\name{allShortestPaths}\alias{allShortestPaths}\alias{extractPath}\title{Find Shortest Paths Between All Nodes in a Directed Graph}\description{  \code{allShortestPaths} finds all shortest paths in a directed (or  undirected) graph using Floyd's algorithm. \code{extractPath} can be  used to actually extract the path between a given pair of nodes.}\usage{allShortestPaths(x)extractPath(obj, start, end)}\arguments{  \item{x}{matrix or distance object}  \item{obj}{return value of \code{allShortestPaths}}  \item{start}{integer, starting point of path}  \item{end}{integer, end point of path}}\details{  If \code{x} is a matrix, then \code{x[i,j]} has to be the length of the  direct path from point \code{i} to point \code{j}. If no direct  connection from point \code{i} to point \code{j} exist, then  \code{x[i,j]} should be either \code{NA} or \code{Inf}. Note that the  graph can be directed, hence \code{x[i,j]} need not be the same as  \code{x[j,i]}. The main diagonal of \code{x} is ignored.  Alternatively, \code{x} can be a distance object as returned by  \code{\link{dist}} (corresponding to an undirected graph).}\value{  \code{allShortestPaths} returns a list with components  \item{length}{A matrix with the total lengths of the shortest    path between each pair of points.}  \item{middlePoints}{A matrix giving a point in the middle of each    shortest path (or 0 if the direct connection is the shortest path),    this is mainly used as input for \code{extractPath}.}  \code{extractPath} returns a vector of node numbers giving with the  shortest path between two points.}\references{Kumar, V., Grama, A., Gupta, A. and Karypis, G. Introduction  to Parallel Programming - Design and Analysis of  Algorithms, Benjamin Cummings Publishing, 1994, ISBN 0-8053-3170-0}\author{Friedrich Leisch}\examples{## build a graph with 5 nodesx <- matrix(NA, 5, 5)diag(x) <- 0x[1,2] <- 30; x[1,3] <- 10x[2,4] <- 70; x[2,5] <- 40x[3,4] <- 50; x[3,5] <- 20x[4,5] <- 60x[5,4] <- 10print(x)## compute all path lengthsz <- allShortestPaths(x)print(z)## the following should give 1 -> 3 -> 5 -> 4extractPath(z, 1, 4)}\keyword{optimize}