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Shortest path distances of all node pairs

`d = distances(G)`

`d = distances(G,s)`

`d = distances(G,s,t)`

`d = distances(___,'Method',algorithm)`

returns a matrix, `d`

= distances(`G`

)`d`

, where `d(i,j)`

is the
length of the shortest path between node `i`

and node
`j`

. If the graph is weighted (that is,
`G.Edges`

contains a variable `Weight`

), then
those weights are used as the distances along the edges in the graph. Otherwise, all
edge distances are taken to be `1`

.

optionally specifies the algorithm to use in computing the shortest path using any
of the input arguments in previous syntaxes. For example, if `d`

= distances(___,'Method',`algorithm`

)`G`

is
a weighted graph, then `distances(G,'Method','unweighted')`

ignores
the edge weights in `G`

and instead treats all edge weights as
`1`

.

The

`shortestpath`

,`shortestpathtree`

, and`distances`

functions do not support undirected graphs with negative edge weights, or more generally any graph containing a negative cycle, for these reasons:A

*negative cycle*is a path that leads from a node back to itself, with the sum of the edge weights on the path being negative. If a negative cycle is on a path between two nodes, then no shortest path exists between the nodes, since a shorter path can always be found by traversing the negative cycle.A single negative edge weight in an undirected graph creates a negative cycle.

`digraph`

| `graph`

| `nearest`

| `shortestpath`

| `shortestpathtree`