Find Shortest Paths Between All Nodes in a Directed Graph
shortestPaths.RdallShortestPaths 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.
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).
Value
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
Examples
## 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)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 30 10 NA NA
#> [2,] NA 0 NA 70 40
#> [3,] NA NA 0 50 20
#> [4,] NA NA NA 0 60
#> [5,] NA NA NA 10 0
## compute all path lengths
z <- allShortestPaths(x)
print(z)
#> $length
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 30 10 40 30
#> [2,] NA 0 NA 50 40
#> [3,] NA NA 0 30 20
#> [4,] NA NA NA 0 60
#> [5,] NA NA NA 10 0
#>
#> $middlePoints
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0 0 5 3
#> [2,] 0 0 0 5 0
#> [3,] 0 0 0 5 0
#> [4,] 0 0 0 0 0
#> [5,] 0 0 0 0 0
#>
## the following should give 1 -> 3 -> 5 -> 4
extractPath(z, 1, 4)
#> [1] 1 3 5 4