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, Problem 5P
(a)
Program Plan Intro
To argue that a minimum cut of graph
(b)
Program Plan Intro
To explains the finding of augmenting path having capacity of at least
(c)
Program Plan Intro
To argue the statement that MAX-FLOW-BY-SCALING returns a maximum flow.
(d)
Program Plan Intro
To explain the capacity of a minimum cut of the residual network
(e)
Program Plan Intro
To argue the statement that the inner while loop of line 5-6 executes
(f)
Program Plan Intro
To explain the MAX-FLOW-BY-SCALING can be implemented so that it runs in
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Let G= (V, E) be an arbitrary flow network with source s and sink t, and a positive integer capacity c(u, v) for each edge (u, v)∈E. Let us call a flow even if the flow in each edge is an even number. Suppose all capacities of edges in G are even numbers. Then,G has a maximum flow with an even flow value.
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).
Is it a bottleneck?
Let G=(V,E) be a flow network with source s and t sink. We say that an edge e is a bottleneck if it crosses every minimum-capacity cut separating s from t. Give an efficient algorithm to determine if a given edge e is a bottleneck in G and explain the complexity.
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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- Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part, and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.arrow_forwardLet G = (V, E) be a flow network with source s and sink t. We say that an edge e is a bottleneck if it crosses every minimum-capacity cut separating s from t. Give an efficient algorithm to determine if a given edge e is a bottleneck in G. Analyze the running time of your algorithm.arrow_forwardShow the residual graph for the network flow given in answer to part (a) Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part (a), and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.arrow_forward
- Let (u, v) be a directed edge in arbitrary flow network G. Prove that if there is a minimum (s, t)-cut (S, T) such that u ∈ S and v ∈ T, then there is no minimum cut (S', T' ) such that u ∈ T' , v ∈ S' ). Note that by definition of cut, s ∈ S, t ∈ T, and similarly s ∈ S' , t ∈ T'.arrow_forwardOnly considering Finite graphs, also note that every flow network has a maximum flow. Which of the following statements are true for all flow networks (G, s, t, c)? • IfG = (V, E) has as cycle then it has at least two different maximum flows. (Recall: two flows f, f' are different if they are different as functions V × V -> R. That is, if f (u, u) + f' (u, v) for some u, v EV. The number of maximum flows is at most the number of minimum cuts. The number of maximum flows is at least the number of minimum cuts. If the value of f is O then f(u, v) = O forallu, U. | The number of maximum flows is 1 or infinity. The number of minimum cuts is finite.arrow_forwardrate(F, C) = cap(C): To prove this, it is sufficient to prove that everyedge u, v crossing from U to V has flow in F at capacity (Fu,v = cu,v) andevery edge v, u crossing back from V to U has zero flow in F. These give thatrate(F, C) = u∈Uv∈V [Fu,v − Fv,u] = u∈Uv∈V [cu,v − 0] = cap(C).arrow_forward
- Lets say there is a flow network called G and a flow in G called f, we say that f saturates an edge e if the flow value on that edge is equal to its capacity. The flow sat problem is: given a flow network G and a positive integer k, determine if there exists a flow f in G such that f saturates at least k edges of G. Prove that Flow sat is NP-complete.arrow_forwardSuppose 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.arrow_forwardConsider the following directed network with flows written as the first number and edge capacity as the second on each edge: Part 1 Draw the residual network obtained from this flow. Part 2 Perform two steps of the Ford Fulkerson algorithm on this network, each using the residual graph of the cumulative flow, and the augmenting paths and flow amounts specified below. After each augment, draw two graphs, preferably side by side; these are graphs of: a) The flow values on the edges b) Residual network The augmenting paths and flow amounts are: i) s → b→d c→t with flow amount 7 Units. ii) s → b→ c→ t with 4 units. Note for continuity your second graph should be coming from the one in (i) NOT from the initial graph. Part 3 Exhibit a maximum flow with flow values on the edges, state its value, and exhibit a cut (specified as a set of vertices) with the same value.arrow_forward
- Describe how to construct an incremental network in the Ford-Fulkerson algorithm in order to find the maximal flow through a network flow model with minimal overall cost.arrow_forwardIf all the capacities in the given network are integers, prove that thealgorithm always returns a solution in which the flow through each edge is an integer.For some applications, this fact is crucial.arrow_forwardIn the Erdös-Rényi random network model, suppose N=101 and p=1/20, that is, there are 101 vertices, and every pair of vertices has a probability of 1/20 of being connected by an edge. For the network model given what is the probability that a network generated with those parameters has exactly 400 edges? No need to give the decimal value, the mathematical expression will sufficearrow_forward
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