# scipy.sparse.csgraph.minimum_spanning_tree¶

scipy.sparse.csgraph.minimum_spanning_tree(csgraph, overwrite=False)

Return a minimum spanning tree of an undirected graph

A minimum spanning tree is a graph consisting of the subset of edges which together connect all connected nodes, while minimizing the total sum of weights on the edges. This is computed using the Kruskal algorithm.

New in version 0.11.0.

Parameters: csgraph : array_like or sparse matrix, 2 dimensions The N x N matrix representing an undirected graph over N nodes (see notes below). overwrite : bool, optional if true, then parts of the input graph will be overwritten for efficiency. span_tree : csr matrix The N x N compressed-sparse representation of the undirected minimum spanning tree over the input (see notes below).

Notes

This routine uses undirected graphs as input and output. That is, if graph[i, j] and graph[j, i] are both zero, then nodes i and j do not have an edge connecting them. If either is nonzero, then the two are connected by the minimum nonzero value of the two.

Examples

The following example shows the computation of a minimum spanning tree over a simple four-component graph:

 input graph             minimum spanning tree

(0)                         (0)
/   \                       /
3     8                     3
/       \                   /
(3)---5---(1)               (3)---5---(1)
\       /                           /
6     2                           2
\   /                           /
(2)                         (2)


It is easy to see from inspection that the minimum spanning tree involves removing the edges with weights 8 and 6. In compressed sparse representation, the solution looks like this:

>>> from scipy.sparse import csr_matrix
>>> from scipy.sparse.csgraph import minimum_spanning_tree
>>> X = csr_matrix([[0, 8, 0, 3],
...                 [0, 0, 2, 5],
...                 [0, 0, 0, 6],
...                 [0, 0, 0, 0]])
>>> Tcsr = minimum_spanning_tree(X)
>>> Tcsr.toarray().astype(int)
array([[0, 0, 0, 3],
[0, 0, 2, 5],
[0, 0, 0, 0],
[0, 0, 0, 0]])


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