shortestpathtree
Shortest path tree from node
Syntax
Description
returns a directed graph, TR
= shortestpathtree(G
,s
)TR
, 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
.
uses additional options specified by one or more NameValue pair arguments, using
any of the input argument combinations in previous syntaxes. For example,
TR
= shortestpathtree(___,Name,Value
)shortestpathtree(G,s,'OutputForm','vector')
returns a numeric
vector that describes the shortest path tree.
Examples
Shortest Paths from Specified Source Node
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: [13x1 table] Nodes: [8x0 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.
TR = shortestpathtree(G,1); p = plot(G); highlight(p,TR,'EdgeColor','r')
Since there is no path from node 1 to node 7, node 7 is disconnected from the tree.
Shortest Paths to Specified Target Node
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.
TR = shortestpathtree(G,'all',10);
plot(TR)
Subset of Shortest Paths with Specified Source Node
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]; [TR,D] = shortestpathtree(G,23,target,'OutputForm','cell')
TR=5×1 cell array
{[ 23 22 21 4 5 1]}
{[ 23 22 21 4 5]}
{[23 22 20 16 17 15 14 13]}
{[ 23 22 20 19 18 32]}
{[ 23 24 48 47 46 44]}
D = 1×5
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 = TR{2}
path = 1×5
23 22 21 4 5
path_length = D(2)
path_length = 4
Input Arguments
s
— Source node(s)
node indices  node names  'all'
Source node(s), specified as one or more node indices or node names, or as
all nodes in the graph with 'all'
.
When used alone,
s
must specify a single source node.When used together with
t
, thes
andt
inputs must satisfy:s
can be a single source node, andt
can specify multiple target nodes.s
can specify several source nodes, andt
can specify a single target node.
This table shows the different ways to refer to one or more nodes either by their numeric node indices or by their node names.
Form  Single Node  Multiple Nodes 

Node index  Scalar Example:  Vector Example: 
Node name  Character vector Example:  Cell array of character vectors Example: 
String scalar Example:  String array Example: 
s
must not specify the node name
'all'
, since this node name conflicts with an option
name. Use findnode
to instead pass in the node index for
this case.
Example: shortestpathtree(G,'a')
Example: shortestpathtree(G,[1 2 3],8)
t
— Target node(s)
'all'
(default)  node indices  node names
Target node(s), specified as one or more node indices or node names, or as
all nodes in the graph with 'all'
.
The s
and t
inputs must
satisfy:
s
can be a single source node, andt
can specify multiple target nodes.s
can specify several source nodes, andt
can specify a single target node.
t
must not specify nodes named
'all'
, 'Method'
, or
'OutputForm'
, since these node names conflict with
option names. Use findnode
to instead pass in the node
index for these cases.
Example: shortestpathtree(G,[1 2 3],8)
Example: shortestpathtree(G,{'a','b','c'},{'f'})
NameValue Arguments
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
[TR,D] =
shortestpathtree(G,s,t,'Method','unweighted','OutputForm','vector')
OutputForm
— Format of output
'tree'
(default)  'cell'
 'vector'
Format of output, specified as the commaseparated pair consisting of
'OutputForm'
and one of the options in the
table.
Option  Description 

'tree' (default) 

'cell' 
If If specified, the third output

'vector' 
In each case
If specified, the third output

Example: shortestpathtree(G,s,'OutputForm','vector')
Method
— Shortest path algorithm
'auto'
(default)  'unweighted'
 'positive'
 'mixed'
 'acyclic'
Shortest path algorithm, specified as the commaseparated pair
consisting of 'Method'
and one of the options in the
table.
Option  Description 

'auto' (default) 
The

'unweighted' 
BreadthFirst computation that treats all edge
weights as 
'positive' 
Dijkstra algorithm that requires all edge weights to be nonnegative. 
'mixed' (only for
digraph ) 
BellmanFord algorithm for directed graphs that requires the graph to have no negative cycles. While 
'acyclic' (only for
digraph ) 
Algorithm designed to improve performance for directed, acyclic graphs (DAGs) with weighted edges. Use 
Note
For most graphs, 'unweighted'
is the fastest
algorithm, followed by 'acyclic'
,
'positive'
, and
'mixed'
.
Example: shortestpath(G,s,t,'Method','acyclic')
Output Arguments
TR
— Shortest path tree
digraph
object (default)  cell array  vector
Shortest path tree, returned as a digraph
object, cell
array, or vector, depending on the value of 'OutputForm'
. Use
the highlight
function to
visualize the shortest path tree on top of a plot of the graph, or use
plot(TR)
to visualize the shortest path tree on its
own.
If there are multiple shortest paths between two nodes, then
TR
contains only one of the paths. The path that is
returned can change depending on which algorithm is specified by
Method
. The TR
output is a graph
with zero edges if there are no paths connecting any of the specified
nodes.
D
— Distance between source and target nodes
vector
Distance between source and target nodes, returned as a vector. A value of
Inf
indicates there is no path between two
nodes.
E
— Edges in tree or on path
logical vector (default)  cell array  vector
Edges in tree or on path, returned as a logical vector, cell array, or
vector, depending on the value of 'OutputForm'
:
If you don't specify
'OutputForm'
or specify a value of'tree'
, thenE
is a logical vector indicating whether each graph edge is in directed graphTR
. This output is compatible with the'Edges'
namevalue pair ofhighlight
, for example:highlight(p,'Edges',E)
.If
'OutputForm'
is'cell'
, thenE
is a cell array containing the edges on the corresponding paths inTR
.If
'OutputForm'
is'vector'
, thenE
is a vector which, for each node, gives the index of the edge connecting it to its parent node in the shortest path tree.
Tips
The
shortestpath
,shortestpathtree
, anddistances
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
shortestpath
 distances
 nearest
 graph
 digraph
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