optrees (version 1.0)

getMinimumCutTree: getMinimumCutTree ---------------------------------------------------------- Computes a minimum cut tree

Description

Given a connected weighted undirected graph, getMinimumCutTree computes a minimum cut tree, also called Gomory-Hu tree. This function uses the Gusfield's algorithm to find it.

Usage

getMinimumCutTree(nodes, arcs, algorithm = "Gusfield", show.data = TRUE, show.graph = TRUE, check.graph = FALSE)

Arguments

nodes

vector containing the nodes of the graph, identified by a number that goes from $1$ to the order of the graph.

arcs

matrix with the list of arcs of the graph. Each row represents one arc. The first two columns contain the two endpoints of each arc and the third column contains their weights.

algorithm

denotes the algorithm to use for find a minimum cut tree or Gomory-Hu tree: "Gusfield".

check.graph

logical value indicating if it is necesary to check the graph. Is FALSE by default.

show.data

logical value indicating if the function displays the console output (TRUE) or not (FALSE). The default is TRUE.

show.graph

logical value indicating if the function displays a graphical representation of the graph and its minimum cut tree (TRUE) or not (FALSE). The default is TRUE.

Value

getMinimumCutTree returns a list with:

tree.nodes

vector containing the nodes of the minimum cut tree.

tree.arcs

matrix containing the list of arcs of the minimum cut tree.

weight

value with the sum of weights of the arcs.

stages

number of stages required.

time

time needed to find the minimum cut tree.

This function also represents the graph and the minimum cut tree and prints in console the results whit additional information (number of stages, computational time, etc.).

Details

The minimum cut tree or Gomory-Hu tree was introduced by R. E. Gomory and T. C. Hu in 1961. Given a connected weighted undirected graph, the Gomory-Hu tree is a weighted tree that contains the minimum s-t cuts for all s-t pairs of nodes in the graph. Gomory and Hu developed an algorithm to find this tree, but it involves maximum flow searchs and nodes contractions.

In 1990, Dan Gusfield proposed a new algorithm that can be used to find the Gomory-Hu tree without any nodes contraction and simplifies the implementation.

References

R. E. Gomory, T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, vol. 9, 1961.

Dan Gusfield (1990). "Very Simple Methods for All Pairs Network Flow Analysis". SIAM J. Comput. 19 (1): 143-155.

Examples

# Graph
nodes <- 1:6
arcs <- matrix(c(1,2,1, 1,3,7, 2,3,1, 2,4,3, 2,5,2, 3,5,4, 4,5,1, 4,6,6,
                5,6,2), byrow = TRUE, ncol = 3)
# Minimum cut tree
getMinimumCutTree(nodes, arcs)