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allpaths

Find all paths between two graph nodes

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Syntax

paths = allpaths(G,s,t)
[paths,edgepaths] = allpaths(G,s,t)
[___] = allpaths(G,s,t,Name,Value)

Description

example

paths= allpaths(G,s,t)returns all paths in graphGthat start at source nodesand end at target nodet. The outputpathsis a cell array where the contents of each cellpaths{k}lists nodes that lie on a path.

example

[paths,edgepaths] = allpaths(G,s,t)also returns the edges on each path fromstot. The outputedgepathsis a cell array whereedgepaths{k}gives the edges along the corresponding path,paths{k}.

example

[___] = allpaths(G,s,t,Name,Value)specifies additional options using one or more name-value arguments. You can use any of the output argument combinations in previous syntaxes. For example, you can specifyMaxNumPathsand a scalar to limit the number of paths returned.

Examples

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Open Live Script

Create an adjacency matrix for a complete graph with four nodes, and then create an undirected graph from the adjacency matrix. Plot the graph.

A = ones(4); G = graph(A); plot(G)

Figure contains an axes object. The axes object contains an object of type graphplot.

Calculate all paths in the graph that begin at node 1 and end at node 3.

paths = allpaths(G,1,3)
paths=5×1 cell array{[ 1 2 3]} {[1 2 4 3]} {[ 1 3]} {[1 4 2 3]} {[ 1 4 3]}
Open Live Script

The second output argument ofallpathsreturns the edges that are along each path. This is particularly useful for multigraphs, where the edge index is required to uniquely identify the edges on the path.

Create a directed multigraph with eight nodes and 18 edges. Specify names for the nodes. Plot the graph with labeled nodes and edges.

s = [1 1 2 2 3 3 2 2 4 6 8 6 6 7 3 3 5 3]; t = [2 3 1 3 2 1 4 4 6 2 6 7 8 8 5 5 7 7]; names = {'A','B','C','D','E','F','G','H'}; G = digraph(s,t,[],names); p = plot(G,'EdgeLabel',1:numedges(G));

Figure contains an axes object. The axes object contains an object of type graphplot.

Calculate all paths between node A and node H. Specify two output arguments to also return the edge indices for edges along each path.

[paths,edgepaths] = allpaths(G,'A','H');

View the nodes and edges along the first path.

paths{1}
ans =1x6 cell{'A'} {'B'} {'C'} {'E'} {'G'} {'H'}
edgepaths{1}
ans =1×51 4 9 13 17

Highlight the nodes and edges along the first path.

highlight(p,'Edges',edgepaths{1},'EdgeColor','r','LineWidth',1.5,'NodeColor','r','MarkerSize',6)

Figure contains an axes object. The axes object contains an object of type graphplot.

Open Live Script

Use the'MaxNumPaths','MaxPathLength', and'MinPathLength'options to limit the number of paths returned byallpaths.

Create an adjacency matrix for a complete graph with 20 nodes. Create an undirected graph from the adjacency matrix, omitting self-loops.

A = ones(20); G = graph(A,“omitselfloops”);

Since all of the nodes in the graph are connected to all other nodes, there are a large number of paths in the graph between any two nodes (more than1.7e16). Therefore, it is not feasible to calculate all of the paths between two nodes since the results will not fit in memory. Instead, calculate the first 10 paths from node 2 to node 5.

paths1 = allpaths(G,2,5,'MaxNumPaths',10)
paths1=10×1 cell array{[ 2 1 3 4 5]} {[ 2 1 3 4 6 5]} {[ 2 1 3 4 6 7 5]} {[ 2 1 3 4 6 7 8 5]} {[ 2 1 3 4 6 7 8 9 5]} {[ 2 1 3 4 6 7 8 9 10 5]} {[ 2 1 3 4 6 7 8 9 10 11 5]} {[ 2 1 3 4 6 7 8 9 10 11 12 5]} {[ 2 1 3 4 6 7 8 9 10 11 12 13 5]} {[2 1 3 4 6 7 8 9 10 11 12 13 14 5]}

Now calculate the first 10 paths between node 2 and node 5 that have a path length less than or equal to 2.

paths2 = allpaths(G,2,5,'MaxNumPaths',10,'MaxPathLength',2)
paths2=10×1 cell array{[ 2 1 5]} {[ 2 3 5]} {[ 2 4 5]} {[ 2 5]} {[ 2 6 5]} {[ 2 7 5]} {[ 2 8 5]} {[ 2 9 5]} {[2 10 5]} {[2 11 5]}

Finally, calculate the first 10 paths between node 2 and node 5 that have a path length greater than or equal to 3.

paths3 = allpaths (G, 2、5、'MaxNumPaths',10,'MinPathLength',3)
paths3=10×1 cell array{[ 2 1 3 4 5]} {[ 2 1 3 4 6 5]} {[ 2 1 3 4 6 7 5]} {[ 2 1 3 4 6 7 8 5]} {[ 2 1 3 4 6 7 8 9 5]} {[ 2 1 3 4 6 7 8 9 10 5]} {[ 2 1 3 4 6 7 8 9 10 11 5]} {[ 2 1 3 4 6 7 8 9 10 11 12 5]} {[ 2 1 3 4 6 7 8 9 10 11 12 13 5]} {[2 1 3 4 6 7 8 9 10 11 12 13 14 5]}

Input Arguments

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输入图,specified as either agraphordigraphobject. Usegraphto create an undirected graph ordigraphto create a directed graph.

Example:G = graph(1,2)

Example:G = digraph([1 2],[2 3])

Source and target node IDs, specified as separate arguments of node indices or node names.

Value Example
Scalar node index 1
Character vector node name 'A'
String scalar node name "A"

Example:allpaths(G,2,5)computes all paths between node 2 and node 5.

Example:allpaths(G,'node1','node2')computes all paths between the named nodesnode1andnode2.

Name-Value Arguments

Specify optional pairs of arguments asName1=Value1,...,NameN=ValueN, whereNameis the argument name andValueis the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and encloseNamein quotes.

Example:allpaths(G,s,t,'MaxNumPaths',100)returns only the first 100 results in the lexicographic ordering of paths.

Maximum number of paths, specified as the comma-separated pair consisting of'MaxNumPaths'and a nonnegative integer scalar. This option is useful when the number of paths between two nodes grows large enough to hit memory limits. You can specifyMaxNumPathsto limit the number of paths returned byallpathsso that the results fit within available memory.

Example:allpaths(G,s,t,'MaxNumPaths',100)

Maximum path length, specified as the comma-separated pair consisting of'MaxPathLength'and a nonnegative integer scalar. This option filters the results returned byallpathsso that no paths with length larger than the specified limit are returned. The length of a path is measured by the number of edges in it, ignoring edge weights.

To find paths with a range of lengths, specify both'MaxPathLength'and'MinPathLength'. To find paths with an exact specified length, specify the same value for both'MaxPathLength'and'MinPathLength'.

Example:allpaths(G,s,t,'MaxPathLength',4)returns paths that have a length less than or equal to 4.

Example:allpaths (G s t MinPathLength',3,'MaxPathLength',5)returns paths that have a length of 3, 4, or 5.

Minimum path length, specified as the comma-separated pair consisting of'MinPathLength'and a nonnegative integer scalar. This option filters the results returned byallpathsso that no paths with length smaller than the specified limit are returned. The length of a path is measured by the number of edges in it, ignoring edge weights.

To find paths with a range of lengths, specify both'MaxPathLength'and'MinPathLength'. To find paths with an exact specified length, specify the same value for both'MaxPathLength'and'MinPathLength'.

Example:allpaths (G s t MinPathLength',2)returns paths that have a length greater than or equal to 2.

Example:allpaths (G s t MinPathLength',3,'MaxPathLength',5)returns paths that have a length of 3, 4, or 5.

Output Arguments

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Paths between specified nodes, returned as a cell array. Each elementpaths{k}包含的节点的路径之一between the specified source and target nodes. The paths are returned in lexicographical order. If the source and target nodessandtspecify the same node, then by conventionallpathsreturns a single path containing that node. If nodetis unreachable from nodes, thenpathsis empty.

The data type of the entries inpathsdepends on the waysandtare specified:

  • Ifsandtare specified as numeric node indices, then each elementpaths{k}is a numeric vector of node indices.

  • Ifsandtare specified as string node names, then each elementpaths{k}is a string array of node names.

  • Ifsandtare specified as character vector node names, then each elementpaths{k}is a cell array of character vector node names.

Edges along each path, returned as a cell array. Each elementedgepaths{k}contains the edge indices for edges that lie along the corresponding path,paths{k}. If nodetis unreachable from nodes, thenedgepathsis empty.

Tips

  • The number of paths in a graph depends heavily on the structure of the graph. For some graph structures, the number of paths can grow exponentially with the number of nodes. For example, a complete graph with 12 nodes given byG = graph(ones(12))contains nearly 10 million paths between any two of its nodes. Use theMaxNumPaths,MaxPathLength, andMinPathLengthname-value pairs to control the output ofallpathsin these cases.

Version History

Introduced in R2021a

See Also

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