# Shortest (directed or undirected) paths between vertices

Source:`R/paths.R`

, `R/structural.properties.R`

`distances.Rd`

`distances()`

calculates the length of all the shortest paths from
or to the vertices in the network. `shortest_paths()`

calculates one
shortest path (the path itself, and not just its length) from or to the
given vertex.

## Usage

```
distance_table(graph, directed = TRUE)
mean_distance(
graph,
weights = NULL,
directed = TRUE,
unconnected = TRUE,
details = FALSE
)
distances(
graph,
v = V(graph),
to = V(graph),
mode = c("all", "out", "in"),
weights = NULL,
algorithm = c("automatic", "unweighted", "dijkstra", "bellman-ford", "johnson",
"floyd-warshall")
)
shortest_paths(
graph,
from,
to = V(graph),
mode = c("out", "all", "in"),
weights = NULL,
output = c("vpath", "epath", "both"),
predecessors = FALSE,
inbound.edges = FALSE,
algorithm = c("automatic", "unweighted", "dijkstra", "bellman-ford")
)
all_shortest_paths(
graph,
from,
to = V(graph),
mode = c("out", "all", "in"),
weights = NULL
)
```

## Arguments

- graph
The graph to work on.

- directed
Whether to consider directed paths in directed graphs, 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). In a weighted graph, the length of a path is the sum of the weights of its constituent edges.- 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 considered as having infinite length, making the mean distance infinite as well.

- details
Whether to provide additional details in the result. Functions accepting this argument (like

`mean_distance()`

) return additional information like the number of disconnected vertex pairs in the result when this parameter is set to`TRUE`

.- v
Numeric vector, the vertices from which the shortest paths will be calculated.

- to
Numeric vector, the vertices to which the shortest paths will be calculated. By default it includes all vertices. Note that for

`distances()`

every vertex must be included here at most once. (This is not required for`shortest_paths()`

.- 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 graph is treated as undirected, i.e. edge directions are not taken into account. This argument is ignored for undirected graphs.- 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.

- output
Character scalar, defines how to report the shortest paths. “vpath” means that the vertices along the paths are reported, this form was used prior to igraph version 0.6. “epath” means that the edges along the paths are reported. “both” means that both forms are returned, in a named list with components “vpath” and “epath”.

- predecessors
Logical scalar, whether to return the predecessor vertex for each vertex. The predecessor of vertex

`i`

in the tree is the vertex from which vertex`i`

was reached. The predecessor of the start vertex (in the`from`

argument) is itself by definition. If the predecessor is zero, it means that the given vertex was not reached from the source during the search. Note that the search terminates if all the vertices in`to`

are reached.- inbound.edges
Logical scalar, whether to return the inbound edge for each vertex. The inbound edge of vertex

`i`

in the tree is the edge via which vertex`i`

was reached. The start vertex and vertices that were not reached during the search will have zero in the corresponding entry of the vector. Note that the search terminates if all the vertices in`to`

are reached.

## Value

For `distances()`

a numeric matrix with `length(to)`

columns and `length(v)`

rows. The shortest path length from a vertex to
itself is always zero. For unreachable vertices `Inf`

is included.

For `shortest_paths()`

a named list with four entries is returned:

- vpath
This itself is a list, of length

`length(to)`

; list element`i`

contains the vertex ids on the path from vertex`from`

to vertex`to[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. If this output is not requested in the`output`

argument, then it will be`NULL`

.- epath
This is a list similar to

`vpath`

, but the vectors of the list contain the edge ids along the shortest paths, instead of the vertex ids. This entry is set to`NULL`

if it is not requested in the`output`

argument.- predecessors
Numeric vector, the predecessor of each vertex in the

`to`

argument, or`NULL`

if it was not requested.- inbound_edges
Numeric vector, the inbound edge for each vertex, or

`NULL`

, if it was not requested.

For `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 `mean_distance()`

a single number is returned if `details=FALSE`

,
or a named list with two entries: `res`

is the mean distance as a numeric
scalar and `unconnected`

is the number of unconnected vertex pairs,
also as a numeric scalar.

`distance_table()`

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. In undirected and directed graphs, unorderde
and ordered pairs are considered, respectively. Therefore the sum of the
two entries is always \(n(n-1)\) for directed graphs and \(n(n-1)/2\)
for undirected graphs.

## Details

The shortest path, or geodesic between two pair of vertices is a path with the minimal number of vertices. The functions documented in this manual page all calculate shortest paths between vertex pairs.

`distances()`

calculates the lengths of pairwise shortest paths from
a set of vertices (`from`

) to another set of vertices (`to`

). It
uses different algorithms, depending on the `algorithm`

argument and
the `weight`

edge attribute of the graph. The implemented algorithms
are breadth-first search (‘`unweighted`

’), this only works for
unweighted graphs; the Dijkstra algorithm (‘`dijkstra`

’), this
works for graphs with non-negative edge weights; the Bellman-Ford algorithm
(‘`bellman-ford`

’); Johnson's algorithm
(‘`johnson`

’); and a faster version of the Floyd-Warshall algorithm
with expected quadratic running time (‘`floyd-warshall`

’). The latter
three algorithms work with arbitrary
edge weights, but (naturally) only for graphs that don't have a negative
cycle. Note that a negative-weight edge in an undirected graph implies
such a cycle. Johnson's algorithm performs better than the Bellman-Ford
one when many source (and target) vertices are given, with all-pairs
shortest path length calculations being the typical use case.

igraph can choose automatically between algorithms, and chooses the most
efficient one that is appropriate for the supplied weights (if any). For
automatic algorithm selection, supply ‘`automatic`

’ as the
`algorithm`

argument. (This is also the default.)

`shortest_paths()`

calculates a single shortest path (i.e. the path
itself, not just its length) between the source vertex given in `from`

,
to the target vertices given in `to`

. `shortest_paths()`

uses
breadth-first search for unweighted graphs and Dijkstra's algorithm for
weighted graphs. The latter only works if the edge weights are non-negative.

`all_shortest_paths()`

calculates *all* shortest paths between
pairs of vertices, including several shortest paths of the same length.
More precisely, it computerd all shortest path starting at `from`

, and
ending at any vertex given in `to`

. It uses a breadth-first search for
unweighted graphs and Dijkstra's algorithm for weighted ones. The latter
only supports non-negative edge weights. Caution: in multigraphs, the
result size is exponentially large in the number of vertex pairs with
multiple edges between them.

`mean_distance()`

calculates the average path length in a graph, by
calculating the shortest paths between all pairs of vertices (both ways for
directed graphs). It uses a breadth-first search for unweighted graphs and
Dijkstra's algorithm for weighted ones. The latter only supports non-negative
edge weights.

`distance_table()`

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.

## References

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

## See also

Other structural.properties:
`bfs()`

,
`component_distribution()`

,
`connect()`

,
`constraint()`

,
`coreness()`

,
`degree()`

,
`dfs()`

,
`edge_density()`

,
`feedback_arc_set()`

,
`girth()`

,
`is_acyclic()`

,
`is_dag()`

,
`is_matching()`

,
`k_shortest_paths()`

,
`knn()`

,
`reciprocity()`

,
`subcomponent()`

,
`subgraph()`

,
`topo_sort()`

,
`transitivity()`

,
`unfold_tree()`

,
`which_multiple()`

,
`which_mutual()`

Other paths:
`all_simple_paths()`

,
`diameter()`

,
`eccentricity()`

,
`graph_center()`

,
`radius()`

## Author

Gabor Csardi csardi.gabor@gmail.com

## Related documentation in the C library

`igraph_path_length_hist()`

, `igraph_average_path_length_dijkstra()`

.

## Examples

```
g <- make_ring(10)
distances(g)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 0 1 2 3 4 5 4 3 2 1
#> [2,] 1 0 1 2 3 4 5 4 3 2
#> [3,] 2 1 0 1 2 3 4 5 4 3
#> [4,] 3 2 1 0 1 2 3 4 5 4
#> [5,] 4 3 2 1 0 1 2 3 4 5
#> [6,] 5 4 3 2 1 0 1 2 3 4
#> [7,] 4 5 4 3 2 1 0 1 2 3
#> [8,] 3 4 5 4 3 2 1 0 1 2
#> [9,] 2 3 4 5 4 3 2 1 0 1
#> [10,] 1 2 3 4 5 4 3 2 1 0
shortest_paths(g, 5)
#> $vpath
#> $vpath[[1]]
#> + 5/10 vertices, from d7e5357:
#> [1] 5 4 3 2 1
#>
#> $vpath[[2]]
#> + 4/10 vertices, from d7e5357:
#> [1] 5 4 3 2
#>
#> $vpath[[3]]
#> + 3/10 vertices, from d7e5357:
#> [1] 5 4 3
#>
#> $vpath[[4]]
#> + 2/10 vertices, from d7e5357:
#> [1] 5 4
#>
#> $vpath[[5]]
#> + 1/10 vertex, from d7e5357:
#> [1] 5
#>
#> $vpath[[6]]
#> + 2/10 vertices, from d7e5357:
#> [1] 5 6
#>
#> $vpath[[7]]
#> + 3/10 vertices, from d7e5357:
#> [1] 5 6 7
#>
#> $vpath[[8]]
#> + 4/10 vertices, from d7e5357:
#> [1] 5 6 7 8
#>
#> $vpath[[9]]
#> + 5/10 vertices, from d7e5357:
#> [1] 5 6 7 8 9
#>
#> $vpath[[10]]
#> + 6/10 vertices, from d7e5357:
#> [1] 5 4 3 2 1 10
#>
#>
#> $epath
#> NULL
#>
#> $predecessors
#> NULL
#>
#> $inbound_edges
#> NULL
#>
all_shortest_paths(g, 1, 6:8)
#> $vpaths
#> $vpaths[[1]]
#> + 6/10 vertices, from d7e5357:
#> [1] 1 10 9 8 7 6
#>
#> $vpaths[[2]]
#> + 6/10 vertices, from d7e5357:
#> [1] 1 2 3 4 5 6
#>
#> $vpaths[[3]]
#> + 5/10 vertices, from d7e5357:
#> [1] 1 10 9 8 7
#>
#> $vpaths[[4]]
#> + 4/10 vertices, from d7e5357:
#> [1] 1 10 9 8
#>
#>
#> $epaths
#> $epaths[[1]]
#> [1] 10 9 8 7 6
#>
#> $epaths[[2]]
#> [1] 1 2 3 4 5
#>
#> $epaths[[3]]
#> [1] 10 9 8 7
#>
#> $epaths[[4]]
#> [1] 10 9 8
#>
#>
#> $nrgeo
#> [1] 1 1 1 1 1 2 1 1 1 1
#>
#> $res
#> $res[[1]]
#> + 6/10 vertices, from d7e5357:
#> [1] 1 10 9 8 7 6
#>
#> $res[[2]]
#> + 6/10 vertices, from d7e5357:
#> [1] 1 2 3 4 5 6
#>
#> $res[[3]]
#> + 5/10 vertices, from d7e5357:
#> [1] 1 10 9 8 7
#>
#> $res[[4]]
#> + 4/10 vertices, from d7e5357:
#> [1] 1 10 9 8
#>
#>
mean_distance(g)
#> [1] 2.777778
## Weighted shortest paths
el <- matrix(
ncol = 3, byrow = TRUE,
c(
1, 2, 0,
1, 3, 2,
1, 4, 1,
2, 3, 0,
2, 5, 5,
2, 6, 2,
3, 2, 1,
3, 4, 1,
3, 7, 1,
4, 3, 0,
4, 7, 2,
5, 6, 2,
5, 8, 8,
6, 3, 2,
6, 7, 1,
6, 9, 1,
6, 10, 3,
8, 6, 1,
8, 9, 1,
9, 10, 4
)
)
g2 <- add_edges(make_empty_graph(10), t(el[, 1:2]), weight = el[, 3])
distances(g2, mode = "out")
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 0 0 0 1 5 2 1 13 3 5
#> [2,] Inf 0 0 1 5 2 1 13 3 5
#> [3,] Inf 1 0 1 6 3 1 14 4 6
#> [4,] Inf 1 0 0 6 3 1 14 4 6
#> [5,] Inf 5 4 5 0 2 3 8 3 5
#> [6,] Inf 3 2 3 8 0 1 16 1 3
#> [7,] Inf Inf Inf Inf Inf Inf 0 Inf Inf Inf
#> [8,] Inf 4 3 4 9 1 2 0 1 4
#> [9,] Inf Inf Inf Inf Inf Inf Inf Inf 0 4
#> [10,] Inf Inf Inf Inf Inf Inf Inf Inf Inf 0
```