__init__.py 34.1 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) 2007-2011 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
60
           "label_biconnected_components", "shortest_distance",
           "shortest_path", "is_planar"]
61

Tiago Peixoto's avatar
Tiago Peixoto committed
62

63
def isomorphism(g1, g2, isomap=False):
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
    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

82
    """
83
84
    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
85
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
86
                             _prop("v", g1, imap))
87
88
89
90
91
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
92

93
def subgraph_isomorphism(sub, g, max_n=0, random=True):
94
    r"""
95
96
    Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n`
    subgraphs, if `max_n > 0`).
97

98
99
100
    If `random` = True, the vertices of `g` are indexed in random order before
    the search.

101
102
103
104
105
106
107
108
109
110
111
112
    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
113
    93
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
    >>> 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
    -----
140
141
142
143
    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`.
144
145
146

    References
    ----------
147
    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
Tiago Peixoto's avatar
Tiago Peixoto committed
148
       isomorphism", Journal of the ACM 23 (1): 31–42, 1976, :doi:`10.1145/321921.321925`
149
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
150
151
152
153

    """
    # 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
154
155
    vlabels=(None, None)
    elabels=(None, None)
156
157
    vmaps = []
    emaps = []
158
159
160
161
    if random:
        seed = numpy.random.randint(0, sys.maxint)
    else:
        seed = 42
162
163
164
165
166
167
    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]),
168
                                vmaps, emaps, max_n, seed)
169
170
171
172
173
    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
174

175
176
177
178
179
180
181
182
183
184
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
185
    `sub`.
186
    """
187
    if vmask is None:
188
        vmask = g.new_vertex_property("bool")
189
    if emask is None:
190
191
192
193
194
195
196
197
198
199
200
201
202
203
        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
204

Tiago Peixoto's avatar
Tiago Peixoto committed
205

206
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
207
208
209
210
211
212
213
214
215
216
217
    """
    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)
218
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
        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
236
    >>> from numpy.random import seed, random
237
    >>> seed(42)
238
239
240
    >>> 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
241
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
242
    >>> tree = gt.min_spanning_tree(g, weights=weight)
Tiago Peixoto's avatar
Tiago Peixoto committed
243
    >>> gt.graph_draw(g, pos=pos, pin=True, output="triang_orig.png")
244
245
    <...>
    >>> g.set_edge_filter(tree)
Tiago Peixoto's avatar
Tiago Peixoto committed
246
    >>> gt.graph_draw(g, pos=pos, pin=True, output="triang_min_span_tree.png")
247
248
249
250
    <...>


    .. image:: triang_orig.png
Tiago Peixoto's avatar
Tiago Peixoto committed
251
252
253
        :width: 400px
    .. image:: triang_min_span_tree.png
        :width: 400px
254
255

    *Left:* Original graph, *Right:* The minimum spanning tree.
256
257
258
259
260

    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
261
262
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
263
264
265
266
267
    .. [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
    """
268
    if tree_map is None:
269
270
271
272
        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.")

273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
    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)
288
    return tree_map
289

Tiago Peixoto's avatar
Tiago Peixoto committed
290

Tiago Peixoto's avatar
Tiago Peixoto committed
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
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)
323
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
324
325
    >>> dom = gt.dominator_tree(g, root[0])
    >>> print dom.a
Tiago Peixoto's avatar
Tiago Peixoto committed
326
327
328
329
    [ 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
330
331
332

    References
    ----------
333
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
334
335

    """
336
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
337
338
339
        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" +
340
341
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
342
        raise ValueError("dominator tree requires a directed graph.")
343
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
344
345
346
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
347

Tiago Peixoto's avatar
Tiago Peixoto committed
348

349
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
    """
    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
371
372
    [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
373
374
375

    References
    ----------
376
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
377
378
379
380
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

381
382
383
    topological_order = Vector_int32_t()
    libgraph_tool_topology.\
               topological_sort(g._Graph__graph, topological_order)
Tiago Peixoto's avatar
Tiago Peixoto committed
384
    return numpy.array(topological_order)
385

Tiago Peixoto's avatar
Tiago Peixoto committed
386

387
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
    """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
    ----------
408
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
409
410
411
412
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

413
414
415
416
417
418
419
    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
420

421
422
def label_components(g, vprop=None, directed=None):
    """
423
    Label the components to which each vertex in the graph belongs. If the
424
425
    graph is directed, it finds the strongly connected components.

426
427
428
    A property map with the component labels is returned, together with an
    histogram of component labels.

429
430
    Parameters
    ----------
431
    g : :class:`~graph_tool.Graph`
432
        Graph to be used.
433
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: None)
434
435
436
437
438
439
440
441
        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
    -------
442
    comp : :class:`~graph_tool.PropertyMap`
443
        Vertex property map with component labels.
444
445
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
446
447
448
449
450
451

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

452
    The algorithm runs in :math:`O(V + E)` time.
453
454
455

    Examples
    --------
456
457
458
    >>> from numpy.random import seed
    >>> seed(43)
    >>> g = gt.random_graph(100, lambda: (1, 1))
459
    >>> comp, hist = gt.label_components(g)
460
    >>> print comp.get_array()
Tiago Peixoto's avatar
Tiago Peixoto committed
461
462
463
    [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]
464
465
    >>> print hist
    [81 15  4]
466
467
    """

468
    if vprop is None:
469
470
471
472
473
        vprop = g.new_vertex_property("int32_t")

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

474
475
    if directed is not None:
        g = GraphView(g, directed=directed)
476

477
478
479
480
481
482
483
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
    return vprop, hist


def label_largest_component(g, directed=None):
    """
484
485
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
486
487
488
489
490
491
492
493
494
495
496
497
498
499

    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`
500
         Boolean vertex property map which labels the largest component.
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524

    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)
    label.a = c.a == h.argmax()
    return label
525

Tiago Peixoto's avatar
Tiago Peixoto committed
526

527
def label_biconnected_components(g, eprop=None, vprop=None):
528
529
530
531
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

532
533
534
535
    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.

536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
    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
580
    >>> seed(43)
581
    >>> g = gt.random_graph(100, lambda: 2, directed=False)
582
    >>> comp, art, hist = gt.label_biconnected_components(g)
583
    >>> print comp.a
Tiago Peixoto's avatar
Tiago Peixoto committed
584
585
586
    [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]
587
588
589
590
    >>> 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]
591
592
    >>> print hist
    [77 13  6  4]
593
594

    """
595

596
    if vprop is None:
597
        vprop = g.new_vertex_property("bool")
598
    if eprop is None:
599
600
601
602
603
604
605
        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")

606
607
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
608
609
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
610
    return eprop, vprop, hist
611

Tiago Peixoto's avatar
Tiago Peixoto committed
612

613
def shortest_distance(g, source=None, weights=None, max_dist=None,
614
615
                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
616
617
618
619
620
621
622
623
624
    """
    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)
625
        Source vertex of the search. If unspecified, the all pairs shortest
626
627
628
629
630
631
        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
632
        are searched. This parameter has no effect if source is None.
633
634
635
636
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
637
638
        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
639
640
641
642
        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.
643
644
645
    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672

    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
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
    [         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]
690
691
    >>> dist = gt.shortest_distance(g)
    >>> print array(dist[g.vertex(0)])
Tiago Peixoto's avatar
Tiago Peixoto committed
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
    [         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]
709
710
711
712
713

    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
714
715
       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
716
717
    .. [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
718
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
719
720
721
722
    .. [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
    """

723
    if weights is None:
724
725
726
727
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

728
729
    if dist_map is None:
        if source is not None:
730
731
732
733
734
            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")
735
    if source is not None:
736
737
738
739
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

740
    if max_dist is None:
741
742
        max_dist = 0

743
    if directed is not None:
744
745
746
747
        g.stash_filter(directed=True)
        g.set_directed(directed)

    try:
748
        if source is not None:
749
            pmap = g.copy_property(g.vertex_index, value_type="int64_t")
750
751
752
            libgraph_tool_topology.get_dists(g._Graph__graph, int(source),
                                             _prop("v", g, dist_map),
                                             _prop("e", g, weights),
753
                                             _prop("v", g, pmap),
754
755
756
757
758
759
760
                                             float(max_dist))
        else:
            libgraph_tool_topology.get_all_dists(g._Graph__graph,
                                                 _prop("v", g, dist_map),
                                                 _prop("e", g, weights), dense)

    finally:
761
        if directed is not None:
762
            g.pop_filter(directed=True)
763
    if source is not None and pred_map:
764
765
766
767
        return dist_map, pmap
    else:
        return dist_map

Tiago Peixoto's avatar
Tiago Peixoto committed
768

769
770
771
772
773
774
775
776
777
778
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
779
    target : :class:`~graph_tool.Vertex`
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
        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
812
    ['10', '66', '46', '266', '101', '143', '91', '275', '82', '11']
813
    >>> print [str(e) for e in elist]
Tiago Peixoto's avatar
Tiago Peixoto committed
814
    ['(10,66)', '(66,46)', '(46,266)', '(266,101)', '(101,143)', '(143,91)', '(91,275)', '(275,82)', '(82,11)']
815
816
817
818
819

    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
820
821
       Press
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
822
823
    .. [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
824
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
825
826
    """

827
    if pred_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
828
829
        pred_map = shortest_distance(g, source, weights=weights,
                                     pred_map=True)[1]
830

Tiago Peixoto's avatar
Tiago Peixoto committed
831
    if pred_map[target] == int(target):  # no path to source
832
833
834
835
836
        return [], []

    vlist = [target]
    elist = []

837
    if weights is not None:
838
839
840
841
842
843
844
845
846
847
848
849
850
        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:
851
                if weights is not None:
852
853
854
855
856
857
858
859
860
861
862
                    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

863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908

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
909
    [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]
910
911
912
913
914
    >>> 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)
915
    >>> gt.graph_draw(g, output="kuratowski.png")
916
917
918
919
920
921
922
923
924
925
    <...>

    .. 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
926
927
       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
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
    .. [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]
952
    if embed is not None:
953
        ret.append(embed)
954
    if kur is not None:
955
956
957
958
959
        ret.append(kur)
    if len(ret) == 1:
        return ret[0]
    else:
        return tuple(ret)