getMinimumArborescence: Computes a minimum cost arborescence
In optrees: Optimal Trees in Weighted Graphs

Description Usage Arguments Details Value References Examples

Given a connected weighted directed graph, getMinimumArborescence computes a minimum cost arborescence. This function provides a method to find the minimum cost arborescence with Edmonds' algorithm.

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getMinimumArborescence(nodes, arcs, source.node = 1, algorithm = "Edmonds",
  stages.data = FALSE, show.data = TRUE, show.graph = TRUE,
  check.graph = FALSE)

`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.
`source.node`	number pointing to the source node of the graph. It's node 1 by default.
`algorithm`	denotes the algorithm used to find a minimum cost arborescence: "Edmonds".
`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 arborescence (`TRUE`) or not (`FALSE`). The default is `TRUE`.
`stages.data`	logical value indicating if the function returns data of each stage. The default is `FALSE`.

Given a connected weighted directed graph, a minimum cost arborescence is an arborescence such that the sum of the weight of its arcs is minimum. In some cases, it is possible to find several minimum cost arborescences, but the proposed algorithm only finds one of them.

Edmonds' algorithm was developed by the mathematician and computer scientist Jack R. Edmonds in 1967. Although, it was previously proposed in 1965 by Yoeng-jin Chu and Tseng-hong Liu. This algorithm decreases the weights of the arcs in a graph and compacts cycles of zero weight until it can find an arborescence. This arborescence has to be a minimum cost arborescence of the graph.

getMinimumArborescence returns a list with:

`tree.nodes`	vector containing the nodes of the minimum cost arborescence.
`tree.arcs`	matrix containing the list of arcs of the minimum cost arborescence.
`weight`	value with the sum of weights of the arcs.
`stages`	number of stages required.
`time`	time needed to find the minimum cost arborescence.

This function also represents the graph and the minimum arborescence and prints to the console the results with additional information (number of stages, computational time, etc.).

Chu, Y. J., and Liu, T. H., "On the Shortest Arborescence of a Directed Graph", Science Sinica, vol. 14, 1965, pp. 1396-1400.

Edmonds, J., "Optimum Branchings", Journal of Research of the National Bureau of Standards, vol. 71B, No. 4, October-December 1967, pp. 233-240.

# Graph
nodes <- 1:4
arcs <- matrix(c(1,2,2, 1,3,3, 1,4,4, 2,3,3, 2,4,4, 3,2,3,
                 3,4,1, 4,2,1, 4,3,2),byrow = TRUE, ncol = 3)
# Minimum cost arborescence
getMinimumArborescence(nodes, arcs)

Loading required package: igraph

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union


 Minimum cost spanning arborescence 
 Algorithm: Edmonds 
 Stages:  2 | Time:  0.173 
 ------------------------------
      head     tail    weight 
         1        3         3
         3        4         1
         4        2         1
 ------------------------------
                   Total = 5