# Copyright 2019 The Google Research Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Utility functions used by generate_graph.py."""
from __future__ import absolute_import, division, print_function
import hashlib
import itertools
import numpy as np
[docs]def gen_is_edge_fn(bits):
"""Generate a boolean function for the edge connectivity.
Given a bitstring FEDCBA and a 4x4 matrix, the generated matrix is
[[0, A, B, D],
[0, 0, C, E],
[0, 0, 0, F],
[0, 0, 0, 0]]
Note that this function is agnostic to the actual matrix dimension due to
order in which elements are filled out (column-major, starting from least
significant bit). For example, the same FEDCBA bitstring (0-padded) on a 5x5
matrix is
[[0, A, B, D, 0],
[0, 0, C, E, 0],
[0, 0, 0, F, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]]
Args:
bits: integer which will be interpreted as a bit mask.
Returns:
vectorized function that returns True when an edge is present.
"""
def is_edge(x, y):
"""Is there an edge from x to y (0-indexed)?"""
if x >= y:
return 0
# Map x, y to index into bit string
index = x + (y * (y - 1) // 2)
return (bits >> index) % 2 == 1
return np.vectorize(is_edge)
[docs]def is_full_dag(matrix):
"""Full DAG == all vertices on a path from vert 0 to (V-1).
i.e. no disconnected or "hanging" vertices.
It is sufficient to check for:
1) no rows of 0 except for row V-1 (only output vertex has no out-edges)
2) no cols of 0 except for col 0 (only input vertex has no in-edges)
Args:
matrix: V x V upper-triangular adjacency matrix
Returns:
True if the there are no dangling vertices.
"""
shape = np.shape(matrix)
rows = matrix[:shape[0]-1, :] == 0
rows = np.all(rows, axis=1) # Any row with all 0 will be True
rows_bad = np.any(rows)
cols = matrix[:, 1:] == 0
cols = np.all(cols, axis=0) # Any col with all 0 will be True
cols_bad = np.any(cols)
return (not rows_bad) and (not cols_bad)
[docs]def num_edges(matrix):
"""Computes number of edges in adjacency matrix."""
return np.sum(matrix)
[docs]def hash_module(matrix, labeling):
"""Computes a graph-invariance MD5 hash of the matrix and label pair.
Args:
matrix: np.ndarray square upper-triangular adjacency matrix.
labeling: list of int labels of length equal to both dimensions of
matrix.
Returns:
MD5 hash of the matrix and labeling.
"""
vertices = np.shape(matrix)[0]
in_edges = np.sum(matrix, axis=0).tolist()
out_edges = np.sum(matrix, axis=1).tolist()
assert len(in_edges) == len(out_edges) == len(labeling)
hashes = list(zip(out_edges, in_edges, labeling))
hashes = [hashlib.md5(str(h).encode('utf-8')).hexdigest() for h in hashes]
# Computing this up to the diameter is probably sufficient but since the
# operation is fast, it is okay to repeat more times.
for _ in range(vertices):
new_hashes = []
for v in range(vertices):
in_neighbors = [hashes[w] for w in range(vertices) if matrix[w, v]]
out_neighbors = [hashes[w] for w in range(vertices) if matrix[v, w]]
new_hashes.append(hashlib.md5(
(''.join(sorted(in_neighbors)) + '|' +
''.join(sorted(out_neighbors)) + '|' +
hashes[v]).encode('utf-8')).hexdigest())
hashes = new_hashes
fingerprint = hashlib.md5(str(sorted(hashes)).encode('utf-8')).hexdigest()
return fingerprint
[docs]def permute_graph(graph, label, permutation):
"""Permutes the graph and labels based on permutation.
Args:
graph: np.ndarray adjacency matrix.
label: list of labels of same length as graph dimensions.
permutation: a permutation list of ints of same length as graph dimensions.
Returns:
np.ndarray where vertex permutation[v] is vertex v from the original graph
"""
# vertex permutation[v] in new graph is vertex v in the old graph
forward_perm = zip(permutation, list(range(len(permutation))))
inverse_perm = [x[1] for x in sorted(forward_perm)]
edge_fn = lambda x, y: graph[inverse_perm[x], inverse_perm[y]] == 1
new_matrix = np.fromfunction(np.vectorize(edge_fn),
(len(label), len(label)),
dtype=np.int8)
new_label = [label[inverse_perm[i]] for i in range(len(label))]
return new_matrix, new_label
[docs]def is_isomorphic(graph1, graph2):
"""Exhaustively checks if 2 graphs are isomorphic."""
matrix1, label1 = np.array(graph1[0]), graph1[1]
matrix2, label2 = np.array(graph2[0]), graph2[1]
assert np.shape(matrix1) == np.shape(matrix2)
assert len(label1) == len(label2)
vertices = np.shape(matrix1)[0]
# Note: input and output in our constrained graphs always map to themselves
# but this script does not enforce that.
for perm in itertools.permutations(range(0, vertices)):
pmatrix1, plabel1 = permute_graph(matrix1, label1, perm)
if np.array_equal(pmatrix1, matrix2) and plabel1 == label2:
return True
return False
