The networkx_algo_common_subtree Module

Networkx algorithms for maximum common ordered subtree minors (or embedding) and maximum common subtree isomorphism. Contains pure python and cython optimized versions.

At its core the maximum_common_ordered_subtree_embedding function is an implementation of:

Lozano, Antoni, and Gabriel Valiente.
    "On the maximum common embedded subtree problem for ordered trees."
    String Algorithmics (2004): 155-170.
    https://pdfs.semanticscholar.org/0b6e/061af02353f7d9b887f9a378be70be64d165.pdf

And maximum_common_ordered_subtree_isomorphism is a variant of the above algorithm that returns common subtree ismorphism instead of subtree minors.

Demo

Consider two directed ordered trees (the algorithm also works on undirected ordered trees):

>>> import networkx as nx
>>> tree1 = nx.DiGraph()
>>> tree2 = nx.DiGraph()
>>> tree1.add_edges_from([(0, 2), (5, 7), (5, 4), (5, 3), (2, 5), (2, 6), (3, 1)])
>>> tree2.add_edges_from([(0, 6), (0, 1), (1, 5), (6, 2), (5, 4), (5, 3), (5, 7)])
>>> print('Tree 1:')
>>> nx.write_network_text(tree1)
>>> print('Tree 2:')
>>> nx.write_network_text(tree2)
Tree 1:
╙── 0
    └─╼ 2
        ├─╼ 5
        │   ├─╼ 7
        │   ├─╼ 4
        │   └─╼ 3
        │       └─╼ 1
        └─╼ 6
Tree 2:
╙── 0
    ├─╼ 6
    │   └─╼ 2
    └─╼ 1
        └─╼ 5
            ├─╼ 4
            ├─╼ 3
            └─╼ 7

The maximum common ordered isomorphism (a strict common subtree structure) is:

>>> import networkx_algo_common_subtree
>>> subtree1, subtree2, score = networkx_algo_common_subtree.maximum_common_ordered_subtree_isomorphism(tree1, tree2)
>>> print(f'{score=}')
>>> print('Isomorphic Subtree 1:')
>>> nx.write_network_text(subtree1)
>>> print('Isomorphic Subtree 2:')
>>> nx.write_network_text(subtree2)
score=3
Isomorphic Subtree 1:
╙── 5
    ├─╼ 4
    └─╼ 3
Isomorphic Subtree 2:
╙── 5
    ├─╼ 4
    └─╼ 3

The biggest common structure between both of the original trees is the subtree involving 5, 3, 4. Notice that the (5, 7) edge is not included because these are ordered subtrees. In tree 1, the (5, 7) edge is before the (5, 4) edge and in the second it is after it, so it is not included because the edges are not in the same order.

This substructure can be generalized by allowing edges in the original graphs to be collapsed, which can produce larger common substructures. This is the maximum common ordered embedding.

>>> import networkx_algo_common_subtree
>>> subtree1, subtree2, score = networkx_algo_common_subtree.maximum_common_ordered_subtree_embedding(tree1, tree2)
>>> print(f'{score=}')
>>> print('Embedded Subtree 1:')
>>> nx.write_network_text(subtree1)
>>> print('Embedded Subtree 2:')
>>> nx.write_network_text(subtree2)
score=4
Embedded Subtree 1:
╙── 0
    └─╼ 5
        ├─╼ 4
        └─╼ 3
Embedded Subtree 2:
╙── 0
    └─╼ 5
        ├─╼ 4
        └─╼ 3

In this example, the edges (0, 2) and (2, 5) in first tree were collapsed into (0, 5). Similarly in the second tree the edges (0, 1) and (1, 5) were collapsed into (0, 5), thus increasing the size of the common ordered subtree.

Other Information

Standalone versions of code were originally submitted as PRs to networkx proper:

networkx/networkx#4350 networkx/networkx#4327

However, these algorithms are roughly O(N⁴), they require a a fast binary (e.g. C / cython) implementation to work on graphs of reasonable size. Thus they are unlikely to be added to mainline networkx.

These algorithms are components of algorithms in torch_liberator, see related information:

TorchLiberator	https://gitlab.kitware.com/computer-vision/torch_liberator
Torch Hackathon 2021	Youtube Video and Google Slides