__init__.py 57.7 KB
Newer Older
1
#! /usr/bin/env python
2
# -*- coding: utf-8 -*-
3
#
4
5
# graph_tool -- a general graph manipulation python module
#
Tiago Peixoto's avatar
Tiago Peixoto committed
6
# Copyright (C) 2007-2012 Tiago de Paula Peixoto <tiago@skewed.de>
7
8
9
10
11
12
13
14
15
16
17
18
19
20
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

21
"""
22
23
``graph_tool.topology`` - Assessing graph topology
--------------------------------------------------
24
25
26
27
28
29
30

Summary
+++++++

.. autosummary::
   :nosignatures:

31
   shortest_distance
Tiago Peixoto's avatar
Tiago Peixoto committed
32
   shortest_path
Tiago Peixoto's avatar
Tiago Peixoto committed
33
   pseudo_diameter
34
   similarity
35
   isomorphism
36
37
   subgraph_isomorphism
   mark_subgraph
38
39
   max_cardinality_matching
   max_independent_vertex_set
40
   min_spanning_tree
41
   random_spanning_tree
42
43
44
   dominator_tree
   topological_sort
   transitive_closure
Tiago Peixoto's avatar
Tiago Peixoto committed
45
   tsp_tour
46
   sequential_vertex_coloring
47
48
   label_components
   label_biconnected_components
49
   label_largest_component
50
   label_out_component
51
   is_bipartite
Tiago Peixoto's avatar
Tiago Peixoto committed
52
   is_DAG
53
   is_planar
54
   make_maximal_planar
Tiago Peixoto's avatar
Tiago Peixoto committed
55
   edge_reciprocity
56
57
58

Contents
++++++++
59

60
61
"""

62
63
from __future__ import division, absolute_import, print_function

Tiago Peixoto's avatar
Tiago Peixoto committed
64
from .. dl_import import dl_import
65
dl_import("from . import libgraph_tool_topology")
66

67
from .. import _prop, Vector_int32_t, _check_prop_writable, \
68
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView, _get_rng
69
import random, sys, numpy
70
__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
71
           "max_cardinality_matching", "max_independent_vertex_set",
72
           "min_spanning_tree", "random_spanning_tree", "dominator_tree",
Tiago Peixoto's avatar
Tiago Peixoto committed
73
           "topological_sort", "transitive_closure", "tsp_tour",
74
75
76
           "sequential_vertex_coloring", "label_components",
           "label_largest_component", "label_biconnected_components",
           "label_out_component", "shortest_distance", "shortest_path",
Tiago Peixoto's avatar
Tiago Peixoto committed
77
           "pseudo_diameter", "is_bipartite", "is_DAG", "is_planar",
78
           "make_maximal_planar", "similarity", "edge_reciprocity"]
79
80
81
82
83
84
85
86
87
88


def similarity(g1, g2, label1=None, label2=None, norm=True):
    r"""Return the adjacency similarity between the two graphs.

    Parameters
    ----------
    g1 : :class:`~graph_tool.Graph`
        First graph to be compared.
    g2 : :class:`~graph_tool.Graph`
Tiago Peixoto's avatar
Tiago Peixoto committed
89
        Second graph to be compared.
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
    label1 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Vertex labels for the first graph to be used in comparison. If not
        supplied, the vertex indexes are used.
    label2 : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Vertex labels for the second graph to be used in comparison. If not
        supplied, the vertex indexes are used.
    norm : bool (optional, default: ``True``)
        If ``True``, the returned value is normalized by the total number of
        edges.

    Returns
    -------
    similarity : float
        Adjacency similarity value.

    Notes
    -----
    The adjacency similarity is the sum of equal entries in the adjacency
    matrix, given a vertex ordering determined by the vertex labels. In other
    words it counts the number of edges which have the same source and target
    labels in both graphs.

    The algorithm runs with complexity :math:`O(E_1 + V_1 + E_2 + V_2)`.

    Examples
    --------
    >>> from numpy.random import seed
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (3,3))
    >>> u = g.copy()
    >>> gt.similarity(u, g)
    1.0
    >>> gt.random_rewire(u);
    >>> gt.similarity(u, g)
    0.03333333333333333
    """

    if label1 is None:
        label1 = g1.vertex_index
    if label2 is None:
        label2 = g2.vertex_index
    if label1.value_type() != label2.value_type():
        raise ValueError("label property maps must be of the same type")
    s = libgraph_tool_topology.\
           similarity(g1._Graph__graph, g2._Graph__graph,
                      _prop("v", g1, label1), _prop("v", g1, label2))
    if not g1.is_directed() or not g2.is_directed():
        s /= 2
    if norm:
        s /= float(max(g1.num_edges(), g2.num_edges()))
    return s
141

Tiago Peixoto's avatar
Tiago Peixoto committed
142

143
def isomorphism(g1, g2, isomap=False):
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
    r"""Check whether two graphs are isomorphic.

    If `isomap` is True, a vertex :class:`~graph_tool.PropertyMap` with the
    isomorphism mapping is returned as well.

    Examples
    --------
    >>> from numpy.random import seed
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (3,3))
    >>> g2 = gt.Graph(g)
    >>> gt.isomorphism(g, g2)
    True
    >>> g.add_edge(g.vertex(0), g.vertex(1))
    <...>
    >>> gt.isomorphism(g, g2)
    False

162
    """
163
164
    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
165
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
166
                             _prop("v", g1, imap))
167
168
169
170
171
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
172

173
def subgraph_isomorphism(sub, g, max_n=0, random=False):
174
    r"""
175
176
    Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n`
    subgraphs, if `max_n > 0`).
177

178

Tiago Peixoto's avatar
Tiago Peixoto committed
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
    Parameters
    ----------
    sub : :class:`~graph_tool.Graph`
        Subgraph for which to be searched.
    g : :class:`~graph_tool.Graph`
        Graph in which the search is performed.
    max_n : int (optional, default: 0)
        Maximum number of matches to find. If `max_n == 0`, all matches are
        found.
    random : bool (optional, default: False)
        If `True`, the vertices of `g` are indexed in random order before
        the search.

    Returns
    -------
    vertex_maps : list of :class:`~graph_tool.PropertyMap` objects
        List containing vertex property map objects which indicate different
        isomorphism mappings. The property maps vertices in `sub` to the
        corresponding vertex index in `g`.
    edge_maps : list of :class:`~graph_tool.PropertyMap` objects
        List containing edge property map objects which indicate different
        isomorphism mappings. The property maps edges in `sub` to the
        corresponding edge index in `g`.

    Notes
    -----
    The algorithm used is described in [ullmann-algorithm-1976]_. It has a
    worse-case complexity of :math:`O(N_g^{N_{sub}})`, but for random graphs it
    typically has a complexity of :math:`O(N_g^\gamma)` with :math:`\gamma`
    depending sub-linearly on the size of `sub`.
209
210
211
212
213
214
215
216

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(42)
    >>> g = gt.random_graph(30, lambda: (poisson(6),poisson(6)))
    >>> sub = gt.random_graph(10, lambda: (poisson(1.8), poisson(1.9)))
    >>> vm, em = gt.subgraph_isomorphism(sub, g)
217
    >>> print(len(vm))
218
    102
219
    >>> for i in range(len(vm)):
220
221
222
223
224
225
226
227
228
229
    ...   g.set_vertex_filter(None)
    ...   g.set_edge_filter(None)
    ...   vmask, emask = gt.mark_subgraph(g, sub, vm[i], em[i])
    ...   g.set_vertex_filter(vmask)
    ...   g.set_edge_filter(emask)
    ...   assert(gt.isomorphism(g, sub))
    >>> g.set_vertex_filter(None)
    >>> g.set_edge_filter(None)
    >>> ewidth = g.copy_property(emask, value_type="double")
    >>> ewidth.a += 0.5
Tiago Peixoto's avatar
Tiago Peixoto committed
230
231
232
    >>> ewidth.a *= 2
    >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
    ...               edge_pen_width=ewidth, output_size=(200, 200),
233
    ...               output="subgraph-iso-embed.pdf")
234
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
235
    >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf")
236
237
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
238
239
    .. image:: subgraph-iso.*
    .. image:: subgraph-iso-embed.*
240

241

Tiago Peixoto's avatar
Tiago Peixoto committed
242
    **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph.
243
244
245

    References
    ----------
246
    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
Tiago Peixoto's avatar
Tiago Peixoto committed
247
       isomorphism", Journal of the ACM 23 (1): 31–42, 1976, :doi:`10.1145/321921.321925`
248
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
249
250
251
252

    """
    # vertex and edge labels disabled for the time being, until GCC is capable
    # of compiling all the variants using reasonable amounts of memory
Tiago Peixoto's avatar
Tiago Peixoto committed
253
254
    vlabels=(None, None)
    elabels=(None, None)
255
256
    vmaps = []
    emaps = []
257
    if random:
258
        seed = numpy.random.randint(0, sys.maxsize)
259
260
    else:
        seed = 42
261
262
263
264
265
266
    libgraph_tool_topology.\
           subgraph_isomorphism(sub._Graph__graph, g._Graph__graph,
                                _prop("v", sub, vlabels[0]),
                                _prop("v", g, vlabels[1]),
                                _prop("e", sub, elabels[0]),
                                _prop("e", g, elabels[1]),
267
                                vmaps, emaps, max_n, seed)
268
    for i in range(len(vmaps)):
269
270
271
272
        vmaps[i] = PropertyMap(vmaps[i], sub, "v")
        emaps[i] = PropertyMap(emaps[i], sub, "e")
    return vmaps, emaps

Tiago Peixoto's avatar
Tiago Peixoto committed
273

274
275
276
277
278
279
280
281
282
283
def mark_subgraph(g, sub, vmap, emap, vmask=None, emask=None):
    r"""
    Mark a given subgraph `sub` on the graph `g`.

    The mapping must be provided by the `vmap` and `emap` parameters,
    which map vertices/edges of `sub` to indexes of the corresponding
    vertices/edges in `g`.

    This returns a vertex and an edge property map, with value type 'bool',
    indicating whether or not a vertex/edge in `g` corresponds to the subgraph
284
    `sub`.
285
    """
286
    if vmask is None:
287
        vmask = g.new_vertex_property("bool")
288
    if emask is None:
289
290
291
292
293
294
295
296
297
298
299
300
301
302
        emask = g.new_edge_property("bool")

    vmask.a = False
    emask.a = False

    for v in sub.vertices():
        w = g.vertex(vmap[v])
        vmask[w] = True
        for ew in w.out_edges():
            for ev in v.out_edges():
                if emap[ev] == g.edge_index[ew]:
                    emask[ew] = True
                    break
    return vmask, emask
303

Tiago Peixoto's avatar
Tiago Peixoto committed
304

305
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
306
307
308
309
310
311
312
    """
    Return the minimum spanning tree of a given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
313
    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
314
315
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
316
    root : :class:`~graph_tool.Vertex` (optional, default: `None`)
317
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
318
        is used. Otherwise, Kruskal's algorithm is used.
319
    tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
        If provided, the edge tree map will be written in this property map.

    Returns
    -------
    tree_map : :class:`~graph_tool.PropertyMap`
        Edge property map with mark the tree edges: 1 for tree edge, 0
        otherwise.

    Notes
    -----
    The algorithm runs with :math:`O(E\log E)` complexity, or :math:`O(E\log V)`
    if `root` is specified.

    Examples
    --------
Tiago Peixoto's avatar
Tiago Peixoto committed
335
    >>> from numpy.random import seed, random
336
    >>> seed(42)
337
338
339
    >>> g, pos = gt.triangulation(random((400, 2)) * 10, type="delaunay")
    >>> weight = g.new_edge_property("double")
    >>> for e in g.edges():
Tiago Peixoto's avatar
Tiago Peixoto committed
340
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
341
    >>> tree = gt.min_spanning_tree(g, weights=weight)
342
    >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf")
343
344
    <...>
    >>> g.set_edge_filter(tree)
345
    >>> gt.graph_draw(g, pos=pos, output="triang_min_span_tree.pdf")
346
347
348
    <...>


349
    .. image:: triang_orig.*
Tiago Peixoto's avatar
Tiago Peixoto committed
350
        :width: 400px
351
    .. image:: triang_min_span_tree.*
Tiago Peixoto's avatar
Tiago Peixoto committed
352
        :width: 400px
353
354

    *Left:* Original graph, *Right:* The minimum spanning tree.
355
356
357
358
359

    References
    ----------
    .. [kruskal-shortest-1956] J. B. Kruskal.  "On the shortest spanning subtree
       of a graph and the traveling salesman problem",  In Proceedings of the
Tiago Peixoto's avatar
Tiago Peixoto committed
360
361
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
362
363
364
365
366
    .. [prim-shortest-1957] R. Prim.  "Shortest connection networks and some
       generalizations",  Bell System Technical Journal, 36:1389-1401, 1957.
    .. [boost-mst] http://www.boost.org/libs/graph/doc/graph_theory_review.html#sec:minimum-spanning-tree
    .. [mst-wiki] http://en.wikipedia.org/wiki/Minimum_spanning_tree
    """
367
    if tree_map is None:
368
369
370
371
        tree_map = g.new_edge_property("bool")
    if tree_map.value_type() != "bool":
        raise ValueError("edge property 'tree_map' must be of value type bool.")

372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
    try:
        g.stash_filter(directed=True)
        g.set_directed(False)
        if root is None:
            libgraph_tool_topology.\
                   get_kruskal_spanning_tree(g._Graph__graph,
                                             _prop("e", g, weights),
                                             _prop("e", g, tree_map))
        else:
            libgraph_tool_topology.\
                   get_prim_spanning_tree(g._Graph__graph, int(root),
                                          _prop("e", g, weights),
                                          _prop("e", g, tree_map))
    finally:
        g.pop_filter(directed=True)
387
    return tree_map
388

Tiago Peixoto's avatar
Tiago Peixoto committed
389

390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
def random_spanning_tree(g, weights=None, root=None, tree_map=None):
    """
    Return a random spanning tree of a given graph, which can be directed or
    undirected.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        The edge weights. If provided, the probability of a particular spanning
        tree being selected is the product of its edge weights.
    root : :class:`~graph_tool.Vertex` (optional, default: `None`)
        Root of the spanning tree. If not provided, it will be selected randomly.
    tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        If provided, the edge tree map will be written in this property map.

    Returns
    -------
    tree_map : :class:`~graph_tool.PropertyMap`
        Edge property map with mark the tree edges: 1 for tree edge, 0
        otherwise.

    Notes
    -----
    The typical running time for random graphs is :math:`O(N\log N)`.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> g, pos = gt.triangulation(random((400, 2)) * 10, type="delaunay")
    >>> weight = g.new_edge_property("double")
    >>> for e in g.edges():
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
    >>> tree = gt.random_spanning_tree(g, weights=weight)
    >>> gt.graph_draw(g, pos=pos, output="rtriang_orig.pdf")
    <...>
    >>> g.set_edge_filter(tree)
Tiago Peixoto's avatar
Tiago Peixoto committed
429
    >>> gt.graph_draw(g, pos=pos, output="triang_random_span_tree.pdf")
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
    <...>


    .. image:: rtriang_orig.*
        :width: 400px
    .. image:: triang_random_span_tree.*
        :width: 400px

    *Left:* Original graph, *Right:* A random spanning tree.

    References
    ----------

    .. [wilson-generating-1996] David Bruce Wilson, "Generating random spanning
       trees more quickly than the cover time", Proceedings of the twenty-eighth
       annual ACM symposium on Theory of computing, Pages 296-303, ACM New York,
       1996, :doi:`10.1145/237814.237880`
    .. [boost-rst] http://www.boost.org/libs/graph/doc/random_spanning_tree.html
    """
    if tree_map is None:
        tree_map = g.new_edge_property("bool")
    if tree_map.value_type() != "bool":
        raise ValueError("edge property 'tree_map' must be of value type bool.")

    if root is None:
        root = g.vertex(numpy.random.randint(0, g.num_vertices()),
                        use_index=False)

    # we need to restrict ourselves to the in-component of root
    l = label_out_component(GraphView(g, reversed=True), root)
    g = GraphView(g, vfilt=l)

    seed = numpy.random.randint(0, sys.maxsize)
    libgraph_tool_topology.\
        random_spanning_tree(g._Graph__graph, int(root),
                             _prop("e", g, weights),
                             _prop("e", g, tree_map), seed)
    return tree_map


Tiago Peixoto's avatar
Tiago Peixoto committed
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
def dominator_tree(g, root, dom_map=None):
    """Return a vertex property map the dominator vertices for each vertex.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    root : :class:`~graph_tool.Vertex`
        The root vertex.
    dom_map : :class:`~graph_tool.PropertyMap` (optional, default: None)
        If provided, the dominator map will be written in this property map.

    Returns
    -------
    dom_map : :class:`~graph_tool.PropertyMap`
        The dominator map. It contains for each vertex, the index of its
        dominator vertex.

    Notes
    -----
    A vertex u dominates a vertex v, if every path of directed graph from the
    entry to v must go through u.

    The algorithm runs with :math:`O((V+E)\log (V+E))` complexity.

    Examples
    --------
    >>> from numpy.random import seed
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (2, 2))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
502
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
503
    >>> dom = gt.dominator_tree(g, root[0])
504
    >>> print(dom.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
505
506
507
    [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
508
509
510

    References
    ----------
511
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
512
513

    """
514
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
515
516
517
        dom_map = g.new_vertex_property("int32_t")
    if dom_map.value_type() != "int32_t":
        raise ValueError("vertex property 'dom_map' must be of value type" +
518
519
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
520
        raise ValueError("dominator tree requires a directed graph.")
521
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
522
523
524
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
525

Tiago Peixoto's avatar
Tiago Peixoto committed
526

527
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
    """
    Return the topological sort of the given graph. It is returned as an array
    of vertex indexes, in the sort order.

    Notes
    -----
    The topological sort algorithm creates a linear ordering of the vertices
    such that if edge (u,v) appears in the graph, then v comes before u in the
    ordering. The graph must be a directed acyclic graph (DAG).

    The time complexity is :math:`O(V + E)`.

    Examples
    --------
    >>> from numpy.random import seed
    >>> seed(42)
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
    >>> sort = gt.topological_sort(g)
548
    >>> print(sort)
Tiago Peixoto's avatar
Tiago Peixoto committed
549
550
    [ 3 20  9 29 15  0 10 23  1  2 21  7  4 12 11  5 26 27  6  8 13 14 22 16 17
     28 18 19 24 25]
Tiago Peixoto's avatar
Tiago Peixoto committed
551
552
553

    References
    ----------
554
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
555
556
557
558
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

559
    topological_order = Vector_int32_t()
Tiago Peixoto's avatar
Tiago Peixoto committed
560
561
562
563
564
    is_DAG = libgraph_tool_topology.\
        topological_sort(g._Graph__graph, topological_order)
    if not is_DAG:
        raise ValueError("Graph is not a directed acylic graph (DAG).");
    return topological_order.a.copy()
565

Tiago Peixoto's avatar
Tiago Peixoto committed
566

567
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
    """Return the transitive closure graph of g.

    Notes
    -----
    The transitive closure of a graph G = (V,E) is a graph G* = (V,E*) such that
    E* contains an edge (u,v) if and only if G contains a path (of at least one
    edge) from u to v. The transitive_closure() function transforms the input
    graph g into the transitive closure graph tc.

    The time complexity (worst-case) is :math:`O(VE)`.

    Examples
    --------
    >>> from numpy.random import seed
    >>> seed(42)
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tc = gt.transitive_closure(g)

    References
    ----------
588
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
589
590
591
592
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

593
594
595
596
597
598
599
    if not g.is_directed():
        raise ValueError("graph must be directed for transitive closure.")
    tg = Graph()
    libgraph_tool_topology.transitive_closure(g._Graph__graph,
                                              tg._Graph__graph)
    return tg

Tiago Peixoto's avatar
Tiago Peixoto committed
600

601
602
def label_components(g, vprop=None, directed=None):
    """
603
    Label the components to which each vertex in the graph belongs. If the
604
605
    graph is directed, it finds the strongly connected components.

606
607
608
    A property map with the component labels is returned, together with an
    histogram of component labels.

609
610
    Parameters
    ----------
611
    g : :class:`~graph_tool.Graph`
612
        Graph to be used.
613
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
614
615
616
617
618
619
620
621
        Vertex property to store the component labels. If none is supplied, one
        is created.
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.

    Returns
    -------
622
    comp : :class:`~graph_tool.PropertyMap`
623
        Vertex property map with component labels.
624
625
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
626
627
628
629
630
631

    Notes
    -----
    The components are arbitrarily labeled from 0 to N-1, where N is the total
    number of components.

632
    The algorithm runs in :math:`O(V + E)` time.
633
634
635

    Examples
    --------
636
637
638
    >>> from numpy.random import seed
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: (1, 1))
639
    >>> comp, hist = gt.label_components(g)
640
    >>> print(comp.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
641
642
643
    [0 0 0 1 0 2 0 0 0 0 2 0 0 0 2 1 0 2 0 1 2 0 1 0 0 1 0 2 0 2 1 0 2 0 0 0 0
     0 0 1 0 0 2 2 2 0 0 0 0 0 0 2 0 0 1 1 0 0 2 0 1 0 0 0 2 0 0 2 2 1 2 1 0 0
     2 0 0 1 2 1 2 2 0 0 0 0 0 2 0 0 0 1 1 0 0 0 1 1 2 2]
644
    >>> print(hist)
Tiago Peixoto's avatar
Tiago Peixoto committed
645
    [58 18 24]
646
647
    """

648
    if vprop is None:
649
650
651
652
653
        vprop = g.new_vertex_property("int32_t")

    _check_prop_writable(vprop, name="vprop")
    _check_prop_scalar(vprop, name="vprop")

654
655
    if directed is not None:
        g = GraphView(g, directed=directed)
656

657
658
659
660
661
662
663
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
    return vprop, hist


def label_largest_component(g, directed=None):
    """
664
665
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
666
667
668
669
670
671
672
673
674
675
676
677
678
679

    A property map with a boolean label is returned.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.

    Returns
    -------
    comp : :class:`~graph_tool.PropertyMap`
680
         Boolean vertex property map which labels the largest component.
681
682
683
684
685
686
687
688
689
690
691

    Notes
    -----
    The algorithm runs in :math:`O(V + E)` time.

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: poisson(1), directed=False)
    >>> l = gt.label_largest_component(g)
692
    >>> print(l.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
693
694
695
    [1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1
     1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0
     0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0]
696
    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
697
    >>> print(u.num_vertices())
Tiago Peixoto's avatar
Tiago Peixoto committed
698
    31
699
700
701
702
    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
703
704
705
706
707
    vfilt, inv = g.get_vertex_filter()
    if vfilt is None:
        label.a = c.a == h.argmax()
    else:
        label.a = (c.a == h.argmax()) & (vfilt.a ^ inv)
708
    return label
709

Tiago Peixoto's avatar
Tiago Peixoto committed
710

711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
def label_out_component(g, root):
    """
    Label the out-component (or simply the component for undirected graphs) of a
    root vertex.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    root : :class:`~graph_tool.Vertex`
        The root vertex.

    Returns
    -------
    comp : :class:`~graph_tool.PropertyMap`
         Boolean vertex property map which labels the out-component.

    Notes
    -----
    The algorithm runs in :math:`O(V + E)` time.

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: poisson(1), directed=False)
    >>> l = gt.label_out_component(g, g.vertex(0))
    >>> print(l.a)
    [1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1
     1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0
     0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0]

    The in-component can be obtained by reversing the graph.

Tiago Peixoto's avatar
Tiago Peixoto committed
745
746
    >>> l = gt.label_out_component(gt.GraphView(g, reversed=True, directed=True),
    ...                            g.vertex(0))
747
748
749
750
751
752
753
754
755
756
757
758
759
    >>> print(l.a)
    [1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1
     1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0
     0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0]
    """

    label = g.new_vertex_property("bool")
    libgraph_tool_topology.\
             label_out_component(g._Graph__graph, int(root),
                                 _prop("v", g, label))
    return label


760
def label_biconnected_components(g, eprop=None, vprop=None):
761
762
763
764
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

765
766
767
768
    An edge property map with the component labels is returned, together a
    boolean vertex map marking the articulation points, and an histogram of
    component labels.

769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.

    eprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
        Edge property to label the biconnected components.

    vprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
        Vertex property to mark the articulation points. If none is supplied,
        one is created.


    Returns
    -------
    bicomp : :class:`~graph_tool.PropertyMap`
        Edge property map with the biconnected component labels.
    articulation : :class:`~graph_tool.PropertyMap`
        Boolean vertex property map which has value 1 for each vertex which is
        an articulation point, and zero otherwise.
    nc : int
        Number of biconnected components.

    Notes
    -----

    A connected graph is biconnected if the removal of any single vertex (and
    all edges incident on that vertex) can not disconnect the graph. More
    generally, the biconnected components of a graph are the maximal subsets of
    vertices such that the removal of a vertex from a particular component will
    not disconnect the component. Unlike connected components, vertices may
    belong to multiple biconnected components: those vertices that belong to
    more than one biconnected component are called "articulation points" or,
    equivalently, "cut vertices". Articulation points are vertices whose removal
    would increase the number of connected components in the graph. Thus, a
    graph without articulation points is biconnected. Vertices can be present in
    multiple biconnected components, but each edge can only be contained in a
    single biconnected component.

    The algorithm runs in :math:`O(V + E)` time.

    Examples
    --------
    >>> from numpy.random import seed
Tiago Peixoto's avatar
Tiago Peixoto committed
813
    >>> seed(43)
814
    >>> g = gt.random_graph(100, lambda: 2, directed=False)
815
    >>> comp, art, hist = gt.label_biconnected_components(g)
816
    >>> print(comp.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
817
818
819
    [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
     0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1
     0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0]
820
    >>> print(art.a)
821
822
823
    [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
824
    >>> print(hist)
Tiago Peixoto's avatar
Tiago Peixoto committed
825
    [87 13]
826
    """
827

828
    if vprop is None:
829
        vprop = g.new_vertex_property("bool")
830
    if eprop is None:
831
832
833
834
835
836
837
        eprop = g.new_edge_property("int32_t")

    _check_prop_writable(vprop, name="vprop")
    _check_prop_scalar(vprop, name="vprop")
    _check_prop_writable(eprop, name="eprop")
    _check_prop_scalar(eprop, name="eprop")

838
839
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
840
841
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
842
    return eprop, vprop, hist
843

Tiago Peixoto's avatar
Tiago Peixoto committed
844

845
def shortest_distance(g, source=None, weights=None, max_dist=None,
846
847
                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
848
849
850
851
852
853
854
855
856
    """
    Calculate the distance of all vertices from a given source, or the all pairs
    shortest paths, if the source is not specified.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    source : :class:`~graph_tool.Vertex` (optional, default: None)
857
        Source vertex of the search. If unspecified, the all pairs shortest
858
859
860
861
862
863
        distances are computed.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
    max_dist : scalar value (optional, default: None)
        If specified, this limits the maximum distance of the vertices
864
        are searched. This parameter has no effect if source is None.
865
866
867
868
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
869
870
        If true, and source is None, the Floyd-Warshall algorithm is used,
        otherwise the Johnson algorithm is used. If source is not None, this option
871
872
873
874
        has no effect.
    dist_map : :class:`~graph_tool.PropertyMap` (optional, default: None)
        Vertex property to store the distances. If none is supplied, one
        is created.
875
876
877
    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903

    Returns
    -------
    dist_map : :class:`~graph_tool.PropertyMap`
        Vertex property map with the distances from source. If source is 'None',
        it will have a vector value type, with the distances to every vertex.

    Notes
    -----

    If a source is given, the distances are calculated with a breadth-first
    search (BFS) or Dijkstra's algorithm [dijkstra]_, if weights are given. If
    source is not given, the distances are calculated with Johnson's algorithm
    [johnson-apsp]_. If dense=True, the Floyd-Warshall algorithm
    [floyd-warshall-apsp]_ is used instead.

    If source is specified, the algorithm runs in :math:`O(V + E)` time, or
    :math:`O(V \log V)` if weights are given. If source is not specified, it
    runs in :math:`O(VE\log V)` time, or :math:`O(V^3)` if dense == True.

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(42)
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
    >>> dist = gt.shortest_distance(g, source=g.vertex(0))
904
    >>> print(dist.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
    [         0          3          6          4 2147483647          3
              4          3          4          2          3          4
              3          4          2          4          2          5
              4          4 2147483647          4 2147483647          6
              4          7          5 2147483647          3          4
              2          3          5          5          4          5
              1          5          6          1 2147483647          8
              4          2          1          5          5          6
              7          4          5          3          4          4
              5          3          3          5          4          5
              4          3          5          4          2 2147483647
              6          5          4          5          1 2147483647
              5          5          4          2          5          4
              6          3          5          3          4 2147483647
              4          4          7          4          3          5
              5          2          7          3          4          4
              4          3          4          4]
922
    >>> dist = gt.shortest_distance(g)
923
    >>> print(dist[g.vertex(0)].a)
Tiago Peixoto's avatar
Tiago Peixoto committed
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
    [         0          3          6          4 2147483647          3
              4          3          4          2          3          4
              3          4          2          4          2          5
              4          4 2147483647          4 2147483647          6
              4          7          5 2147483647          3          4
              2          3          5          5          4          5
              1          5          6          1 2147483647          8
              4          2          1          5          5          6
              7          4          5          3          4          4
              5          3          3          5          4          5
              4          3          5          4          2 2147483647
              6          5          4          5          1 2147483647
              5          5          4          2          5          4
              6          3          5          3          4 2147483647
              4          4          7          4          3          5
              5          2          7          3          4          4
              4          3          4          4]
941
942
943
944
945

    References
    ----------
    .. [bfs] Edward Moore, "The shortest path through a maze", International
       Symposium on the Theory of Switching (1959), Harvard University
Tiago Peixoto's avatar
Tiago Peixoto committed
946
947
       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
948
949
    .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with
       graphs." Numerische Mathematik, 1:269-271, 1959.
Tiago Peixoto's avatar
Tiago Peixoto committed
950
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
951
952
953
954
    .. [johnson-apsp] http://www.boost.org/libs/graph/doc/johnson_all_pairs_shortest.html
    .. [floyd-warshall-apsp] http://www.boost.org/libs/graph/doc/floyd_warshall_shortest.html
    """

955
    if weights is None:
956
957
958
959
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

960
961
    if dist_map is None:
        if source is not None:
962
963
964
965
966
            dist_map = g.new_vertex_property(dist_type)
        else:
            dist_map = g.new_vertex_property("vector<%s>" % dist_type)

    _check_prop_writable(dist_map, name="dist_map")
967
    if source is not None:
968
969
970
971
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

972
    if max_dist is None:
973
974
        max_dist = 0

975
    if directed is not None:
976
977
978
979
        g.stash_filter(directed=True)
        g.set_directed(directed)

    try:
980
        if source is not None:
981
            pmap = g.copy_property(g.vertex_index, value_type="int64_t")
982
983
984
            libgraph_tool_topology.get_dists(g._Graph__graph, int(source),
                                             _prop("v", g, dist_map),
                                             _prop("e", g, weights),
985
                                             _prop("v", g, pmap),
986
987
988
989
990
991
992
                                             float(max_dist))
        else:
            libgraph_tool_topology.get_all_dists(g._Graph__graph,
                                                 _prop("v", g, dist_map),
                                                 _prop("e", g, weights), dense)

    finally:
993
        if directed is not None:
994
            g.pop_filter(directed=True)
995
    if source is not None and pred_map:
996
997
998
999
        return dist_map, pmap
    else:
        return dist_map

Tiago Peixoto's avatar
Tiago Peixoto committed
1000

For faster browsing, not all history is shown. View entire blame