__init__.py 44.1 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
41
42
43
44
45
   min_spanning_tree
   dominator_tree
   topological_sort
   transitive_closure
   label_components
   label_biconnected_components
46
   label_largest_component
47
   is_planar
48
49
50

Contents
++++++++
51

52
53
"""

Tiago Peixoto's avatar
Tiago Peixoto committed
54
from .. dl_import import dl_import
55
dl_import("import libgraph_tool_topology")
56

57
from .. import _prop, Vector_int32_t, _check_prop_writable, \
58
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView
59
60
from .. flow import libgraph_tool_flow
import random, sys, numpy
61
__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
62
           "max_cardinality_matching", "max_independent_vertex_set",
63
           "min_spanning_tree", "dominator_tree", "topological_sort",
64
           "transitive_closure", "label_components", "label_largest_component",
65
66
           "label_biconnected_components", "shortest_distance", "shortest_path",
           "pseudo_diameter", "is_planar", "similarity"]
67
68
69
70
71
72
73
74
75
76


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
77
        Second graph to be compared.
78
79
80
81
82
83
84
85
86
87
88
89
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
    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
129

Tiago Peixoto's avatar
Tiago Peixoto committed
130

131
def isomorphism(g1, g2, isomap=False):
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
    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

150
    """
151
152
    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
153
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
154
                             _prop("v", g1, imap))
155
156
157
158
159
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
160

161
def subgraph_isomorphism(sub, g, max_n=0, random=False):
162
    r"""
163
164
    Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n`
    subgraphs, if `max_n > 0`).
165

166
167
168
    If `random` = True, the vertices of `g` are indexed in random order before
    the search.

169
170
171
172
173
174
175
176
177
178
179
180
    It returns two lists, containing the vertex and edge property maps for `sub`
    with the isomorphism mappings. The value of the properties are the
    vertex/edge index of the corresponding vertex/edge in `g`.

    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)
    >>> print len(vm)
181
    102
182
183
184
185
186
187
188
189
190
191
192
    >>> for i in xrange(len(vm)):
    ...   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
193
194
195
    >>> ewidth.a *= 2
    >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
    ...               edge_pen_width=ewidth, output_size=(200, 200),
196
    ...               output="subgraph-iso-embed.pdf")
197
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
198
    >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf")
199
200
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
201
202
    .. image:: subgraph-iso.*
    .. image:: subgraph-iso-embed.*
203

204

Tiago Peixoto's avatar
Tiago Peixoto committed
205
    **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph.
206
207
208

    Notes
    -----
209
210
211
212
    The algorithm used is described in [ullmann-algorithm-1976]. It has
    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`.
213
214
215

    References
    ----------
216
    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
Tiago Peixoto's avatar
Tiago Peixoto committed
217
       isomorphism", Journal of the ACM 23 (1): 31–42, 1976, :doi:`10.1145/321921.321925`
218
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
219
220
221
222

    """
    # 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
223
224
    vlabels=(None, None)
    elabels=(None, None)
225
226
    vmaps = []
    emaps = []
227
228
229
230
    if random:
        seed = numpy.random.randint(0, sys.maxint)
    else:
        seed = 42
231
232
233
234
235
236
    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]),
237
                                vmaps, emaps, max_n, seed)
238
239
240
241
242
    for i in xrange(len(vmaps)):
        vmaps[i] = PropertyMap(vmaps[i], sub, "v")
        emaps[i] = PropertyMap(emaps[i], sub, "e")
    return vmaps, emaps

Tiago Peixoto's avatar
Tiago Peixoto committed
243

244
245
246
247
248
249
250
251
252
253
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
254
    `sub`.
255
    """
256
    if vmask is None:
257
        vmask = g.new_vertex_property("bool")
258
    if emask is None:
259
260
261
262
263
264
265
266
267
268
269
270
271
272
        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
273

Tiago Peixoto's avatar
Tiago Peixoto committed
274

275
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
276
277
278
279
280
281
282
283
284
285
286
    """
    Return the minimum spanning tree of a given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
    root : :class:`~graph_tool.Vertex` (optional, default: None)
287
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
        is used. Otherwise, Kruskal's algorithm is used.
    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 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
305
    >>> from numpy.random import seed, random
306
    >>> seed(42)
307
308
309
    >>> 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
310
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
311
    >>> tree = gt.min_spanning_tree(g, weights=weight)
312
    >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf")
313
314
    <...>
    >>> g.set_edge_filter(tree)
315
    >>> gt.graph_draw(g, pos=pos, output="triang_min_span_tree.pdf")
316
317
318
    <...>


319
    .. image:: triang_orig.*
Tiago Peixoto's avatar
Tiago Peixoto committed
320
        :width: 400px
321
    .. image:: triang_min_span_tree.*
Tiago Peixoto's avatar
Tiago Peixoto committed
322
        :width: 400px
323
324

    *Left:* Original graph, *Right:* The minimum spanning tree.
325
326
327
328
329

    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
330
331
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
332
333
334
335
336
    .. [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
    """
337
    if tree_map is None:
338
339
340
341
        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.")

342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
    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)
357
    return tree_map
358

Tiago Peixoto's avatar
Tiago Peixoto committed
359

Tiago Peixoto's avatar
Tiago Peixoto committed
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
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)
392
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
393
394
    >>> dom = gt.dominator_tree(g, root[0])
    >>> print dom.a
Tiago Peixoto's avatar
Tiago Peixoto committed
395
396
397
    [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
398
399
400

    References
    ----------
401
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
402
403

    """
404
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
405
406
407
        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" +
408
409
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
410
        raise ValueError("dominator tree requires a directed graph.")
411
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
412
413
414
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
415

Tiago Peixoto's avatar
Tiago Peixoto committed
416

417
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
    """
    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)
    >>> print sort
Tiago Peixoto's avatar
Tiago Peixoto committed
439
440
    [ 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
441
442
443

    References
    ----------
444
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
445
446
447
448
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

449
450
451
    topological_order = Vector_int32_t()
    libgraph_tool_topology.\
               topological_sort(g._Graph__graph, topological_order)
Tiago Peixoto's avatar
Tiago Peixoto committed
452
    return numpy.array(topological_order)
453

Tiago Peixoto's avatar
Tiago Peixoto committed
454

455
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
    """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
    ----------
476
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
477
478
479
480
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

481
482
483
484
485
486
487
    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
488

489
490
def label_components(g, vprop=None, directed=None):
    """
491
    Label the components to which each vertex in the graph belongs. If the
492
493
    graph is directed, it finds the strongly connected components.

494
495
496
    A property map with the component labels is returned, together with an
    histogram of component labels.

497
498
    Parameters
    ----------
499
    g : :class:`~graph_tool.Graph`
500
        Graph to be used.
501
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
502
503
504
505
506
507
508
509
        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
    -------
510
    comp : :class:`~graph_tool.PropertyMap`
511
        Vertex property map with component labels.
512
513
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
514
515
516
517
518
519

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

520
    The algorithm runs in :math:`O(V + E)` time.
521
522
523

    Examples
    --------
524
525
526
    >>> from numpy.random import seed
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: (1, 1))
527
    >>> comp, hist = gt.label_components(g)
528
    >>> print comp.a
Tiago Peixoto's avatar
Tiago Peixoto committed
529
530
531
    [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]
532
    >>> print hist
Tiago Peixoto's avatar
Tiago Peixoto committed
533
    [58 18 24]
534
535
    """

536
    if vprop is None:
537
538
539
540
541
        vprop = g.new_vertex_property("int32_t")

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

542
543
    if directed is not None:
        g = GraphView(g, directed=directed)
544

545
546
547
548
549
550
551
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
    return vprop, hist


def label_largest_component(g, directed=None):
    """
552
553
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
554
555
556
557
558
559
560
561
562
563
564
565
566
567

    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`
568
         Boolean vertex property map which labels the largest component.
569
570
571
572
573
574
575
576
577
578
579
580

    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)
    >>> print l.a
Tiago Peixoto's avatar
Tiago Peixoto committed
581
582
583
    [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]
584
585
    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
    >>> print u.num_vertices()
Tiago Peixoto's avatar
Tiago Peixoto committed
586
    31
587
588
589
590
    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
591
592
593
594
595
    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)
596
    return label
597

Tiago Peixoto's avatar
Tiago Peixoto committed
598

599
def label_biconnected_components(g, eprop=None, vprop=None):
600
601
602
603
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

604
605
606
607
    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.

608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
    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
652
    >>> seed(43)
653
    >>> g = gt.random_graph(100, lambda: 2, directed=False)
654
    >>> comp, art, hist = gt.label_biconnected_components(g)
655
    >>> print comp.a
Tiago Peixoto's avatar
Tiago Peixoto committed
656
657
658
    [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]
659
660
661
662
    >>> print art.a
    [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]
663
    >>> print hist
Tiago Peixoto's avatar
Tiago Peixoto committed
664
    [87 13]
665
    """
666

667
    if vprop is None:
668
        vprop = g.new_vertex_property("bool")
669
    if eprop is None:
670
671
672
673
674
675
676
        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")

677
678
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
679
680
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
681
    return eprop, vprop, hist
682

Tiago Peixoto's avatar
Tiago Peixoto committed
683

684
def shortest_distance(g, source=None, weights=None, max_dist=None,
685
686
                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
687
688
689
690
691
692
693
694
695
    """
    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)
696
        Source vertex of the search. If unspecified, the all pairs shortest
697
698
699
700
701
702
        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
703
        are searched. This parameter has no effect if source is None.
704
705
706
707
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
708
709
        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
710
711
712
713
        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.
714
715
716
    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
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

    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))
743
    >>> print dist.a
Tiago Peixoto's avatar
Tiago Peixoto committed
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
    [         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]
761
    >>> dist = gt.shortest_distance(g)
762
    >>> print dist[g.vertex(0)].a
Tiago Peixoto's avatar
Tiago Peixoto committed
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
    [         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]
780
781
782
783
784

    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
785
786
       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
787
788
    .. [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
789
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
790
791
792
793
    .. [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
    """

794
    if weights is None:
795
796
797
798
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

799
800
    if dist_map is None:
        if source is not None:
801
802
803
804
805
            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")
806
    if source is not None:
807
808
809
810
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

811
    if max_dist is None:
812
813
        max_dist = 0

814
    if directed is not None:
815
816
817
818
        g.stash_filter(directed=True)
        g.set_directed(directed)

    try:
819
        if source is not None:
820
            pmap = g.copy_property(g.vertex_index, value_type="int64_t")
821
822
823
            libgraph_tool_topology.get_dists(g._Graph__graph, int(source),
                                             _prop("v", g, dist_map),
                                             _prop("e", g, weights),
824
                                             _prop("v", g, pmap),
825
826
827
828
829
830
831
                                             float(max_dist))
        else:
            libgraph_tool_topology.get_all_dists(g._Graph__graph,
                                                 _prop("v", g, dist_map),
                                                 _prop("e", g, weights), dense)

    finally:
832
        if directed is not None:
833
            g.pop_filter(directed=True)
834
    if source is not None and pred_map:
835
836
837
838
        return dist_map, pmap
    else:
        return dist_map

Tiago Peixoto's avatar
Tiago Peixoto committed
839

840
841
842
843
844
845
846
847
848
849
def shortest_path(g, source, target, weights=None, pred_map=None):
    """
    Return the shortest path from `source` to `target`.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    source : :class:`~graph_tool.Vertex`
        Source vertex of the search.
Tiago Peixoto's avatar
Tiago Peixoto committed
850
    target : :class:`~graph_tool.Vertex`
851
852
        Target vertex of the search.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
Tiago Peixoto's avatar
Tiago Peixoto committed
853
        The edge weights.
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
    pred_map :  :class:`~graph_tool.PropertyMap` (optional, default: None)
        Vertex property map with the predecessors in the search tree. If this is
        provided, the shortest paths are not computed, and are obtained directly
        from this map.

    Returns
    -------
    vertex_list : list of :class:`~graph_tool.Vertex`
        List of vertices from `source` to `target` in the shortest path.
    edge_list : list of :class:`~graph_tool.Edge`
        List of edges from `source` to `target` in the shortest path.

    Notes
    -----

    The paths are computed with a breadth-first search (BFS) or Dijkstra's
    algorithm [dijkstra]_, if weights are given.

    The algorithm runs in :math:`O(V + E)` time, or :math:`O(V \log V)` if
    weights are given.

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(42)
    >>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3)))
    >>> vlist, elist = gt.shortest_path(g, g.vertex(10), g.vertex(11))
    >>> print [str(v) for v in vlist]
Tiago Peixoto's avatar
Tiago Peixoto committed
882
    ['10', '222', '246', '0', '50', '257', '12', '242', '11']
883
    >>> print [str(e) for e in elist]
884
    ['(10, 222)', '(222, 246)', '(246, 0)', '(0, 50)', '(50, 257)', '(257, 12)', '(12, 242)', '(242, 11)']
885
886
887
888
889

    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
890
891
       Press
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
892
893
    .. [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
894
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
895
896
    """

897
    if pred_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
898
899
        pred_map = shortest_distance(g, source, weights=weights,
                                     pred_map=True)[1]
900

Tiago Peixoto's avatar
Tiago Peixoto committed
901
    if pred_map[target] == int(target):  # no path to source
902
903
904
905
906
        return [], []

    vlist = [target]
    elist = []

907
    if weights is not None:
908
909
910
911
912
913
914
915
916
917
918
919
920
        max_w = weights.a.max() + 1
    else:
        max_w = None

    v = target
    while v != source:
        p = g.vertex(pred_map[v])
        min_w = max_w
        pe = None
        s = None
        for e in v.in_edges() if g.is_directed() else v.out_edges():
            s = e.source() if g.is_directed() else e.target()
            if s == p:
921
                if weights is not None:
922
923
924
925
926
927
928
929
930
931
932
                    if weights[e] < min_w:
                        min_w = weights[e]
                        pe = e
                else:
                    pe = e
                    break
        elist.insert(0, pe)
        vlist.insert(0, p)
        v = p
    return vlist, elist

933

Tiago Peixoto's avatar
Tiago Peixoto committed
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
def pseudo_diameter(g, source=None, weights=None):
    """
    Compute the pseudo-diameter of the graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    source : :class:`~graph_tool.Vertex` (optional, default: `None`)
        Source vertex of the search. If not supplied, the first vertex
        in the graph will be chosen.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        The edge weights.

    Returns
    -------
    pseudo_diameter : int
        The pseudo-diameter of the graph.
    end_points : pair of :class:`~graph_tool.Vertex`
        The two vertices which correspond to the pseudo-diameter found.

    Notes
    -----

    The pseudo-diameter is an approximate graph diameter. It is obtained by
    starting from a vertex `source`, and finds a vertex `target` that is
    farthest away from `source`. This process is repeated by treating
    `target` as the new starting vertex, and ends when the graph distance no
    longer increases. A vertex from the last level set that has the smallest
    degree is chosen as the final starting vertex u, and a traversal is done
    to see if the graph distance can be increased. This graph distance is
    taken to be the pseudo-diameter.

    The paths are computed with a breadth-first search (BFS) or Dijkstra's
    algorithm [dijkstra]_, if weights are given.

    The algorithm runs in :math:`O(V + E)` time, or :math:`O(V \log V)` if
    weights are given.

    Examples
    --------
    >>> from numpy.random import seed, poisson
    >>> seed(42)
    >>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3)))
    >>> dist, ends = gt.pseudo_diameter(g)
    >>> print dist
980
981
982
    9.0
    >>> print int(ends[0]), int(ends[1])
    0 255
Tiago Peixoto's avatar
Tiago Peixoto committed
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005

    References
    ----------
    .. [pseudo-diameter] http://en.wikipedia.org/wiki/Distance_%28graph_theory%29
    """

    if source is None:
        source = g.vertex(0)
    dist, target = 0, source
    while True:
        new_source = target
        new_target, new_dist = libgraph_tool_topology.get_diam(g._Graph__graph,
                                                               int(new_source),
                                                               _prop("e", g, weights))
        if new_dist > dist:
            target = new_target
            source = new_source
            dist = new_dist
        else:
            break
    return dist, (g.vertex(source), g.vertex(target))


1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
def is_planar(g, embedding=False, kuratowski=False):
    """
    Test if the graph is planar.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    embedding : bool (optional, default: False)
        If true, return a mapping from vertices to the clockwise order of
        out-edges in the planar embedding.
    kuratowski : bool (optional, default: False)
        If true, the minimal set of edges that form the obstructing Kuratowski
        subgraph will be returned as a property map, if the graph is not planar.

    Returns
    -------
    is_planar : bool
        Whether or not the graph is planar.
    embedding : :class:`~graph_tool.PropertyMap` (only if `embedding=True`)
        A vertex property map with the out-edges indexes in clockwise order in
        the planar embedding,
    kuratowski : :class:`~graph_tool.PropertyMap` (only if `kuratowski=True`)
        An edge property map with the minimal set of edges that form the
        obstructing Kuratowski subgraph (if the value of kuratowski[e] is 1,
        the edge belongs to the set)

    Notes
    -----

    A graph is planar if it can be drawn in two-dimensional space without any of
    its edges crossing. This algorithm performs the Boyer-Myrvold planarity
    testing [boyer-myrvold]_. See [boost-planarity]_ for more details.

    This algorithm runs in :math:`O(V)` time.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
    >>> g = gt.triangulation(random((100,2)))[0]
    >>> p, embed_order = gt.is_planar(g, embedding=True)
    >>> print p
    True
    >>> print list(embed_order[g.vertex(0)])
Tiago Peixoto's avatar
Tiago Peixoto committed
1051
    [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
1052
1053
1054
1055
1056
    >>> g = gt.random_graph(100, lambda: 4, directed=False)
    >>> p, kur = gt.is_planar(g, kuratowski=True)
    >>> print p
    False
    >>> g.set_edge_filter(kur, True)
Tiago Peixoto's avatar
Tiago Peixoto committed
1057
    >>> gt.graph_draw(g, output_size=(300, 300), output="kuratowski.pdf")
1058
1059
    <...>

1060
    .. figure:: kuratowski.*
1061
1062
1063
1064
1065
1066
1067
        :align: center

        Obstructing Kuratowski subgraph of a random graph.

    References
    ----------
    .. [boyer-myrvold] John M. Boyer and Wendy J. Myrvold, "On the Cutting Edge:
Tiago Peixoto's avatar
Tiago Peixoto committed
1068
1069
       Simplified O(n) Planarity by Edge Addition" Journal of Graph Algorithms
       and Applications, 8(2): 241-273, 2004. http://www.emis.ams.org/journals/JGAA/accepted/2004/BoyerMyrvold2004.8.3.pdf
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
    .. [boost-planarity] http://www.boost.org/libs/graph/doc/boyer_myrvold.html
    """

    g.stash_filter(directed=True)
    g.set_directed(False)

    if embedding:
        embed = g.new_vertex_property("vector<int>")
    else:
        embed = None

    if kuratowski:
        kur = g.new_edge_property("bool")
    else:
        kur = None

    try:
        is_planar = libgraph_tool_topology.is_planar(g._Graph__graph,
                                                     _prop("v", g, embed),
                                                     _prop("e", g, kur))
    finally:
        g.pop_filter(directed=True)

    ret = [is_planar]
1094
    if embed is not None:
1095
        ret.append(embed)
1096
    if kur is not None:
1097
1098
1099
1100
1101
        ret.append(kur)
    if len(ret) == 1:
        return ret[0]
    else:
        return tuple(ret)
1102
1103
1104
1105


def max_cardinality_matching(g, heuristic=False, weight=None, minimize=True,
                             match=None):
Tiago Peixoto's avatar
Tiago Peixoto committed
1106
    r"""Find a maximum cardinality matching in the graph.
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    heuristic : bool (optional, default: `False`)
        If true, a random heuristic will be used, which runs in linear time.
    weight : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        If provided, the matching will minimize the edge weights (or maximize
        if ``minimize == False``. This option has no effect if
        ``heuristic == False``.
    minimize : bool (optional, default: `True`)
        If `True`, the matching will minimize the weights, otherwise they will
        be maximized. This option has no effect if ``heuristic == False``.
    match : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        Edge property map where the matching will be specified.

    Returns
    -------
    match : :class:`~graph_tool.PropertyMap`
        Boolean edge property map where the matching is specified.
    is_maximal : bool
        True if the matching is indeed maximal, or False otherwise. This is only
        returned if ``heuristic == False``.

    Notes
    -----
    A *matching* is a subset of the edges of a graph such that no two edges
    share a common vertex. A *maximum cardinality matching* has maximum size
    over all matchings in the graph.

Tiago Peixoto's avatar
Tiago Peixoto committed
1138
1139
1140
1141
    This algorithm runs in time :math:`O(EV\times\alpha(E,V))`, where
    :math:`\alpha(m,n)` is a slow growing function that is at most 4 for any
    feasible input. If `heuristic == True`, the algorithm runs in time :math:`O(V + E)`.

1142
1143
1144
1145
    For a more detailed description, see [boost-max-matching]_.

    Examples
    --------
Tiago Peixoto's avatar
Tiago Peixoto committed
1146
    >>> from numpy.random import seed
1147
    >>> seed(43)
Tiago Peixoto's avatar
Tiago Peixoto committed
1148
    >>> g = gt.GraphView(gt.price_network(300), directed=False)
1149
1150
1151
    >>> res = gt.max_cardinality_matching(g)
    >>> print res[1]
    True
Tiago Peixoto's avatar
Tiago Peixoto committed
1152
1153
1154
1155
    >>> w = res[0].copy("double")
    >>> w.a = 2 * w.a + 2
    >>> gt.graph_draw(g, edge_color=res[0], edge_pen_width=w, vertex_fill_color="grey",
    ...               output="max_card_match.pdf")
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
    <...>

    .. figure:: max_card_match.*
        :align: center

        Edges belonging to the matching are in red.

    References
    ----------
    .. [boost-max-matching] http://www.boost.org/libs/graph/doc/maximum_matching.html
    .. [matching-heuristic] B. Hendrickson and R. Leland. "A Multilevel Algorithm
       for Partitioning Graphs." In S. Karin, editor, Proc. Supercomputing ’95,
       San Diego. ACM Press, New York, 1995, :doi:`10.1145/224170.224228`

    """
    if match is None:
        match = g.new_edge_property("bool")
    _check_prop_scalar(match, "match")
    _check_prop_writable(match, "match")
    if weight is not None:
        _check_prop_scalar(weight, "weight")

    seed = numpy.random.randint(0, sys.maxint)
    u = GraphView(g, directed=False)
    if not heuristic:
        check = libgraph_tool_flow.\
                max_cardinality_matching(u._Graph__graph, _prop("e", u, match))
        return match, check
    else:
        libgraph_tool_topology.\
                random_matching(u._Graph__graph, _prop("e", u, weight),
                                 _prop("e", u, match), minimize, seed)
        return match
1189
1190
1191


def max_independent_vertex_set(g, high_deg=False, mivs=None):
Tiago Peixoto's avatar
Tiago Peixoto committed
1192
    r"""Find a maximal independent vertex set in the graph.
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    high_deg : bool (optional, default: `False`)
        If `True`, vertices with high degree will be included first in the set,
        otherwise they will be included last.
    mivs : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        Vertex property map where the vertex set will be specified.

    Returns
    -------
Tiago Peixoto's avatar
Tiago Peixoto committed
1206
1207
    mivs : :class:`~graph_tool.PropertyMap`
        Boolean vertex property map where the set is specified.
1208
1209
1210

    Notes
    -----
Tiago Peixoto's avatar
Tiago Peixoto committed
1211
1212
1213
    A maximal independent vertex set is an independent set such that adding any
    other vertex to the set forces the set to contain an edge between two
    vertices of the set.
1214

Tiago Peixoto's avatar
Tiago Peixoto committed
1215
1216
    This implements the algorithm described in [mivs-luby]_, which runs in time
    :math:`O(V + E)`.
1217
1218
1219

    Examples
    --------
Tiago Peixoto's avatar
Tiago Peixoto committed
1220
    >>> from numpy.random import seed
1221
    >>> seed(43)
Tiago Peixoto's avatar
Tiago Peixoto committed
1222
1223
1224
    >>> g = gt.GraphView(gt.price_network(300), directed=False)
    >>> res = gt.max_independent_vertex_set(g)
    >>> gt.graph_draw(g, vertex_fill_color=res, output="mivs.pdf")
1225
1226
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
1227
    .. figure:: mivs.*
1228
1229
        :align: center

Tiago Peixoto's avatar
Tiago Peixoto committed
1230
        Vertices belonging to the set are in red.
1231
1232
1233

    References
    ----------
Tiago Peixoto's avatar
Tiago Peixoto committed
1234
1235
1236
1237
    .. [mivs-wikipedia] http://en.wikipedia.org/wiki/Independent_set_%28graph_theory%29
    .. [mivs-luby] Luby, M., "A simple parallel algorithm for the maximal independent set problem",
       Proc. 17th Symposium on Theory of Computing, Association for Computing Machinery, pp. 1–10, (1985)
       :doi:`10.1145/22145.22146`.
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251

    """
    if mivs is None:
        mivs = g.new_vertex_property("bool")
    _check_prop_scalar(mivs, "mivs")
    _check_prop_writable(mivs, "mivs")

    seed = numpy.random.randint(0, sys.maxint)
    u = GraphView(g, directed=False)
    libgraph_tool_topology.\
        maximal_vertex_set(u._Graph__graph, _prop("v", u, mivs), high_deg,
                           seed)
    mivs = g.own_property(mivs)
    return mivs