getMinimumSpanningTree: Computes a minimum cost spanning tree

Description

Given a connected weighted undirected graph, getMinimumSpanningTree computes a minimum cost spanning tree. This function provides methods to find a minimum cost spanning tree with the three most commonly used algorithms: "Prim", "Kruskal" and "Boruvka".

Usage

getMinimumSpanningTree(nodes, arcs, algorithm, start.node = 1, 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 used to find a minimum spanning tree: "Prim", "Kruskal" or "Boruvka".

check.graph

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

start.node

number which indicates the first node in Prim's algorithm. If none is specified node 1 is used 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 spanning tree (TRUE) or not (FALSE). The default is TRUE.

Value

tree.nodes: vector containing the nodes of the minimum cost spanning tree.
tree.arcs: matrix containing the list of arcs of the minimum cost spanning tree.
weight: value with the sum of weights of the arcs.
stages: number of stages required.
stages.arcs: stages in which each arc was added.
time: time needed to find the minimum cost spanning tree.

Details

Given a connected weighted undirected graph, a minimum spanning tree is a spanning tree such that the sum of the weights of the arcs is minimum. There may be several minimum spanning trees of the same weight in a graph. Several algorithms were proposed to find a minimum spanning tree in a graph.

Prim's algorithm was developed in 1930 by the mathematician Vojtech Jarnik, independently proposed by the computer scientist Robert C. Prim in 1957 and rediscovered by Edsger Dijkstra in 1959. This is a greedy algorithm that can find a minimum spanning tree in a connected weighted undirected graph by adding minimum cost arcs leaving visited nodes recursively.

Kruskal's algorithm was published for first time in 1956 by mathematician Joseph Kruskal. This is a greedy algorithm that finds a minimum cost spanning tree in a connected weighted undirected graph by adding, without form cycles, the minimum weight arc of the graph in each iteration.

Boruvka's algorithm was published for first time in 1926 by mathematician Otakar Boruvka. This algorithm go through a connected weighted undirected graph, reviewing each component and adding the minimum weight arcs without repeat it until one minimum spanning tree is complete.

References

Prim, R. C. (1957), "Shortest Connection Networks And Some Generalizations", Bell System Technical Journal, 36 (1957), pp. 1389-1401.

Kruskal, Joshep B. (1956), "On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem", Proceedings of the American Mathematical Society, Vol. 7, No. 1 (Feb., 1956), pp. 48-50.

Boruvka, Otakar (1926). "O jistem problemu minimalnim (About a certain minimal problem)". Prace mor. prirodoved. spol. v Brne III (in Czech, German summary) 3: 37-58.

Examples

Run this code

# Graph
nodes <- 1:4
arcs <- matrix(c(1,2,2, 1,3,15, 1,4,3, 2,3,1, 2,4,9, 3,4,1),
               ncol = 3, byrow = TRUE)
# Minimum cost spanning tree with several algorithms
getMinimumSpanningTree(nodes, arcs, algorithm = "Prim")
getMinimumSpanningTree(nodes, arcs, algorithm = "Kruskal")
getMinimumSpanningTree(nodes, arcs, algorithm = "Boruvka")

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