__init__.py 36.3 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) 2007-2011 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`` - Important functions for 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
33
   isomorphism
34
35
   subgraph_isomorphism
   mark_subgraph
36
37
38
39
40
41
   min_spanning_tree
   dominator_tree
   topological_sort
   transitive_closure
   label_components
   label_biconnected_components
42
   label_largest_component
43
   is_planar
44
45
46

Contents
++++++++
47

48
49
"""

Tiago Peixoto's avatar
Tiago Peixoto committed
50
from .. dl_import import dl_import
51
dl_import("import libgraph_tool_topology")
52

53
from .. import _prop, Vector_int32_t, _check_prop_writable, \
54
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView
55
56
57
import random, sys, numpy, weakref
__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
           "min_spanning_tree", "dominator_tree", "topological_sort",
58
           "transitive_closure", "label_components", "label_largest_component",
59
           "label_biconnected_components", "shortest_distance",
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
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
           "shortest_path", "is_planar", "similarity"]


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`
        second graph to be compared.
    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
123

Tiago Peixoto's avatar
Tiago Peixoto committed
124

125
def isomorphism(g1, g2, isomap=False):
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
    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

144
    """
145
146
    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
147
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
148
                             _prop("v", g1, imap))
149
150
151
152
153
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
154

155
def subgraph_isomorphism(sub, g, max_n=0, random=True):
156
    r"""
157
158
    Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n`
    subgraphs, if `max_n > 0`).
159

160
161
162
    If `random` = True, the vertices of `g` are indexed in random order before
    the search.

163
164
165
166
167
168
169
170
171
172
173
174
    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)
Tiago Peixoto's avatar
Tiago Peixoto committed
175
    93
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
    >>> 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 *= 1.5
    >>> ewidth.a += 0.5
    >>> gt.graph_draw(g, vcolor=vmask, ecolor=emask, penwidth=ewidth,
    ...               output="subgraph-iso-embed.png")
    <...>
    >>> gt.graph_draw(sub, output="subgraph-iso.png")
    <...>

    .. image:: subgraph-iso.png
    .. image:: subgraph-iso-embed.png

    *Left:* Subgraph searched, *Right:* One isomorphic subgraph found in main
     graph.

    Notes
    -----
202
203
204
205
    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`.
206
207
208

    References
    ----------
209
    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
Tiago Peixoto's avatar
Tiago Peixoto committed
210
       isomorphism", Journal of the ACM 23 (1): 31–42, 1976, :doi:`10.1145/321921.321925`
211
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
212
213
214
215

    """
    # 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
216
217
    vlabels=(None, None)
    elabels=(None, None)
218
219
    vmaps = []
    emaps = []
220
221
222
223
    if random:
        seed = numpy.random.randint(0, sys.maxint)
    else:
        seed = 42
224
225
226
227
228
229
    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]),
230
                                vmaps, emaps, max_n, seed)
231
232
233
234
235
    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
236

237
238
239
240
241
242
243
244
245
246
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
247
    `sub`.
248
    """
249
    if vmask is None:
250
        vmask = g.new_vertex_property("bool")
251
    if emask is None:
252
253
254
255
256
257
258
259
260
261
262
263
264
265
        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
266

Tiago Peixoto's avatar
Tiago Peixoto committed
267

268
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
269
270
271
272
273
274
275
276
277
278
279
    """
    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)
280
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
        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
298
    >>> from numpy.random import seed, random
299
    >>> seed(42)
300
301
302
    >>> 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
303
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
304
    >>> tree = gt.min_spanning_tree(g, weights=weight)
Tiago Peixoto's avatar
Tiago Peixoto committed
305
    >>> gt.graph_draw(g, pos=pos, pin=True, output="triang_orig.png")
306
307
    <...>
    >>> g.set_edge_filter(tree)
Tiago Peixoto's avatar
Tiago Peixoto committed
308
    >>> gt.graph_draw(g, pos=pos, pin=True, output="triang_min_span_tree.png")
309
310
311
312
    <...>


    .. image:: triang_orig.png
Tiago Peixoto's avatar
Tiago Peixoto committed
313
314
315
        :width: 400px
    .. image:: triang_min_span_tree.png
        :width: 400px
316
317

    *Left:* Original graph, *Right:* The minimum spanning tree.
318
319
320
321
322

    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
323
324
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
325
326
327
328
329
    .. [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
    """
330
    if tree_map is None:
331
332
333
334
        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.")

335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
    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)
350
    return tree_map
351

Tiago Peixoto's avatar
Tiago Peixoto committed
352

Tiago Peixoto's avatar
Tiago Peixoto committed
353
354
355
356
357
358
359
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
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)
385
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
386
387
    >>> dom = gt.dominator_tree(g, root[0])
    >>> print dom.a
Tiago Peixoto's avatar
Tiago Peixoto committed
388
389
390
391
    [ 0  0 72  0  0  0  0  0  0  0  0  0  0  0 21  0  0  0  0  0  0  3  0  0  0
      0  0  0  0  0  0 41  0  0  0  0  0  0  0  0  0 11  0  0  0  0  0  0  0  0
      0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  3  0  0
      0  0  0  0  0  2  0  0  0  0  0  0  0 80  0  0  0  0  0  0  0  0  0  0  0]
Tiago Peixoto's avatar
Tiago Peixoto committed
392
393
394

    References
    ----------
395
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
396
397

    """
398
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
399
400
401
        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" +
402
403
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
404
        raise ValueError("dominator tree requires a directed graph.")
405
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
406
407
408
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
409

Tiago Peixoto's avatar
Tiago Peixoto committed
410

411
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
    """
    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
433
434
    [19 27  1  7  0 23  8 16  2 15 24 12  3  4 22  5  6  9 10 11 18 13 21 14 20
     17 25 26 28 29]
Tiago Peixoto's avatar
Tiago Peixoto committed
435
436
437

    References
    ----------
438
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
439
440
441
442
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

443
444
445
    topological_order = Vector_int32_t()
    libgraph_tool_topology.\
               topological_sort(g._Graph__graph, topological_order)
Tiago Peixoto's avatar
Tiago Peixoto committed
446
    return numpy.array(topological_order)
447

Tiago Peixoto's avatar
Tiago Peixoto committed
448

449
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
    """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
    ----------
470
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
471
472
473
474
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

475
476
477
478
479
480
481
    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
482

483
484
def label_components(g, vprop=None, directed=None):
    """
485
    Label the components to which each vertex in the graph belongs. If the
486
487
    graph is directed, it finds the strongly connected components.

488
489
490
    A property map with the component labels is returned, together with an
    histogram of component labels.

491
492
    Parameters
    ----------
493
    g : :class:`~graph_tool.Graph`
494
        Graph to be used.
495
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
496
497
498
499
500
501
502
503
        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
    -------
504
    comp : :class:`~graph_tool.PropertyMap`
505
        Vertex property map with component labels.
506
507
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
508
509
510
511
512
513

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

514
    The algorithm runs in :math:`O(V + E)` time.
515
516
517

    Examples
    --------
518
519
520
    >>> from numpy.random import seed
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: (1, 1))
521
    >>> comp, hist = gt.label_components(g)
522
    >>> print comp.get_array()
Tiago Peixoto's avatar
Tiago Peixoto committed
523
524
525
    [0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 2 0 0 0 1 0 0 0 0 1 1 0 2 0 1 1 0 0 0 0 1 0
     0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 1 0 0 0 0 0 1 0 0 0
     1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0]
526
527
    >>> print hist
    [81 15  4]
528
529
    """

530
    if vprop is None:
531
532
533
534
535
        vprop = g.new_vertex_property("int32_t")

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

536
537
    if directed is not None:
        g = GraphView(g, directed=directed)
538

539
540
541
542
543
544
545
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
    return vprop, hist


def label_largest_component(g, directed=None):
    """
546
547
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
548
549
550
551
552
553
554
555
556
557
558
559
560
561

    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`
562
         Boolean vertex property map which labels the largest component.
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584

    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
    [1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0
     0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1
     0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0]
    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
    >>> print u.num_vertices()
    26
    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
585
586
587
588
589
    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)
590
    return label
591

Tiago Peixoto's avatar
Tiago Peixoto committed
592

593
def label_biconnected_components(g, eprop=None, vprop=None):
594
595
596
597
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

598
599
600
601
    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.

602
603
604
605
606
607
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
    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
646
    >>> seed(43)
647
    >>> g = gt.random_graph(100, lambda: 2, directed=False)
648
    >>> comp, art, hist = gt.label_biconnected_components(g)
649
    >>> print comp.a
Tiago Peixoto's avatar
Tiago Peixoto committed
650
651
652
    [1 0 0 0 2 0 1 0 0 0 0 0 1 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1 0 0 0 0 0
     1 0 1 3 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0
     0 0 0 0 0 0 0 0 0 0 1 3 1 0 2 1 0 0 0 0 0 2 0 0 0 2]
653
654
655
656
    >>> 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]
657
658
    >>> print hist
    [77 13  6  4]
659
660

    """
661

662
    if vprop is None:
663
        vprop = g.new_vertex_property("bool")
664
    if eprop is None:
665
666
667
668
669
670
671
        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")

672
673
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
674
675
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
676
    return eprop, vprop, hist
677

Tiago Peixoto's avatar
Tiago Peixoto committed
678

679
def shortest_distance(g, source=None, weights=None, max_dist=None,
680
681
                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
682
683
684
685
686
687
688
689
690
    """
    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)
691
        Source vertex of the search. If unspecified, the all pairs shortest
692
693
694
695
696
697
        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
698
        are searched. This parameter has no effect if source is None.
699
700
701
702
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
703
704
        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
705
706
707
708
        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.
709
710
711
    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
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

    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))
    >>> print dist.get_array()
Tiago Peixoto's avatar
Tiago Peixoto committed
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
    [         0          3          5          4 2147483647          1
              6          3          4          4          5          4
              4          4          4          4          1          3
              3          1          5          3 2147483647          4
              2          5          5 2147483647          5          5
              4          3          3          2          4          4
              4          4          5          5 2147483647 2147483647
              4          4          3          5          3          4
     2147483647          3          2          4          5          5
              3          3          3          5          4 2147483647
              3          4          5          4          2 2147483647
              4          3          2          4          2 2147483647
              3          3          4          3          4          5
              2          3          6          4          4 2147483647
              6          4          5          1          4          5
              3          4          4          2          4          6
              3          4          2          4]
756
757
    >>> dist = gt.shortest_distance(g)
    >>> print array(dist[g.vertex(0)])
Tiago Peixoto's avatar
Tiago Peixoto committed
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
    [         0          3          5          4 2147483647          1
              6          3          4          4          5          4
              4          4          4          4          1          3
              3          1          5          3 2147483647          4
              2          5          5 2147483647          5          5
              4          3          3          2          4          4
              4          4          5          5 2147483647 2147483647
              4          4          3          5          3          4
     2147483647          3          2          4          5          5
              3          3          3          5          4 2147483647
              3          4          5          4          2 2147483647
              4          3          2          4          2 2147483647
              3          3          4          3          4          5
              2          3          6          4          4 2147483647
              6          4          5          1          4          5
              3          4          4          2          4          6
              3          4          2          4]
775
776
777
778
779

    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
780
781
       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
782
783
    .. [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
784
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
785
786
787
788
    .. [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
    """

789
    if weights is None:
790
791
792
793
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

794
795
    if dist_map is None:
        if source is not None:
796
797
798
799
800
            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")
801
    if source is not None:
802
803
804
805
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

806
    if max_dist is None:
807
808
        max_dist = 0

809
    if directed is not None:
810
811
812
813
        g.stash_filter(directed=True)
        g.set_directed(directed)

    try:
814
        if source is not None:
815
            pmap = g.copy_property(g.vertex_index, value_type="int64_t")
816
817
818
            libgraph_tool_topology.get_dists(g._Graph__graph, int(source),
                                             _prop("v", g, dist_map),
                                             _prop("e", g, weights),
819
                                             _prop("v", g, pmap),
820
821
822
823
824
825
826
                                             float(max_dist))
        else:
            libgraph_tool_topology.get_all_dists(g._Graph__graph,
                                                 _prop("v", g, dist_map),
                                                 _prop("e", g, weights), dense)

    finally:
827
        if directed is not None:
828
            g.pop_filter(directed=True)
829
    if source is not None and pred_map:
830
831
832
833
        return dist_map, pmap
    else:
        return dist_map

Tiago Peixoto's avatar
Tiago Peixoto committed
834

835
836
837
838
839
840
841
842
843
844
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
845
    target : :class:`~graph_tool.Vertex`
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
        Target vertex of the search.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
    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
878
    ['10', '66', '46', '266', '101', '143', '91', '275', '82', '11']
879
    >>> print [str(e) for e in elist]
Tiago Peixoto's avatar
Tiago Peixoto committed
880
    ['(10,66)', '(66,46)', '(46,266)', '(266,101)', '(101,143)', '(143,91)', '(91,275)', '(275,82)', '(82,11)']
881
882
883
884
885

    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
886
887
       Press
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
888
889
    .. [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
890
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
891
892
    """

893
    if pred_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
894
895
        pred_map = shortest_distance(g, source, weights=weights,
                                     pred_map=True)[1]
896

Tiago Peixoto's avatar
Tiago Peixoto committed
897
    if pred_map[target] == int(target):  # no path to source
898
899
900
901
902
        return [], []

    vlist = [target]
    elist = []

903
    if weights is not None:
904
905
906
907
908
909
910
911
912
913
914
915
916
        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:
917
                if weights is not None:
918
919
920
921
922
923
924
925
926
927
928
                    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

929
930
931
932
933
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

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
975
    [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]
976
977
978
979
980
    >>> 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)
981
    >>> gt.graph_draw(g, output="kuratowski.png")
982
983
984
985
986
987
988
989
990
991
    <...>

    .. figure:: kuratowski.png
        :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
992
993
       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
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
    .. [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]
1018
    if embed is not None:
1019
        ret.append(embed)
1020
    if kur is not None:
1021
1022
1023
1024
1025
        ret.append(kur)
    if len(ret) == 1:
        return ret[0]
    else:
        return tuple(ret)