shortestpathtree

Shortest path tree from node

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Syntax

  • tree = shortestpathtree(G,s)
    example
  • tree = shortestpathtree(G,s,t)
    example
  • tree = shortestpathtree(___,Name,Value)
    example
  • [tree,D] = shortestpathtree(___)
    example

Description

example

tree = shortestpathtree(G,s) returns a directed graph, tree, that contains the tree of shortest paths from source node s to all other nodes in the graph. 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.

example

tree = shortestpathtree(G,s,t) computes the tree of shortest paths between multiple source or target nodes:

  • s can be a single source node, and t can specify multiple target nodes.

  • s can specify several source nodes, and t can specify a single target node.

example

tree = shortestpathtree(___,Name,Value) uses additional options specified by one or more Name-Value pair arguments, using any of the input argument combinations in previous syntaxes. For example, shortestpathtree(G,s,'OutputForm','vector') returns a numeric vector that describes the shortest path tree.

example

[tree,D] = shortestpathtree(___) additionally returns the shortest path distance between nodes in the tree.

Examples

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Open Script

Find the shortest paths from a source node to each of the other reachable nodes in a graph, and plot the results.

Create a directed graph.

s = [1 1 2 3 3 4 4 6 6 7 8 7 5];
t = [2 3 4 4 5 5 6 1 8 1 3 2 8];
G = digraph(s,t)

G = 

  digraph with properties:

    Edges: [13×1 table]
    Nodes: [8×0 table]

Calculate the shortest paths from node 1 to each of the other reachable nodes in the graph. Then, plot the resulting tree on top of the graph.

tree = shortestpathtree(G,1);
p = plot(G);
highlight(p,tree,'EdgeColor','r')

Since there is no path from node 1 to node 7, node 7 is disconnected from the tree.

Open Script

Find the shortest paths from each node in a graph to a target node, and plot the results.

Create and plot a graph.

s = [1 1 1 1 1 1 1 2 2 7 7 7 7 9 9 3 3 1 6 4 8 10 6 8 4 5];
t = [2 3 4 5 6 8 7 6 7 5 6 8 9 6 8 6 10 10 10 10 10 11 11 11 8 8];
G = graph(s,t);
x = [0 0.5 -0.5 -0.5 0.5 0 1.5 0 2 -1.5 -2];
y = [0 0.5 0.5 -0.5 -0.5 2 0 -2 0 0 0];
plot(G,'XData',x,'YData',y)

Find the shortest paths from each node in the graph to node 10. Plot the resulting tree.

tree = shortestpathtree(G,'all',10);
plot(tree)

Open Script

Find the shortest paths and path lengths from a single source node to several target nodes.

Create and plot a graph.

G = digraph(bucky);
plot(G)

Find the shortest paths from node 23 to several other nodes. Specify OutputForm as cell to return the shortest paths in a cell array. Specify two outputs to also return the shortest path distances.

target = [1 5 13 32 44];
[tree,D] = shortestpathtree(G,23,target,'OutputForm','cell')

tree =

  5×1 cell array

    [1×6 double]
    [1×5 double]
    [1×8 double]
    [1×6 double]
    [1×6 double]


D =

     5     4     7     5     5

tree{j} is the shortest path from node 23 to node target(j) with length D(j).

Find the path and path length from node 21 to node 5.

path = tree{2}

path =

    23    22    21     4     5


path_length = D(2)

path_length =

     4

Input Arguments

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Input graph, specified as either a graph or digraph object. Use the graph function to create an undirected graph or the digraph function to create a directed graph.

For more information, see graph or digraph.

Example: G = graph(1,2)

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

Source node(s), specified as a single node with a scalar node index or character vector node name, as multiple nodes with a numeric vector or cell array of character vectors, or as all nodes in the graph with 'all'.

  • When used alone, s must specify a single source node.

  • When used together with t, the s and t inputs must satisfy:

    • s can be a single source node, and t can specify multiple target nodes.

    • s can specify several source nodes, and t can specify a single target node.

Example: shortestpathtree(G,'a')

Example: shortestpathtree(G,[1 2 3],8)

Target node(s), specified as a single node with a scalar node index, as multiple nodes with a numeric vector or cell array of character vectors, or as all nodes in the graph with 'all'.

The s and t inputs must satisfy:

  • s can be a single source node, and t can specify multiple target nodes.

  • s can specify several source nodes, and t can specify a single target node.

Example: shortestpathtree(G,[1 2 3],8)

Example: shortestpathtree(G,{'a','b','c'},{'f'})

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [tree,D] = shortestpathtree(G,s,t,'Method','unweighted','OutputForm','vector')

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Format of output, specified as the comma-separated pair consisting of 'OutputForm' and one of the options in the table.

OptionDescription
'tree' (default)

tree is a directed graph representing the shortest path tree.

'cell'

tree is a cell array, and tree{k} contains the path from s to t(k) or from s(k) to t. If there is no path between the nodes, then tree{k} is empty.

If s and t are node names, then tree{k} is a cell array of character vectors. Otherwise, tree{k} is a numeric vector.

'vector'

tree is a vector that describes the tree:

  • If s contains a single source node, then tree(k) is the ID of the node that precedes node k on the path from s to k. By convention, tree(s) = 0.

  • If s contains multiple source nodes, then tree(k) is the ID of the node that succeeds node k on the path from k to t. By convention, tree(t) = 0.

In each case tree(k) is NaN if node k is not part of the tree.

Example: shortestpathtree(G,s,'OutputForm','vector')

Shortest path algorithm, specified as the comma-separated pair consisting of 'Method' and one of the options in the table.

OptionDescription
'auto' (default)

The 'auto' option automatically selects the algorithm:

  • 'unweighted' is used for graph and digraph inputs with no edge weights.

  • 'positive' is used for all graph inputs that have edge weights, and requires the weights to be nonnegative. This option is also used for digraph inputs with nonnegative edge weights.

  • 'mixed' is used for digraph inputs whose edge weights contain some negative values. The graph cannot have negative cycles.

'unweighted'

Breadth-First computation that treats all edge weights as 1.

'positive'

Dijkstra algorithm that requires all edge weights to be nonnegative.

'mixed' (only for digraph)

Bellman-Ford algorithm for directed graphs that requires the graph to have no negative cycles.

While 'mixed' is slower than 'positive' for the same problem, 'mixed' is more versatile as it allows some edge weights to be negative.

'acyclic' (only for digraph)

Algorithm designed to improve performance for directed, acyclic graphs (DAGs) with weighted edges.

Use isdag to confirm if a directed graph is acyclic.

    Note:   For most graphs, 'unweighted' is the fastest algorithm, followed by 'acyclic', 'positive', and 'mixed'.

Example: shortestpath(G,s,t,'Method','acyclic')

Output Arguments

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Shortest path tree, returned as a digraph object. Use the highlight function to visualize the shortest path tree on top of a plot of the graph, or use plot(tree) to visualize the shortest path tree on its own.

If there are multiple shortest paths between two nodes, then tree contains only one of the paths. The path that is returned can change depending on which algorithm is specified by Method. The treeoutput is a graph with zero edges if there are no paths connecting any of the specified nodes.

Distance between source and target nodes, returned as a vector. A value of Inf indicates there is no path between two nodes.

More About

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Tips

  • 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.

See Also

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Introduced in R2015b

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