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 29, Problem 5P
a.
Program Plan Intro
To formulate the minimum cost circulation problem as a linear program.
b.
Program Plan Intro
To find the optimal solution for minimum cost circulation.
c.
Program Plan Intro
To find the maximum flow problem as minimum cost circulation.
d.
Program Plan Intro
To find the Single Source Shortest Path Problem as Minimum Cost Circulation Problem.
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Given a flow network as below with S and T as source and sink (destination). The pair of integers on each edge corresponds to
the flow value and the capacity of that edge. For instance, the edge (S.A) has capacity 16 and currently is assigned a flow of 5
(units). Assume that we are using the Ford-Fullkerson's method to find a maximum flow for this problem. Fill in the blanks below
with your answers.
a) An augmenting path in the corresponding residual network is
Note: give you answer by listing the vertices along the path, starting with S and ending with T, e.g., SADT (note that this is for
demonstration purpose only and may not be a valid answer), with no spaces or punctuation marks, i.e., no commas "," or full
stops ".". If there are more than one augmenting path, then you can choose one arbitrarily.
b) The maximum increase of the flow value that can be applied along the augmenting path identified in Part a) is
c) The value of a maximum flow is
Note: your answers for Part b) and Part…
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.
Consider 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.
Chapter 29 Solutions
Introduction to Algorithms
Ch. 29.1 - Prob. 1ECh. 29.1 - Prob. 2ECh. 29.1 - Prob. 3ECh. 29.1 - Prob. 4ECh. 29.1 - Prob. 5ECh. 29.1 - Prob. 6ECh. 29.1 - Prob. 7ECh. 29.1 - Prob. 8ECh. 29.1 - Prob. 9ECh. 29.2 - Prob. 1E
Ch. 29.2 - Prob. 2ECh. 29.2 - Prob. 3ECh. 29.2 - Prob. 4ECh. 29.2 - Prob. 5ECh. 29.2 - Prob. 6ECh. 29.2 - Prob. 7ECh. 29.3 - Prob. 1ECh. 29.3 - Prob. 2ECh. 29.3 - Prob. 3ECh. 29.3 - Prob. 4ECh. 29.3 - Prob. 5ECh. 29.3 - Prob. 6ECh. 29.3 - Prob. 7ECh. 29.3 - Prob. 8ECh. 29.4 - Prob. 1ECh. 29.4 - Prob. 2ECh. 29.4 - Prob. 3ECh. 29.4 - Prob. 4ECh. 29.4 - Prob. 5ECh. 29.4 - Prob. 6ECh. 29.5 - Prob. 1ECh. 29.5 - Prob. 2ECh. 29.5 - Prob. 3ECh. 29.5 - Prob. 4ECh. 29.5 - Prob. 5ECh. 29.5 - Prob. 6ECh. 29.5 - Prob. 7ECh. 29.5 - Prob. 8ECh. 29.5 - Prob. 9ECh. 29 - Prob. 1PCh. 29 - Prob. 2PCh. 29 - Prob. 3PCh. 29 - Prob. 4PCh. 29 - Prob. 5P
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Similar questions
- 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_forwardAlgorithm: Network Flow(Maximu Flow, Ford-Fulkerson) and Application of Flow (Minimum Cuts, Bipartite Matching) Consider a flow network and an arbitrary s, t-cut (S, T). We know that by definition s must always be on the S "side" of a cut and t is always going to be on the T "side" of the cut. Obviously, this is true for any cut. Now, consider minimum cuts. This is obviously still true for s and t, but what about other vertices in the flow network? Are there vertices that will always be on one side or the other in every minimum cut? Let's define these notions more concretely. • We say a vertex v is source-docked if v ∈ S for all minimum cuts (S, T). • We say a vertex v is sink-docked if v ∈ T for all minimum cuts (S, T). • We say a vertex v is undocked if v is neither source-docked nor sink-docked. That is, there exist minimum cuts (S, T) and (S 0 , T0 ) such that v ∈ S and v ∈ T' Give an algorithm that takes as input a flow network G and assigns each vertex to one of the three…arrow_forwardLet G = (V, E) be a flow network with source s and sink t. We say that an edge e is a bottleneck in G if it belongs to every minimum capacity cut separating s from t. Give a polynomial-time algorithm to determine if a given edge e is a bottleneck in G.arrow_forward
- Consider the following graph. Node I is the source node and node 8 is the terminal node. (a) Solve the maximal flow problem from node 1 to node 8 using the Ford-Fulkerson method. As an initial path, use 1 3→7 8 and use the path 12→5→8at the second iteration. At every iteration, please show the path and the residual graph. Finally, show the optimal flow path. 12 10 1 13 5 3 (b) Formulate the above maximal flow problem as a linear (integer) programming problem. 2. 3.arrow_forward1. 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. Need help, as you can see the checked boxes is not the right answer, something is missing…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_forward
- 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.arrow_forwardTrue 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_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
- 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_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_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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