PCh. 26 - Prob. 3PCh. 26 - Prob. 4PCh. 26 - Prob. 5PCh. 26 - Prob. 6P
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- Among all pairs of nodes in a directed network that are connected by an edge, half are connected in only one direction and the rest are connected in both directions. What is the reciprocity of the network?arrow_forwarda) Show the residual graph for the network flow given in answer to part b) What is the bottleneck edge of the path (?, ?3, ?4, ?) in the residual graph you have given in answer to part (a) Show the network with the flow that results from augmenting the flow based on the path (?, ?3, ?4, ?) of the residual graph you have given in answer to part (a)arrow_forwardSuppose that the road network is defined by the undirected graph, where the vertices represent cities and edges represent the road between two cities. The Department of Highways (DOH) decides to install the cameras to detect the bad driver. To reduce the cost, the cameras are strategically installed in the cities that a driver must pass through in order to go from one city to another city. For example, if there are two cities A and B such that the path that goes from A to B and the path that goes from B to A must pass the city C, then C must install the camera. Suppose that there are m cities and n roads. Write an O (m + n) to list all cities where cameras should be installed.arrow_forward
- Consider a network that is a rooted tree, with the root as its source, the leaves as its sinks, and all the edges directed along the paths from the root to the leaves. Design an efficient algorithm for finding a maximum flow in such a network. What is the time efficiency of your algorithm? Describe your algorithm step by step.arrow_forwardConsider an arbitrary connected undirected graph network, with unique identifiers for nodes. We run a deterministic distributed algorithm, such that each node retains an edge only to its lowest indexed neighbour. An edge is retained, if at least one of its two endpoints retains it. A node communicates to that neighbour with which it is retaining its edge, to that effect. Then, the retained sub networkarrow_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_forward
- g) Show the residual graph for the network flow given in answer to part (c). What is the bottleneck edge of the path (?, ?2, ?3, ?1, ?4, ?) in the residual graph you have given in answer to part (g) ? Show the network with the flow that results from augmenting the flow based on the path (?, ?2, ?3, ?1, ?4, ?) of the residual graph you have given in answer to part (g).arrow_forwardProve the following claim : Given a graph G and two vertices a,b of it, there is a walk between a and b if and only if there is a trail between a and b if and only if there is a (simple) path between a and b.arrow_forwardIf a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.arrow_forward
- Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approacharrow_forwardTrue or False Let (u, v) be an edge in a maximum flow graph with capacity greater than 0.Then it is possible that the residual graph contains (v, u), but not (u, v).arrow_forwardConsider the network depicted in Figure 1; suppose that each node starts with the behavior B, and each node has a threshold of q = 1/2 for switching to behavior A. Find a cluster of density greater than 1 − q = 1/2 in the part of the graph outside S that blocks behavior A from spreading to all nodes, starting from S, at threshold qarrow_forward
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