shortest.paths {igraph} | R Documentation

## Shortest (directed or undirected) paths between vertices

### Description

`shortest.paths` calculates the length of all the shortest paths from or to the vertices in the network. `get.shortest.paths` calculates one shortest path (the path itself, and not just its length) from or to the given vertex.

### Usage

    shortest.paths(graph, v=V(graph), mode = c("all", "out", "in"),
          weights = NULL, algorithm = c("automatic", "unweighted",
                                        "dijkstra", "bellman-ford",
                                        "johnson"))
    get.shortest.paths(graph, from, to=V(graph), mode = c("all", "out",
          "in"), weights = NULL)
    get.all.shortest.paths(graph, from, to = V(graph), mode = c("all", "out", "in"))
    average.path.length(graph, directed=TRUE, unconnected=TRUE)
    path.length.hist (graph, directed = TRUE, verbose = igraph.par("verbose"))

### Arguments

`graph` | The graph to work on.
`v` | Numeric vector, the vertices from or to which the shortest paths will be calculated.
`mode` | Character constant, gives whether the shortest paths to or from the given vertices should be calculated for directed graphs. If `out` then the shortest paths from the vertex, if `in` then to it will be considered. If `all`, the default, then the corresponding undirected graph will be used, ie. not directed paths are searched. This argument is ignored for undirected graphs.
`weights` | Possibly a numeric vector giving edge weights. If this is `NULL` and the graph has a `weight` edge attribute, then the attribute is used. If this is `NA` then no weights are used (even if the graph has a `weight` attribute).

`algorithm`

Which algorithm to use for the calculation. By default igraph tries to select the fastest suitable algorithm. If there are no weights, then an unweighted breadth-first search is used, otherwise if all weights are positive, then Dijkstra's algorithm is used. If there are negative weights and we do the calculation for more than 100 sources, then Johnson's algorithm is used. Otherwise the Bellman-Ford algorithm is used. You can override igraph's choice by explicitly giving this parameter. Note that the igraph C core might still override your choice in obvious cases, i.e. if there are no edge weights, then the unweighted algorithm will be used, regardless of this argument.

`from` | Numeric constant, the vertex from or to the shortest paths will be calculated. Note that right now this is not a vector of vertex ids, but only a single vertex.
`to` | Numeric vector, only the shortest paths to these vertices will be calculated. Defaults to all vertices.
`directed` | Whether to consider directed paths in directed graphs, this argument is ignored for undirected graphs.
`unconnected` | What to do if the graph is unconnected (not strongly connected if directed paths are considered). If TRUE only the lengths of the existing paths are considered and averaged; if FALSE the length of the missing paths are counted having length `vcount(graph)`, one longer than the longest possible geodesic in the network.
`verbose` | Logical scalar, whether to draw a progress meter while the calculation is running.

### Details

The shortest paths (also called geodesics) are calculated by using breath-first search in the graph. If no edge weights were specified, then a breadth-first search is used to calculate the shortest paths. If edge weigths are given and all of them are non-zero, then Dijkstra's algorithm is used. Otherwise the Bellman-Ford algorithm is used for `shortest.paths`.

Please do NOT call `get.shortest.paths` and `get.all.shortest.paths` with negative edge weights, it will not work, these functions do not use the Belmann-Ford algotithm.

Note that `shortest.paths` is able to calculate the path length from or to many vertices at the same time, but `get.shortest.paths` works from one source only. This might change in the future.

Also note that `get.shortest.paths` gives only one shortest path, however, more than one might exist between two vertices.

`get.all.shortest.paths` calculates all shortest paths from a vertex to other vertices given in the `to` argument.

`path.length.hist` calculates a histogram, by calculating the shortest path length between each pair of vertices. For directed graphs both directions are considered, so every pair of vertices appears twice in the histogram.

### Value

For `shortest.paths` a numeric matrix with `vcount(graph)` columns and `length(v)` rows. The shortest path length from a vertex to itself is always zero. For unreachable vertices `Inf` is included.
For `get.shortest.paths` a list of length `vcount(graph)`. List element `i` contains the vertex ids on the path from vertex `from` to vertex `i` (or the other way for directed graphs depending on the `mode` argument). The vector also contains `from` and `i` as the first and last elements. If `from` is the same as `i` then it is only included once. If there is no path between two vertices then a numeric vector of length zero is returned as the list element.
For `get.all.shortest.paths` a list is returned, each list element contains a shortest path from `from` to a vertex in `to`. The shortest paths to the same vertex are collected into consecutive elements of the list.
For `average.path.length` a single number is returned.
`path.length.hist` returns a named list with two entries: `res` is a numeric vector, the histogram of distances, `unconnected` is a numeric scalar, the number of pairs for which the first vertex is not reachable from the second. The sum of the two entries is always n(n-1) for directed graphs and n(n-1)/2 for undirected graphs.

### Author(s)

Gabor Csardi csardi@rmki.kfki.hu

### References

West, D.B. (1996). Introduction to Graph Theory. Upper Saddle River, N.J.: Prentice Hall.

### Examples

    g <- graph.ring(10)
    shortest.paths(g)
    get.shortest.paths(g, 5)
    get.all.shortest.paths(g, 0, 5:7)
    average.path.length(g)
    ## Weighted shortest paths
    el <- matrix(nc=3, byrow=TRUE,
                 c(0,1,0, 0,2,2, 0,3,1, 1,2,0, 1,4,5, 1,5,2, 2,1,1, 2,3,1,
                   2,6,1, 3,2,0, 3,6,2, 4,5,2, 4,7,8, 5,2,2, 5,6,1, 5,8,1,
                   5,9,3, 7,5,1, 7,8,1, 8,9,4) )
    g2 <- add.edges(graph.empty(10), t(el[,1:2]), weight=el[,3])
    shortest.paths(g2, mode="out")

[Package igraph version 0.5.2 Index]