209 lines
7.6 KiB
Python
209 lines
7.6 KiB
Python
|
r"""
|
||
|
|
Compressed sparse graph routines (:mod:`scipy.sparse.csgraph`)
|
||
|
|
==============================================================
|
||
|
|
|
||
|
|
.. currentmodule:: scipy.sparse.csgraph
|
||
|
|
|
||
|
|
Fast graph algorithms based on sparse matrix representations.
|
||
|
|
|
||
|
|
Contents
|
||
|
|
--------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
connected_components -- determine connected components of a graph
|
||
|
|
laplacian -- compute the laplacian of a graph
|
||
|
|
shortest_path -- compute the shortest path between points on a positive graph
|
||
|
|
dijkstra -- use Dijkstra's algorithm for shortest path
|
||
|
|
floyd_warshall -- use the Floyd-Warshall algorithm for shortest path
|
||
|
|
bellman_ford -- use the Bellman-Ford algorithm for shortest path
|
||
|
|
johnson -- use Johnson's algorithm for shortest path
|
||
|
|
breadth_first_order -- compute a breadth-first order of nodes
|
||
|
|
depth_first_order -- compute a depth-first order of nodes
|
||
|
|
breadth_first_tree -- construct the breadth-first tree from a given node
|
||
|
|
depth_first_tree -- construct a depth-first tree from a given node
|
||
|
|
minimum_spanning_tree -- construct the minimum spanning tree of a graph
|
||
|
|
reverse_cuthill_mckee -- compute permutation for reverse Cuthill-McKee ordering
|
||
|
|
maximum_flow -- solve the maximum flow problem for a graph
|
||
|
|
maximum_bipartite_matching -- compute a maximum matching of a bipartite graph
|
||
|
|
min_weight_full_bipartite_matching - compute a minimum weight full matching of a bipartite graph
|
||
|
|
structural_rank -- compute the structural rank of a graph
|
||
|
|
NegativeCycleError
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
construct_dist_matrix
|
||
|
|
csgraph_from_dense
|
||
|
|
csgraph_from_masked
|
||
|
|
csgraph_masked_from_dense
|
||
|
|
csgraph_to_dense
|
||
|
|
csgraph_to_masked
|
||
|
|
reconstruct_path
|
||
|
|
|
||
|
|
Graph Representations
|
||
|
|
---------------------
|
||
|
|
This module uses graphs which are stored in a matrix format. A
|
||
|
|
graph with N nodes can be represented by an (N x N) adjacency matrix G.
|
||
|
|
If there is a connection from node i to node j, then G[i, j] = w, where
|
||
|
|
w is the weight of the connection. For nodes i and j which are
|
||
|
|
not connected, the value depends on the representation:
|
||
|
|
|
||
|
|
- for dense array representations, non-edges are represented by
|
||
|
|
G[i, j] = 0, infinity, or NaN.
|
||
|
|
|
||
|
|
- for dense masked representations (of type np.ma.MaskedArray), non-edges
|
||
|
|
are represented by masked values. This can be useful when graphs with
|
||
|
|
zero-weight edges are desired.
|
||
|
|
|
||
|
|
- for sparse array representations, non-edges are represented by
|
||
|
|
non-entries in the matrix. This sort of sparse representation also
|
||
|
|
allows for edges with zero weights.
|
||
|
|
|
||
|
|
As a concrete example, imagine that you would like to represent the following
|
||
|
|
undirected graph::
|
||
|
|
|
||
|
|
G
|
||
|
|
|
||
|
|
(0)
|
||
|
|
/ \
|
||
|
|
1 2
|
||
|
|
/ \
|
||
|
|
(2) (1)
|
||
|
|
|
||
|
|
This graph has three nodes, where node 0 and 1 are connected by an edge of
|
||
|
|
weight 2, and nodes 0 and 2 are connected by an edge of weight 1.
|
||
|
|
We can construct the dense, masked, and sparse representations as follows,
|
||
|
|
keeping in mind that an undirected graph is represented by a symmetric matrix::
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> G_dense = np.array([[0, 2, 1],
|
||
|
|
... [2, 0, 0],
|
||
|
|
... [1, 0, 0]])
|
||
|
|
>>> G_masked = np.ma.masked_values(G_dense, 0)
|
||
|
|
>>> from scipy.sparse import csr_matrix
|
||
|
|
>>> G_sparse = csr_matrix(G_dense)
|
||
|
|
|
||
|
|
This becomes more difficult when zero edges are significant. For example,
|
||
|
|
consider the situation when we slightly modify the above graph::
|
||
|
|
|
||
|
|
G2
|
||
|
|
|
||
|
|
(0)
|
||
|
|
/ \
|
||
|
|
0 2
|
||
|
|
/ \
|
||
|
|
(2) (1)
|
||
|
|
|
||
|
|
This is identical to the previous graph, except nodes 0 and 2 are connected
|
||
|
|
by an edge of zero weight. In this case, the dense representation above
|
||
|
|
leads to ambiguities: how can non-edges be represented if zero is a meaningful
|
||
|
|
value? In this case, either a masked or sparse representation must be used
|
||
|
|
to eliminate the ambiguity::
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> G2_data = np.array([[np.inf, 2, 0 ],
|
||
|
|
... [2, np.inf, np.inf],
|
||
|
|
... [0, np.inf, np.inf]])
|
||
|
|
>>> G2_masked = np.ma.masked_invalid(G2_data)
|
||
|
|
>>> from scipy.sparse.csgraph import csgraph_from_dense
|
||
|
|
>>> # G2_sparse = csr_matrix(G2_data) would give the wrong result
|
||
|
|
>>> G2_sparse = csgraph_from_dense(G2_data, null_value=np.inf)
|
||
|
|
>>> G2_sparse.data
|
||
|
|
array([ 2., 0., 2., 0.])
|
||
|
|
|
||
|
|
Here we have used a utility routine from the csgraph submodule in order to
|
||
|
|
convert the dense representation to a sparse representation which can be
|
||
|
|
understood by the algorithms in submodule. By viewing the data array, we
|
||
|
|
can see that the zero values are explicitly encoded in the graph.
|
||
|
|
|
||
|
|
Directed vs. undirected
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
Matrices may represent either directed or undirected graphs. This is
|
||
|
|
specified throughout the csgraph module by a boolean keyword. Graphs are
|
||
|
|
assumed to be directed by default. In a directed graph, traversal from node
|
||
|
|
i to node j can be accomplished over the edge G[i, j], but not the edge
|
||
|
|
G[j, i]. Consider the following dense graph::
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> G_dense = np.array([[0, 1, 0],
|
||
|
|
... [2, 0, 3],
|
||
|
|
... [0, 4, 0]])
|
||
|
|
|
||
|
|
When ``directed=True`` we get the graph::
|
||
|
|
|
||
|
|
---1--> ---3-->
|
||
|
|
(0) (1) (2)
|
||
|
|
<--2--- <--4---
|
||
|
|
|
||
|
|
In a non-directed graph, traversal from node i to node j can be
|
||
|
|
accomplished over either G[i, j] or G[j, i]. If both edges are not null,
|
||
|
|
and the two have unequal weights, then the smaller of the two is used.
|
||
|
|
|
||
|
|
So for the same graph, when ``directed=False`` we get the graph::
|
||
|
|
|
||
|
|
(0)--1--(1)--3--(2)
|
||
|
|
|
||
|
|
Note that a symmetric matrix will represent an undirected graph, regardless
|
||
|
|
of whether the 'directed' keyword is set to True or False. In this case,
|
||
|
|
using ``directed=True`` generally leads to more efficient computation.
|
||
|
|
|
||
|
|
The routines in this module accept as input either scipy.sparse representations
|
||
|
|
(csr, csc, or lil format), masked representations, or dense representations
|
||
|
|
with non-edges indicated by zeros, infinities, and NaN entries.
|
||
|
|
"""
|
||
|
|
|
||
|
|
__docformat__ = "restructuredtext en"
|
||
|
|
|
||
|
|
__all__ = ['connected_components',
|
||
|
|
'laplacian',
|
||
|
|
'shortest_path',
|
||
|
|
'floyd_warshall',
|
||
|
|
'dijkstra',
|
||
|
|
'bellman_ford',
|
||
|
|
'johnson',
|
||
|
|
'breadth_first_order',
|
||
|
|
'depth_first_order',
|
||
|
|
'breadth_first_tree',
|
||
|
|
'depth_first_tree',
|
||
|
|
'minimum_spanning_tree',
|
||
|
|
'reverse_cuthill_mckee',
|
||
|
|
'maximum_flow',
|
||
|
|
'maximum_bipartite_matching',
|
||
|
|
'min_weight_full_bipartite_matching',
|
||
|
|
'structural_rank',
|
||
|
|
'construct_dist_matrix',
|
||
|
|
'reconstruct_path',
|
||
|
|
'csgraph_masked_from_dense',
|
||
|
|
'csgraph_from_dense',
|
||
|
|
'csgraph_from_masked',
|
||
|
|
'csgraph_to_dense',
|
||
|
|
'csgraph_to_masked',
|
||
|
|
'NegativeCycleError']
|
||
|
|
|
||
|
|
from ._laplacian import laplacian
|
||
|
|
from ._shortest_path import (
|
||
|
|
shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson,
|
||
|
|
NegativeCycleError
|
||
|
|
)
|
||
|
|
from ._traversal import (
|
||
|
|
breadth_first_order, depth_first_order, breadth_first_tree,
|
||
|
|
depth_first_tree, connected_components
|
||
|
|
)
|
||
|
|
from ._min_spanning_tree import minimum_spanning_tree
|
||
|
|
from ._flow import maximum_flow
|
||
|
|
from ._matching import (
|
||
|
|
maximum_bipartite_matching, min_weight_full_bipartite_matching
|
||
|
|
)
|
||
|
|
from ._reordering import reverse_cuthill_mckee, structural_rank
|
||
|
|
from ._tools import (
|
||
|
|
construct_dist_matrix, reconstruct_path, csgraph_from_dense,
|
||
|
|
csgraph_to_dense, csgraph_masked_from_dense, csgraph_from_masked,
|
||
|
|
csgraph_to_masked
|
||
|
|
)
|
||
|
|
|
||
|
|
from scipy._lib._testutils import PytestTester
|
||
|
|
test = PytestTester(__name__)
|
||
|
|
del PytestTester
|