1d:T790,# Find Shortest Paths Between All Nodes in a Directed Graph `allShortestPaths` finds all shortest paths in a directed (or undirected) graph using Floyd's algorithm. `extractPath` can be used to actually extract the path between a given pair of nodes. ```r allShortestPaths(x) extractPath(obj, start, end) ``` ## Arguments - `x`: matrix or distance object - `obj`: return value of `allShortestPaths` - `start`: integer, starting point of path - `end`: integer, end point of path ## Details If `x` is a matrix, then `x[i,j]` has to be the length of the direct path from point `i` to point `j`. If no direct connection from point `i` to point `j` exist, then `x[i,j]` should be either `NA` or `Inf`. Note that the graph can be directed, hence `x[i,j]` need not be the same as `x[j,i]`. The main diagonal of `x` is ignored. Alternatively, `x` can be a distance object as returned by `dist` (corresponding to an undirected graph). ## Returns `allShortestPaths` returns a list with components - **length**: A matrix with the total lengths of the shortest path between each pair of points. - **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 `extractPath`. `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(s) Friedrich Leisch ## Examples ```r ## build a graph with 5 nodes x <- matrix(NA, 5, 5) diag(x) <- 0 x[1,2] <- 30; x[1,3] <- 10 x[2,4] <- 70; x[2,5] <- 40 x[3,4] <- 50; x[3,5] <- 20 x[4,5] <- 60 x[5,4] <- 10 print(x) ## compute all path lengths z <- allShortestPaths(x) print(z) ## the following should give 1 -> 3 -> 5 -> 4 extractPath(z, 1, 4) ```

shortestPaths function