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 26.2, Problem 5E
Program Plan Intro
To prove that any flow has a finite value if the edge of the network with multiple sources and sinks has finite capacity.
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Problem 2.
Find the maximum flow in the flow network shown in figure 1. In the flow network ‘s’ is the source vertex and ‘t’ is the destination vertex. The capacity of each of the edges are given in the figure.
Let f be a flow of flow network G and f' a flow of residual network Gf . Show that f +f' is a flow of G.
1. Recall that a flow network is a directed graph G = (V, E) with a source s, a sink t, and a capacity function c: V x
V + Rj that is positive on E and 0 outside E.We only consider finite graphs here. Also, note that every flow
network has a maximum flow. This sounds obvious but requires a proof (and we did not prove it in the video
lecture).
Which of the following statements are true for all flow networks (G, s,t,c)?
O 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 x V R. That is, if f(u, v) # f'(u, v) for some u, ve V.
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 0 then f(u, v) = 0 for all u, v.
The number of maximum flows is 1 or infinity.
The number of minimum cuts is finite.
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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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- 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).arrow_forwardConsider the flow network G shown in figure 1 with source s and sink t. The edge capacities are the numbers given near each edge. (a) Find a maximum flow in this network. Once you have done this, draw a copy of the original network G and clearly indicate the flow on each edge of G in your maximum flow. (b) Find a minimum s-t cut in the network, i.e. name the two (non-empty) sets of vertices that define a minimum cut. Also, say what its capacity is. g C 4 3 6 d b 3 Fig.1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.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_forward
- Consider the flow network G shown in figure 1 with source s and sink t. The edge capacities are the numbers given near each edge. (a) Find a maximum flow in this network. Once you have done this, draw a copy of the original network G and clearly indicate the flow on each edge of G in your maximum flow. (b) Find a minimum s-t cut in the network, i.e. name the two (non-empty) sets of vertices that define a minimum cut. Also, say what its capacity is. 3 2 b 6 Fig. 1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.arrow_forwardTrue or false: For any flow network G and any maximum flow on G, there is always an edge e such that increasing the capacity of e increases the maximum flow of the network. Justify your answer. Doğru veya yanlış: Herhangi bir akış ağı G ve G üzerindeki herhangi bir maksimum akış için, her zaman bir e kenarı vardır, öyle ki e'nin kapasitesini artırmak ağın maksimum akışını arttırır. Cevabınızı gerekçelendirin.arrow_forward4. Find the maximum flow from source (node 0) to destination (node 5) from the following flow graph. Show the residual network at each step. 12 1 16 20 10||4 7 13 4 14 4. 2.arrow_forward
- c.) Figure 2 shows a flow network on which an s-t flow has been computed. The capacity of each edge appears as a label next to the edge, and the numbers in boxes give the amount of flow sent on each edge. (Edges without boxed numbers have no flow being sent on them.) What is the value of the flow in Figure 2? Is this a maximum (s,t) flow in this graph? d). Find a maximum flow from s to t in Figure 2 (draw a picture like Figure 2, and specify the amount of flow in the box for each edge), and also say what its capacity is.arrow_forwardCompute the maximum flow in the following flow network using Ford-Fulkerson's algorithm. 12 8 14 15 11 2 5 1 3 d 7 f 6 What is the maximum flow value? A minimum cut has exactly which vertices on one side? 4.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_forward
- Let G=(V,E) be a flow network and suppose that you are given a cut (S,T) of minimum capacity x. What else can be said about G? a. The maximum flow of G is x b. There are no augmenting paths in c. All valid cuts of G have capacity x d. A and B e. All of the above. G, the residual network of G 'ƒ'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_forwardQuestion 1Draw the residual network obtained from this flow. Question2Perform 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. Question 3Exhibit 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
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