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 26.3, Problem 1E
Program Plan Intro
To explain the running of FORD-FULKERSON
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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.
Show the residual graph for the network flow given in answer to part.
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 (i), and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?).
True or false. If true, give a brief explanation justifying the statement. If false, provide a counterexample.
- If you are given a flow network (labeled F), let (L,R) be a minimum capacity cut in the flow graph. If the capacity of all of the graph edges is increased by 1, then (L,R) is still a minimum capacity cut in our new modified graph.
Question 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.
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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Similar questions
Modify the Chebyshev center coding with julia in a simple style using vectors, matrices and for loops # Given matrix A and vector bA = [2 -1 2; -1 2 4; 1 2 -2; -1 0 0; 0 -1 0; 0 0 -1]b = [2; 16; 8; 0; 0; 0]
A small sample:Let t_(l),t_(o),t_(m),t_(n),t_(t),t_(s) be starttimes of the associated tasks.Now use the graph to write thedependency constraints:Tasks o,m, and n can't start until task I is finished, and task Itakes 3 days to finish. So the constraints are:t_(l)+3<=t_(o),t_(l)+3<=t_(m),t_(l)+3<=t_(n)Task t can't start until tasks m and n are finished. Therefore:t_(m)+1<=t_(t),t_(n)+2<=t_(t),Task s can't start until tasks o and t are finished. Therefore:t_(o)+3<=t_(s),t_(t)+3<=t_(s)
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a) 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)
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a) Now suppose the graph above is a residual graph after processing two augmenting paths. What were those paths and what were their flows?
b) Draw the original directed acyclic graph and provide the maximum flow of this graph below. You may use either Ford-Fulkerson or Edmunds Karp to find the solution. Please show your work
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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.
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Show 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.
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Show that the loop invariant, which states that it always retains a legal flow, is maintained by the network flow algorithm described in this section. Show that the flow changes don't breach any edge capacities or cause leaks at any nodes to accomplish this. Show that progress is being made by increasing overall flow as well. Use caution when using the plus and negative symbols.
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Prove that the network flow algorithm presented in this section maintains the loop invariant that it always holds a legal flow. Do this by proving that the changes to the flow do not violate any edge capacities or create leaks at nodes. Also prove that progress is made because the total flow increases. You need to be careful with your plus and minus signs.
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a) Show the residual graph for the network flow given in answer to part (c).
b) What is the bottleneck edge of the path (?, ?3, ?4, ?) in the residual graph you have given in answer to part (d) ?
c) 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 (d).
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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.
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Given the complement of a graph G is a graph G' which contains all the vertices of G, but for each unweighted edge that exists in G, it is not in G', and for each possible edge not in G, it is in G'. What logical operation and operand(s) can be applied to the adjacency matrix of G to produce G'?
AND G's adjacency matrix with 0 to produce G'
XOR G's adjacency matrix with 0 to produce G'
XOR G's adjacency matrix with 1 to produce G'
AND G's adjacency matrix with 1 to produce G'
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Using Ford-Fulkerson algorithm, determine the maximum flow from the source S to sink t in the graph provided.
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a) List all of the paths from 1 to 4 in the graph S with a length of 5.
b) The adjacent matrix for the graph R U S
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