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.

csgrapharray_like or sparse matrix, 2 dimensions

The N x N matrix representing an undirected graph over N nodes (see notes below).

overwritebool, optional

If true, then parts of the input graph will be overwritten for efficiency. Default is False.

span_treecsr matrix

The N x N compressed-sparse representation of the undirected minimum spanning tree over the input (see notes below).


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.

This routine loses precision when users input a dense matrix. Small elements < 1E-8 of the dense matrix are rounded to zero. All users should input sparse matrices if possible to avoid it.

If the graph is not connected, this routine returns the minimum spanning forest, i.e. the union of the minimum spanning trees on each connected component.

If multiple valid solutions are possible, output may vary with SciPy and Python version.


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]])