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 26.1, Problem 2E
Program Plan Intro
To show the multi-source, multiple-sink flow network corresponds to a flow of identical value in the single-source, single-sink network obtained by adding a supersource and supersink and vice versa.
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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.
What is a simple example of a network that has no bottleneck edges and is a valid flow network?
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.
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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- Demonstrate that the network flow technique given in this section keeps the loop invariant that it always has a legal flow. Prove that the changes to the flow do not violate any edge capabilities or cause node leakage. Also demonstrate progress by increasing the total flow. You must exercise caution when using plus and negative signs.arrow_forwardTrue 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.arrow_forwardHow can vector network analyser be used to find SS1 parameters?arrow_forward
- Consider the flow network shown in the following figure (left), where the label next to each arc is its capacity, and the initial s-t flow on right. (a) What must be checked to show that the initial f is a flow?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_forwardProblem 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_forward
- 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,?).arrow_forwardCreate an SVM to calculate the XOR function. Use the values +1 / -1 for inputs and outputs. Map the input [x1,x2] to the space with coordinates [x1,x2x1]. Draw the four possible points of the training set and their maximum linear separator in this space. What is its edge?arrow_forwardDescribe 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_forward
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