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```
#-----------------------------------------------------------------------------#
# optrees Package #
# Minimun Spanning Tree Problem #
#-----------------------------------------------------------------------------#
# msTreeKruskal ---------------------------------------------------------------
#' Minimum cost spanning tree with Kruskal's algorithm
#'
#' \code{msTreeKruskal} computes a minimum cost spanning tree of an undirected
#' graph with Kruskal's algorithm.
#'
#' @details 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 forming cycles, the minimum weight arc of the graph at each
#' stage.
#'
#' @param nodes vector containing the nodes of the graph, identified by a
#' number that goes from \eqn{1} to the order of the graph.
#' @param 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.
#'
#' @return \code{msTreeKruskal} returns a list with:
#' 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.
#' stages number of stages required.
#' stages.arcs stages in which each arc was added.
#'
#' @references 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
#'
#' @seealso A more general function \link{getMinimumSpanningTree}.
#'
#' @export
msTreeKruskal <- function(nodes, arcs) {
# Order arcs by weight
arcs <- matrix(arcs[order(arcs[, 3]), ], ncol = 3)
# Components
components <- matrix(c(nodes, nodes), ncol = 2)
# Initialize tree with first arc
tree.arcs <- matrix(ncol = 3)[-1, ]
stages <- 0 # initialize counter
stages.arcs <- c() # vector to store stage number in wich each arc was added
# Start with first arc
i <- 1
# Repeat until we have |N|-1 arcs
while(nrow(tree.arcs) < length(nodes) - 1) {
# Select arc
min.arc <- arcs[i, ]
# Check components of the two nodes of selected arc
iComp <- components[components[, 1] == min.arc[1], 2]
jComp <- components[components[, 1] == min.arc[2], 2]
if (iComp != jComp) {
# Add arc to msTree
tree.arcs <- rbind(tree.arcs, min.arc)
# Merge components
components[components[, 2] == jComp, 2] <- iComp
}
stages <- stages + 1 # counter
# Save in which stage an arc was added to the tree and update
stages.arcs <- c(stages.arcs,
rep(stages, nrow(tree.arcs) - length(stages.arcs)))
# Continue with next arc
i <- i + 1
}
# Column names
colnames(tree.arcs) <- c("ept1", "ept2", "weight")
# Remove row names
rownames(tree.arcs) <- NULL
output <- list("tree.nodes" = nodes, "tree.arcs" = tree.arcs,
"stages" = stages, "stages.arcs" = stages.arcs)
return(output)
}
#-----------------------------------------------------------------------------#
```

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