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maxflow (biograph)

Calculate maximum flow in biograph object

Syntax

[MaxFlow, FlowMatrix, Cut] = maxflow(BGObj, SNode, TNode)
[...] = maxflow(BGObj, SNode, TNode, ...'Capacity', CapacityValue, ...)
[...] = maxflow(BGObj, SNode, TNode, ...'Method', MethodValue, ...)

Arguments

 BGObj Biograph object created by biograph (object constructor). SNode Node in a directed graph represented by an N-by-N adjacency matrix extracted from biograph object, BGObj. TNode Node in a directed graph represented by an N-by-N adjacency matrix extracted from biograph object, BGObj. CapacityValue Column vector that specifies custom capacities for the edges in the N-by-N adjacency matrix. It must have one entry for every nonzero value (edge) in the N-by-N adjacency matrix. The order of the custom capacities in the vector must match the order of the nonzero values in the N-by-N adjacency matrix when it is traversed column-wise. By default, maxflow gets capacity information from the nonzero entries in the N-by-N adjacency matrix. MethodValue Character vector that specifies the algorithm used to find the minimal spanning tree (MST). Choices are:'Edmonds' — Uses the Edmonds and Karp algorithm, the implementation of which is based on a variation called the labeling algorithm. Time complexity is O(N*E^2), where N and E are the number of nodes and edges respectively.'Goldberg' — Default algorithm. Uses the Goldberg algorithm, which uses the generic method known as preflow-push. Time complexity is O(N^2*sqrt(E)), where N and E are the number of nodes and edges respectively.

Description

Tip

For introductory information on graph theory functions, see Graph Theory Functions.

[MaxFlow, FlowMatrix, Cut] = maxflow(BGObj, SNode, TNode) calculates the maximum flow of a directed graph represented by an N-by-N adjacency matrix extracted from a biograph object, BGObj, from node SNode to node TNode. Nonzero entries in the matrix determine the capacity of the edges. Output MaxFlow is the maximum flow, and FlowMatrix is a sparse matrix with all the flow values for every edge. FlowMatrix(X,Y) is the flow from node X to node Y. Output Cut is a logical row vector indicating the nodes connected to SNode after calculating the minimum cut between SNode and TNode. If several solutions to the minimum cut problem exist, then Cut is a matrix.

Tip

The algorithm that determines Cut, all minimum cuts, has a time complexity of O(2^N), where N is the number of nodes. If this information is not needed, use the maxflow method without the third output.

[...] = maxflow(BGObj, SNode, TNode, ...'PropertyName', PropertyValue, ...) calls maxflow with optional properties that use property name/property value pairs. You can specify one or more properties in any order. Each PropertyName must be enclosed in single quotes and is case insensitive. These property name/property value pairs are as follows:

[...] = maxflow(BGObj, SNode, TNode, ...'Capacity', CapacityValue, ...) lets you specify custom capacities for the edges. CapacityValue is a column vector having one entry for every nonzero value (edge) in the N-by-N adjacency matrix. The order of the custom capacities in the vector must match the order of the nonzero values in the matrix when it is traversed column-wise. By default, graphmaxflow gets capacity information from the nonzero entries in the matrix.

[...] = maxflow(BGObj, SNode, TNode, ...'Method', MethodValue, ...) lets you specify the algorithm used to find the minimal spanning tree (MST). Choices are:

• 'Edmonds' — Uses the Edmonds and Karp algorithm, the implementation of which is based on a variation called the labeling algorithm. Time complexity is O(N*E^2), where N and E are the number of nodes and edges respectively.

• 'Goldberg' — Default algorithm. Uses the Goldberg algorithm, which uses the generic method known as preflow-push. Time complexity is O(N^2*sqrt(E)), where N and E are the number of nodes and edges respectively.

References

[1] Edmonds, J. and Karp, R.M. (1972). Theoretical improvements in the algorithmic efficiency for network flow problems. Journal of the ACM 19, 248-264.

[2] Goldberg, A.V. (1985). A New Max-Flow Algorithm. MIT Technical Report MIT/LCS/TM-291, Laboratory for Computer Science, MIT.

[3] Siek, J.G., Lee, L-Q, and Lumsdaine, A. (2002). The Boost Graph Library User Guide and Reference Manual, (Upper Saddle River, NJ:Pearson Education).