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# graphallshortestpaths

Find all shortest paths in graph

## Syntax

[dist] = graphallshortestpaths(G)

[dist] = graphallshortestpaths(G, ...'Directed', DirectedValue, ...)
[dist] = graphallshortestpaths(G, ...'Weights', WeightsValue, ...)

## Arguments

 G N-by-N sparse matrix that represents a graph. Nonzero entries in matrix G represent the weights of the edges. DirectedValue Property that indicates whether the graph is directed or undirected. Enter false for an undirected graph. This results in the upper triangle of the sparse matrix being ignored. Default is true. WeightsValue Column vector that specifies custom weights for the edges in matrix G. It must have one entry for every nonzero value (edge) in matrix G. The order of the custom weights in the vector must match the order of the nonzero values in matrix G when it is traversed column-wise. This property lets you use zero-valued weights. By default, graphallshortestpaths gets weight information from the nonzero entries in matrix G.

## Description

 Tip   For introductory information on graph theory functions, see Graph Theory Functions.

[dist] = graphallshortestpaths(G) finds the shortest paths between every pair of nodes in the graph represented by matrix G, using Johnson's algorithm. Input G is an N-by-N sparse matrix that represents a graph. Nonzero entries in matrix G represent the weights of the edges.

Output dist is an N-by-N matrix where dist(S,T) is the distance of the shortest path from source node S to target node T. Elements in the diagonal of this matrix are always 0, indicating the source node and target node are the same. A 0 not in the diagonal indicates that the distance between the source node and target node is 0. An Inf indicates there is no path between the source node and the target node.

Johnson's algorithm has a time complexity of O(N*log(N)+N*E), where N and E are the number of nodes and edges respectively.

[...] = graphallshortestpaths (G, 'PropertyName', PropertyValue, ...) calls graphallshortestpaths with optional properties that use property name/property value pairs. You can specify one or more properties in any order. Each PropertyName must be enclosed in single quotes and is case insensitive. These property name/property value pairs are as follows:

[dist] = graphallshortestpaths(G, ...'Directed', DirectedValue, ...)
indicates whether the graph is directed or undirected. Set DirectedValue to false for an undirected graph. This results in the upper triangle of the sparse matrix being ignored. Default is true.

[dist] = graphallshortestpaths(G, ...'Weights', WeightsValue, ...) lets you specify custom weights for the edges. WeightsValue is a column vector having one entry for every nonzero value (edge) in matrix G. The order of the custom weights in the vector must match the order of the nonzero values in matrix G when it is traversed column-wise. This property lets you use zero-valued weights. By default, graphallshortestpaths gets weight information from the nonzero entries in matrix G.

## Examples

Finding All Shortest Paths in a Directed Graph

1. Create and view a directed graph with 6 nodes and 11 edges.

```W = [.41 .99 .51 .32 .15 .45 .38 .32 .36 .29 .21];
DG = sparse([6 1 2 2 3 4 4 5 5 6 1],[2 6 3 5 4 1 6 3 4 3 5],W)

DG =

(4,1)       0.4500
(6,2)       0.4100
(2,3)       0.5100
(5,3)       0.3200
(6,3)       0.2900
(3,4)       0.1500
(5,4)       0.3600
(1,5)       0.2100
(2,5)       0.3200
(1,6)       0.9900
(4,6)       0.3800

view(biograph(DG,[],'ShowWeights','on'))
```

2. Find all the shortest paths between every pair of nodes in the directed graph.

```graphallshortestpaths(DG)

ans =

0    1.3600    0.5300    0.5700    0.2100    0.9500
1.1100         0    0.5100    0.6600    0.3200    1.0400
0.6000    0.9400         0    0.1500    0.8100    0.5300
0.4500    0.7900    0.6700         0    0.6600    0.3800
0.8100    1.1500    0.3200    0.3600         0    0.7400
0.8900    0.4100    0.2900    0.4400    0.7300         0```

The resulting matrix shows the shortest path from node 1 (first row) to node 6 (sixth column) is 0.95. You can see this in the graph by tracing the path from node 1 to node 5 to node 4 to node 6 (0.21 + 0.36 + 0.38 = 0.95).

Finding All Shortest Paths in an Undirected Graph

1. Create and view an undirected graph with 6 nodes and 11 edges.

```UG = tril(DG + DG')

UG =

(4,1)       0.4500
(5,1)       0.2100
(6,1)       0.9900
(3,2)       0.5100
(5,2)       0.3200
(6,2)       0.4100
(4,3)       0.1500
(5,3)       0.3200
(6,3)       0.2900
(5,4)       0.3600
(6,4)       0.3800

view(biograph(UG,[],'ShowArrows','off','ShowWeights','on'))

```

2. Find all the shortest paths between every pair of nodes in the undirected graph.

```graphallshortestpaths(UG,'directed',false)

ans =

0    0.5300    0.5300    0.4500    0.2100    0.8300
0.5300         0    0.5100    0.6600    0.3200    0.7000
0.5300    0.5100         0    0.1500    0.3200    0.5300
0.4500    0.6600    0.1500         0    0.3600    0.3800
0.2100    0.3200    0.3200    0.3600         0    0.7400
0.8300    0.7000    0.5300    0.3800    0.7400         0```

The resulting matrix is symmetrical because it represents an undirected graph. It shows the shortest path from node 1 (first row) to node 6 (sixth column) is 0.83. You can see this in the graph by tracing the path from node 1 to node 4 to node 6 (0.45 + 0. 38 = 0.83). Because UG is an undirected graph, we can use the edge between node 1 and node 4, which we could not do in the directed graph DG.

## References

[1] Johnson, D.B. (1977). Efficient algorithms for shortest paths in sparse networks. Journal of the ACM 24(1), 1-13.

[2] Siek, J.G., Lee, L-Q, and Lumsdaine, A. (2002). The Boost Graph Library User Guide and Reference Manual, (Upper Saddle River, NJ:Pearson Education).