yamada

Python implementation of the Yamada-Kataoka-Watanabe algorithm to find all minimum spanning trees in an undirected graph.

Implementation mostly follows the ALL_MST2 algorithm outlined in the original paper. The implementation differs slightly by performing a breadth-first search in liue of a depth-first search. This modification was made so that more variable spanning trees were returned when capping the total number of trees returned.

Original Paper

Yamada, T. Kataoka, S. Watanabe, K. "Listing all the minimum spanning trees in an undirected graph". International Journal of Computer Mathematics. Vol 87, No. 14. pp. 3175 - 3185. November 2010.

Installation

The module can be installed via pip with the command:

pip install yamada-mst

Tests

Proper implementation was tested using the examples found in the original paper, and implementation of those tests can be found in the test subdirectory. The graph structure used in Figure 3 of the original paper, is used to explicitly test for exact minimum spanning tree membership. Meanwhile, the unit-weight, complete graphs k_i are tested for unique membership and expected length for i in {3, 4, 5, 6}. The Substitute() algorithm is tested using the example found in table 3 of the original paper.

To run the tests simply execute the following command:

python tests/test_yamada.py

Dependencies

This module depends on the numpy, networkx, collections, sortedcontainers, sys, and unittest packages, and was written in Python 3.6. The exact requirements can be found in the requirements.txt file. A yamada.yaml file is also provided for conda environment creation.

Example

import yamada
import networkx as nx
 
example = {1: {2: {'weight': 2},
               3: {'weight': 1}},
           2: {1: {'weight': 2},
               3: {'weight': 3},
               4: {'weight': 1}},
           3: {1: {'weight': 1},
               2: {'weight': 3},
               4: {'weight': 2},
               5: {'weight': 2}},
           4: {2: {'weight': 1},
               3: {'weight': 2},
               5: {'weight': 1},
               6: {'weight': 3}},
           5: {3: {'weight': 2},
               4: {'weight': 1},
               6: {'weight': 3}},
           6: {4: {'weight': 3},
               5: {'weight': 3}}}
graph = nx.Graph(example)

# retrieve all minimum spanning trees 
graph_yamada = yamada.Yamada(graph)
all_msts = graph_yamada.spanning_trees()
print(len(all_msts))

# retrieve fixed number of minimum spanning trees
graph_yamada = yamada.Yamada(graph, n_trees=3)
msts = graph_yamada.spanning_trees()
print(len(msts))