__init__.py 65.6 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) 2006-2013 Tiago de Paula Peixoto <tiago@skewed.de>
7
8
9
10
11
12
13
14
15
16
17
18
19
20
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

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

Summary
+++++++

.. autosummary::
   :nosignatures:

31
   shortest_distance
Tiago Peixoto's avatar
Tiago Peixoto committed
32
   shortest_path
Tiago Peixoto's avatar
Tiago Peixoto committed
33
   pseudo_diameter
34
   similarity
35
   isomorphism
36
37
   subgraph_isomorphism
   mark_subgraph
38
39
   max_cardinality_matching
   max_independent_vertex_set
40
   min_spanning_tree
41
   random_spanning_tree
42
43
44
   dominator_tree
   topological_sort
   transitive_closure
Tiago Peixoto's avatar
Tiago Peixoto committed
45
   tsp_tour
46
   sequential_vertex_coloring
47
48
   label_components
   label_biconnected_components
49
   label_largest_component
50
   label_out_component
Tiago Peixoto's avatar
Tiago Peixoto committed
51
   kcore_decomposition
52
   is_bipartite
Tiago Peixoto's avatar
Tiago Peixoto committed
53
   is_DAG
54
   is_planar
55
   make_maximal_planar
Tiago Peixoto's avatar
Tiago Peixoto committed
56
   edge_reciprocity
57
58
59

Contents
++++++++
60

61
62
"""

63
64
from __future__ import division, absolute_import, print_function

Tiago Peixoto's avatar
Tiago Peixoto committed
65
from .. dl_import import dl_import
66
dl_import("from . import libgraph_tool_topology")
67

68
from .. import _prop, Vector_int32_t, _check_prop_writable, \
69
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView,\
Tiago Peixoto's avatar
Tiago Peixoto committed
70
     libcore, _get_rng, _degree
71
import random, sys, numpy
72
__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
73
           "max_cardinality_matching", "max_independent_vertex_set",
74
           "min_spanning_tree", "random_spanning_tree", "dominator_tree",
Tiago Peixoto's avatar
Tiago Peixoto committed
75
           "topological_sort", "transitive_closure", "tsp_tour",
76
77
           "sequential_vertex_coloring", "label_components",
           "label_largest_component", "label_biconnected_components",
Tiago Peixoto's avatar
Tiago Peixoto committed
78
79
80
           "label_out_component", "kcore_decomposition", "shortest_distance",
           "shortest_path", "pseudo_diameter", "is_bipartite", "is_DAG",
           "is_planar", "make_maximal_planar", "similarity", "edge_reciprocity"]
81
82
83
84
85
86
87
88
89
90


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
91
        Second graph to be compared.
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
    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
    --------
118
119
120
121
122
123
124
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

125
126
127
128
    >>> g = gt.random_graph(100, lambda: (3,3))
    >>> u = g.copy()
    >>> gt.similarity(u, g)
    1.0
Tiago Peixoto's avatar
Tiago Peixoto committed
129
130
    >>> gt.random_rewire(u)
    21
131
    >>> gt.similarity(u, g)
Tiago Peixoto's avatar
Tiago Peixoto committed
132
    0.03
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
    """

    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
149

Tiago Peixoto's avatar
Tiago Peixoto committed
150

151
def isomorphism(g1, g2, isomap=False):
152
153
154
155
156
157
158
    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
    --------
159
160
161
162
163
164
165
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

166
167
168
169
170
171
172
173
174
    >>> 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

175
    """
176
177
    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
178
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
179
                             _prop("v", g1, imap))
180
181
182
183
184
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
185

186
187
188
def subgraph_isomorphism(sub, g, max_n=0, vertex_label=None, edge_label=None,
                         random=False):
    r"""Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n` subgraphs, if `max_n > 0`).
189

190

Tiago Peixoto's avatar
Tiago Peixoto committed
191
192
193
194
195
196
    Parameters
    ----------
    sub : :class:`~graph_tool.Graph`
        Subgraph for which to be searched.
    g : :class:`~graph_tool.Graph`
        Graph in which the search is performed.
197
    max_n : int (optional, default: `0`)
Tiago Peixoto's avatar
Tiago Peixoto committed
198
199
        Maximum number of matches to find. If `max_n == 0`, all matches are
        found.
200
201
202
203
204
205
206
207
    vertex_label : pair of :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        If provided, this should be a pair of :class:`~graph_tool.PropertyMap`
        objects, belonging to `sub` and `g` (in this order), which specify vertex labels
        which should match, in addition to the topological isomorphism.
    edge_label : pair of :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        If provided, this should be a pair of :class:`~graph_tool.PropertyMap`
        objects, belonging to `sub` and `g` (in this order), which specify edge labels
        which should match, in addition to the topological isomorphism.
Tiago Peixoto's avatar
Tiago Peixoto committed
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
    random : bool (optional, default: False)
        If `True`, the vertices of `g` are indexed in random order before
        the search.

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

    Notes
    -----
    The algorithm used is described in [ullmann-algorithm-1976]_. It has a
    worse-case complexity of :math:`O(N_g^{N_{sub}})`, but for random graphs it
    typically has a complexity of :math:`O(N_g^\gamma)` with :math:`\gamma`
    depending sub-linearly on the size of `sub`.
229
230
231

    Examples
    --------
232
233
234
235
236
237
238
239
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(44)
       gt.seed_rng(44)

    >>> from numpy.random import poisson
Tiago Peixoto's avatar
Tiago Peixoto committed
240
    >>> g = gt.random_graph(30, lambda: (poisson(6.1), poisson(6.1)))
241
    >>> sub = gt.random_graph(10, lambda: (poisson(1.9), poisson(1.9)))
242
    >>> vm, em = gt.subgraph_isomorphism(sub, g)
243
    >>> print(len(vm))
Tiago Peixoto's avatar
Tiago Peixoto committed
244
    35
245
    >>> for i in range(len(vm)):
246
247
248
249
250
251
252
253
254
255
    ...   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
256
257
258
    >>> ewidth.a *= 2
    >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
    ...               edge_pen_width=ewidth, output_size=(200, 200),
259
    ...               output="subgraph-iso-embed.pdf")
260
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
261
    >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf")
262
263
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
264
265
266
267
268
269
270
271
    .. testcode::
       :hide:

       gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
                     edge_pen_width=ewidth, output_size=(200, 200),
                     output="subgraph-iso-embed.png")
       gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.png")

Tiago Peixoto's avatar
Tiago Peixoto committed
272
273
    .. image:: subgraph-iso.*
    .. image:: subgraph-iso-embed.*
274

275

Tiago Peixoto's avatar
Tiago Peixoto committed
276
    **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph.
277
278
279

    References
    ----------
280
    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
281
       isomorphism", Journal of the ACM 23 (1): 31-42, 1976, :doi:`10.1145/321921.321925`
282
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
283
284

    """
285
286
    if sub.num_vertices() == 0:
        raise ValueError("Cannot search for an empty subgraph.")
287
288
289
290
291
292
293
294
    if vertex_label is None:
        vertex_label = (None, None)
    elif vertex_label[0].value_type() != vertex_label[1].value_type():
        raise ValueError("Both vertex label property maps must be of the same type!")
    if edge_label is None:
        edge_label = (None, None)
    elif edge_label[0].value_type() != edge_label[1].value_type():
        raise ValueError("Both edge label property maps must be of the same type!")
295
296
    vmaps = []
    emaps = []
297
    if random:
298
        rng = _get_rng()
299
    else:
300
        rng = libcore.rng_t()
301
302
    libgraph_tool_topology.\
           subgraph_isomorphism(sub._Graph__graph, g._Graph__graph,
303
304
305
306
                                _prop("v", sub, vertex_label[0]),
                                _prop("v", g, vertex_label[1]),
                                _prop("e", sub, edge_label[0]),
                                _prop("e", g, edge_label[1]),
307
                                vmaps, emaps, max_n, rng)
308
    for i in range(len(vmaps)):
309
310
311
312
        vmaps[i] = PropertyMap(vmaps[i], sub, "v")
        emaps[i] = PropertyMap(emaps[i], sub, "e")
    return vmaps, emaps

Tiago Peixoto's avatar
Tiago Peixoto committed
313

314
315
316
317
318
319
320
321
322
323
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
324
    `sub`.
325
    """
326
    if vmask is None:
327
        vmask = g.new_vertex_property("bool")
328
    if emask is None:
329
330
331
332
333
334
335
336
337
338
339
340
341
342
        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
343

Tiago Peixoto's avatar
Tiago Peixoto committed
344

345
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
346
347
348
349
350
351
352
    """
    Return the minimum spanning tree of a given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
353
    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
354
355
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
356
    root : :class:`~graph_tool.Vertex` (optional, default: `None`)
357
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
358
        is used. Otherwise, Kruskal's algorithm is used.
359
    tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
        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
    --------
375
376
377
378
379
380
381
382
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

    >>> from numpy.random import random
383
384
385
    >>> 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
386
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
387
    >>> tree = gt.min_spanning_tree(g, weights=weight)
388
    >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf")
389
390
    <...>
    >>> g.set_edge_filter(tree)
391
    >>> gt.graph_draw(g, pos=pos, output="triang_min_span_tree.pdf")
392
393
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
394
395
396
397
398
    .. testcode::
       :hide:

       gt.graph_draw(g, pos=pos, output="triang_orig.png")
       gt.graph_draw(g, pos=pos, output="triang_min_span_tree.png")
399

400
    .. image:: triang_orig.*
Tiago Peixoto's avatar
Tiago Peixoto committed
401
        :width: 400px
402
    .. image:: triang_min_span_tree.*
Tiago Peixoto's avatar
Tiago Peixoto committed
403
        :width: 400px
404
405

    *Left:* Original graph, *Right:* The minimum spanning tree.
406
407
408
409
410

    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
411
412
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
413
414
415
416
417
    .. [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
    """
418
    if tree_map is None:
419
420
421
422
        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.")

423
424
425
426
427
428
429
430
431
432
433
    u = GraphView(g, directed=False)
    if root is None:
        libgraph_tool_topology.\
               get_kruskal_spanning_tree(u._Graph__graph,
                                         _prop("e", g, weights),
                                         _prop("e", g, tree_map))
    else:
        libgraph_tool_topology.\
               get_prim_spanning_tree(u._Graph__graph, int(root),
                                      _prop("e", g, weights),
                                      _prop("e", g, tree_map))
434
    return tree_map
435

Tiago Peixoto's avatar
Tiago Peixoto committed
436

437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
def random_spanning_tree(g, weights=None, root=None, tree_map=None):
    """
    Return a random spanning tree of a given graph, which can be directed or
    undirected.

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

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

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

    Examples
    --------
466
467
468
469
470
471
472
473
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

    >>> from numpy.random import random
474
475
476
477
478
479
480
481
    >>> g, pos = gt.triangulation(random((400, 2)) * 10, type="delaunay")
    >>> weight = g.new_edge_property("double")
    >>> for e in g.edges():
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
    >>> tree = gt.random_spanning_tree(g, weights=weight)
    >>> gt.graph_draw(g, pos=pos, output="rtriang_orig.pdf")
    <...>
    >>> g.set_edge_filter(tree)
Tiago Peixoto's avatar
Tiago Peixoto committed
482
    >>> gt.graph_draw(g, pos=pos, output="triang_random_span_tree.pdf")
483
484
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
485
486
487
488
489
    .. testcode::
       :hide:

       gt.graph_draw(g, pos=pos, output="rtriang_orig.png")
       gt.graph_draw(g, pos=pos, output="triang_random_span_tree.png")
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517

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

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

    References
    ----------

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

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

    # we need to restrict ourselves to the in-component of root
    l = label_out_component(GraphView(g, reversed=True), root)
518
519
520
    u = GraphView(g, vfilt=l)
    if u.num_vertices() != g.num_vertices():
        raise ValueError("There must be a path from all vertices to the root vertex: %d" % int(root) )
521
522
523
524

    libgraph_tool_topology.\
        random_spanning_tree(g._Graph__graph, int(root),
                             _prop("e", g, weights),
525
                             _prop("e", g, tree_map), _get_rng())
526
527
528
    return tree_map


Tiago Peixoto's avatar
Tiago Peixoto committed
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
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
    --------
556
557
558
559
560
561
562
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

Tiago Peixoto's avatar
Tiago Peixoto committed
563
564
565
    >>> g = gt.random_graph(100, lambda: (2, 2))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
566
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
567
    >>> dom = gt.dominator_tree(g, root[0])
568
    >>> print(dom.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
569
    [ 0  0  0  0  0  0 62  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
570
      0  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
571
572
      0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
      0  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
573
574
575

    References
    ----------
576
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
577
578

    """
579
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
580
581
582
        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" +
583
584
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
585
        raise ValueError("dominator tree requires a directed graph.")
586
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
587
588
589
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
590

Tiago Peixoto's avatar
Tiago Peixoto committed
591

592
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
593
594
595
596
597
598
599
600
601
602
603
604
605
606
    """
    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
    --------
607
608
609
610
611
612
613
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

Tiago Peixoto's avatar
Tiago Peixoto committed
614
615
616
617
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
    >>> sort = gt.topological_sort(g)
618
    >>> print(sort)
Tiago Peixoto's avatar
Tiago Peixoto committed
619
620
    [ 1 14  2  7 17  0  3  4  5  6  8  9 22 10 11 12 13 16 23 27 15 18 19 20 21
     24 25 26 28 29]
Tiago Peixoto's avatar
Tiago Peixoto committed
621
622
623

    References
    ----------
624
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
625
626
627
628
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

629
    topological_order = Vector_int32_t()
Tiago Peixoto's avatar
Tiago Peixoto committed
630
631
632
633
634
    is_DAG = libgraph_tool_topology.\
        topological_sort(g._Graph__graph, topological_order)
    if not is_DAG:
        raise ValueError("Graph is not a directed acylic graph (DAG).");
    return topological_order.a.copy()
635

Tiago Peixoto's avatar
Tiago Peixoto committed
636

637
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
638
639
640
641
642
643
644
645
646
647
648
649
650
    """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
    --------
651
652
653
654
655
656
657
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

Tiago Peixoto's avatar
Tiago Peixoto committed
658
659
660
661
662
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tc = gt.transitive_closure(g)

    References
    ----------
663
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
664
665
666
667
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

668
669
670
671
672
673
674
    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
675

676
def label_components(g, vprop=None, directed=None, attractors=False):
677
    """
678
    Label the components to which each vertex in the graph belongs. If the
679
680
    graph is directed, it finds the strongly connected components.

681
682
683
    A property map with the component labels is returned, together with an
    histogram of component labels.

684
685
    Parameters
    ----------
686
    g : :class:`~graph_tool.Graph`
687
        Graph to be used.
688
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
689
690
        Vertex property to store the component labels. If none is supplied, one
        is created.
691
    directed : bool (optional, default: ``None``)
692
693
        Treat graph as directed or not, independently of its actual
        directionality.
694
695
696
697
    attractors : bool (optional, default: ``False``)
        If ``True``, and the graph is directed, an additional array with Boolean
        values is returned, specifying if the strongly connected components are
        attractors or not.
698
699
700

    Returns
    -------
701
    comp : :class:`~graph_tool.PropertyMap`
702
        Vertex property map with component labels.
703
704
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
705
706
707
708
    is_attractor : :class:`~numpy.ndarray`
        A Boolean array specifying if the strongly connected components are
        attractors or not. This returned only if ``attractors == True``, and the
        graph is directed.
709
710
711
712
713
714

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

715
    The algorithm runs in :math:`O(V + E)` time.
716
717
718

    Examples
    --------
719
720
721
722
723
724
    .. testcode::
       :hide:

       numpy.random.seed(43)
       gt.seed_rng(43)

725
726
    >>> g = gt.random_graph(100, lambda: (poisson(2), poisson(2)))
    >>> comp, hist, is_attractor = gt.label_components(g, attractors=True)
727
    >>> print(comp.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
728
729
730
731
    [14 15 14 14 14  5 14 14 18 14 14  8 14 14 13 14 14 21 14 14  7 23 10 14 14
     14 24  4 14 14  0 14 14 14 25 14 14  1 14 26 14 19  9 14 14  3 14 14 27 28
     29 14 14  6 14 14 14 30 14 14 20 14  2 14 22 33 34 14 14 14 35 14 14 16 14
     11 36 37 14 14 31 14 14 17 14 14 14 14 14  0 14 38 39 32 14 12 14 40 14 14]
732
    >>> print(hist)
Tiago Peixoto's avatar
Tiago Peixoto committed
733
734
    [ 2  1  1  1  1  1  1  1  1  1  1  1  1  1 59  1  1  1  1  1  1  1  1  1  1
      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1]
735
    >>> print(is_attractor)
Tiago Peixoto's avatar
Tiago Peixoto committed
736
737
    [ True  True  True False False False  True  True False False False False
      True  True False False False False False False False False  True False
738
     False False False False False False False False False False False False
Tiago Peixoto's avatar
Tiago Peixoto committed
739
     False False False False False]
740
741
    """

742
    if vprop is None:
743
744
745
746
747
        vprop = g.new_vertex_property("int32_t")

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

748
749
    if directed is not None:
        g = GraphView(g, directed=directed)
750

751
752
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
753
754
755
756
757
758
759
760
761

    if attractors and g.is_directed() and directed != False:
        is_attractor = numpy.ones(len(hist), dtype="bool")
        libgraph_tool_topology.\
               label_attractors(g._Graph__graph, _prop("v", g, vprop),
                                is_attractor)
        return vprop, hist, is_attractor
    else:
        return vprop, hist
762
763
764
765


def label_largest_component(g, directed=None):
    """
766
767
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
768
769
770
771
772
773
774
775
776
777
778
779
780
781

    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`
782
         Boolean vertex property map which labels the largest component.
783
784
785
786
787
788
789

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

    Examples
    --------
790
791
792
793
794
795
796
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

797
798
    >>> g = gt.random_graph(100, lambda: poisson(1), directed=False)
    >>> l = gt.label_largest_component(g)
799
    >>> print(l.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
800
801
802
    [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0
     0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
     0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0]
803
    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
804
    >>> print(u.num_vertices())
Tiago Peixoto's avatar
Tiago Peixoto committed
805
    18
806
807
808
809
    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
810
811
812
813
814
    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)
815
    return label
816

Tiago Peixoto's avatar
Tiago Peixoto committed
817

818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
def label_out_component(g, root):
    """
    Label the out-component (or simply the component for undirected graphs) of a
    root vertex.

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

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

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

    Examples
    --------
841
842
843
844
845
846
847
848
849
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

    >>> g = gt.random_graph(100, lambda: poisson(2.2), directed=False)
    >>> l = gt.label_out_component(g, g.vertex(2))
850
    >>> print(l.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
851
852
853
    [1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0
     1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1
     1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0]
854
855
856

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

Tiago Peixoto's avatar
Tiago Peixoto committed
857
    >>> l = gt.label_out_component(gt.GraphView(g, reversed=True, directed=True),
858
    ...                            g.vertex(1))
859
    >>> print(l.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
860
861
862
    [0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
     0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0]
863
864
865
866
867
868
869
870
871
    """

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


872
def label_biconnected_components(g, eprop=None, vprop=None):
873
874
875
876
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

877
878
879
880
    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.

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
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
    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
    --------
924
925
926
927
928
929
930
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

Tiago Peixoto's avatar
Tiago Peixoto committed
931
    >>> g = gt.random_graph(100, lambda: poisson(2), directed=False)
932
    >>> comp, art, hist = gt.label_biconnected_components(g)
933
    >>> print(comp.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
934
935
936
937
    [35 23 35 35 32 32 35 35  1 37 32 28 32 35 32  2 32 29 35 35 13 14 34 12 17
     35 35 35 35 35  3 35 35 35 35 35 35 28 28 35 33 35 19 35 35 35 35  6 35 35
     24 39 35 31 35 10  9 22 32 35  4 25 26 35 35  7 35 35 35 35 35 35 36 35 35
     35 32 35  0 35 35 35 32 35 28 32 35 20 30 27 18 38 16  5 15 11 28 35  8 21]
938
    >>> print(art.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
939
940
941
    [1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0
     0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0
     1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0]
942
    >>> print(hist)
Tiago Peixoto's avatar
Tiago Peixoto committed
943
    [ 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
Tiago Peixoto's avatar
Tiago Peixoto committed
944
      1  1  1  5  1  1  1 10  1  1 48  1  1  1  1]
945
    """
946

947
    if vprop is None:
948
        vprop = g.new_vertex_property("bool")
949
    if eprop is None:
950
951
952
953
954
955
956
        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")

957
958
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
959
960
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
961
    return eprop, vprop, hist
962

Tiago Peixoto's avatar
Tiago Peixoto committed
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
def kcore_decomposition(g, deg="out", vprop=None):
    """
    Perform a k-core decomposition of the given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    deg : string
        Degree to be used for the decomposition. It can be either "in", "out" or
        "total", for in-, out-, or total degree of the vertices.
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Vertex property to store the decomposition. If ``None`` is supplied,
        one is created.

    Returns
    -------
    kval : :class:`~graph_tool.PropertyMap`
        Vertex property map with the k-core decomposition, i.e. a given vertex v
        belongs to the ``kval[v]``-core.

    Notes
    -----

    The k-core is a maximal set of vertices such that its induced subgraph only
    contains vertices with degree larger than or equal to k.

    This algorithm is described in [batagelk-algorithm]_ and runs in :math:`O(V + E)`
    time.

    Examples
    --------

    >>> g = gt.collection.data["netscience"]
    >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
    >>> kcore = gt.kcore_decomposition(g)
    >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=kcore, vertex_text=kcore, output="netsci-kcore.pdf")
    <...>

    .. testcode::
       :hide:

       gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=kcore, vertex_text=kcore, output="netsci-kcore.png")

    .. figure:: netsci-kcore.*
        :align: center

        K-core decomposition of a network of network scientists.

    References
    ----------
    .. [k-core] http://en.wikipedia.org/wiki/Degeneracy_%28graph_theory%29
1015
1016
1017
1018
1019
    .. [batagelk-algorithm]  Vladimir Batagelj, Matjaž Zaveršnik, "Fast
       algorithms for determining (generalized) core groups in social
       networks", Advances in Data Analysis and Classification
       Volume 5, Issue 2, pp 129-145 (2011), :DOI:`10.1007/s11634-010-0079-y`,
       :arxiv:`cs/0310049`
Tiago Peixoto's avatar
Tiago Peixoto committed
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041

    """

    if vprop is None:
        vprop = g.new_vertex_property("int32_t")

    _check_prop_writable(vprop, name="vprop")
    _check_prop_scalar(vprop, name="vprop")
    if deg not in ["in", "out", "total"]:
        raise ValueError("invalid degree: " + str(deg))

    if g.is_directed():
        if deg == "out":
            g = GraphView(g, reversed=True)
        if deg == "total":
            g = GraphView(g, directed=False)

    libgraph_tool_topology.\
               kcore_decomposition(g._Graph__graph, _prop("v", g, vprop),
                                   _degree(g, deg))
    return vprop

Tiago Peixoto's avatar
Tiago Peixoto committed
1042

1043
def shortest_distance(g, source=None, target=None, weights=None, max_dist=None,
1044
1045
                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
1046
    """
1047
1048
1049
    Calculate the distance from a source to a target vertex, or to of all
    vertices from a given source, or the all pairs shortest paths, if the source
    is not specified.
1050
1051
1052
1053
1054
1055

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    source : :class:`~graph_tool.Vertex` (optional, default: None)
1056
        Source vertex of the search. If unspecified, the all pairs shortest
1057
        distances are computed.
1058
1059
1060
    target : :class:`~graph_tool.Vertex` (optional, default: None)
        Target vertex of the search. If unspecified, the distance to all
        vertices from the source will be computed.
1061
1062
1063
1064
1065
    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
Tiago Peixoto's avatar
Tiago Peixoto committed
1066
        searched. This parameter has no effect if source is None.
1067
1068
1069
1070
    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
1071
1072
        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
1073
1074
1075
1076
        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.
1077
1078
1079
    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101

    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
    --------
1102
1103
1104
1105
1106
1107
1108
1109
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

    >>> from numpy.random import poisson
1110
1111
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
    >>> dist = gt.shortest_distance(g, source=g.vertex(0))
1112
    >>> print(dist.a)
Tiago Peixoto's avatar
Tiago Peixoto committed
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
    [         0          6          3          6 2147483647 2147483647
              6          5          2          4          5          6
              6          3          7          5          4          4
              3          4          2          4          3          3
              4          4          6          6          4          1
              5          2          4          5          3          5
              6          5          4          5 2147483647          9
              4          4          4          6          3          4
              6          6          3          2          4          4
              5          4          5          8          6          6
              5          5          4          5          6          3
              4          3          5          5 2147483647 2147483647
              5          5          8          3          7          4
              5          2          7          5          2          5
              5          5          7          7          4          3
              6          5          5          4          5          5
              4          4          6          5]
1130

1131
    >>> dist = gt.shortest_distance(g)
1132
    >>> print(dist[g.vertex(0)].a)
Tiago Peixoto's avatar
Tiago Peixoto committed
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
    [         0          6          3          6 2147483647 2147483647
              6          5          2          4          5          6
              6          3          7          5          4          4
              3          4          2          4          3          3
              4          4          6          6          4          1
              5          2          4          5          3          5
              6          5          4          5 2147483647          9
              4          4          4          6          3          4
              6          6          3          2          4          4
              5          4          5          8          6          6
              5          5          4          5          6          3
              4          3          5          5 2147483647 2147483647
              5          5          8          3          7          4
              5          2          7          5          2          5
              5          5          7          7          4          3
              6          5          5          4          5          5
              4          4          6          5]
1150
1151
1152
1153
1154

    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
1155
1156
       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
1157
1158
    .. [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
1159
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
1160
1161
1162
1163
    .. [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
    """

1164
    if weights is None:
1165
1166
1167
1168
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

1169
1170
    if dist_map is None:
        if source is not None:
1171
1172
1173
1174
1175
            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")
1176
    if source is not None:
1177
1178
1179
1180
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

1181
    if max_dist is None:
1182
1183
        max_dist = 0

1184
    if directed is not None:
1185
1186
1187
        u = GraphView(g, directed=directed)
    else:
        u = g
1188

1189
1190
1191
    if target is None:
        target = -1

1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
    if source is not None:
        pmap = g.copy_property(u.vertex_index, value_type="int64_t")
        libgraph_tool_topology.get_dists(g._Graph__graph,
                                         int(source),
                                         int(target),
                                         _prop("v", g, dist_map),
                                         _prop("e", g, weights),
                                         _prop("v", g, pmap),
                                         float(max_dist))
    else:
        libgraph_tool_topology.get_all_dists(u._Graph__graph,
1203
                                             _prop("v", g, dist_map),
1204
                                             _prop("e", g, weights), dense)
1205

1206
1207
1208
1209

    if source is not None and target != -1:
        dist_map = dist_map[target]

1210
    if source is not None and pred_map:
1211
1212
1213
1214
        return dist_map, pmap
    else:
        return dist_map

Tiago Peixoto's avatar
Tiago Peixoto committed
1215

1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
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
1226
    target : :class:`~graph_tool.Vertex`
1227
1228
        Target vertex of the search.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
Tiago Peixoto's avatar
Tiago Peixoto committed
1229
        The edge weights.
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
    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
    --------
1253
1254
1255
1256
1257
1258
1259
1260
1261
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(43)
       gt.seed_rng(43)

    >>> from numpy.random import poisson
    >>> g = gt.random_graph(300, lambda: (poisson(4), poisson(4)))
1262
    >>> vlist, elist = gt.shortest_path(g, g.vertex(10), g.vertex(11))
1263
    >>> print([str(v) for v in vlist])
Tiago Peixoto's avatar
Tiago Peixoto committed
1264
    ['10', '131', '184', '265', '223', '11']
1265
    >>> print([str(e) for e in elist])
Tiago Peixoto's avatar
Tiago Peixoto committed
1266
    ['(10, 131)', '(131, 184)', '(184, 265)', '(265, 223)', '(223, 11)']
1267
1268
1269
1270
1271

    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
1272
1273
       Press
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
1274
1275
    .. [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
1276
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
1277
1278
    """

1279
    if pred_map is None:
1280
1281
        pred_map = shortest_distance(g, source, target,
                                     weights=weights,
Tiago Peixoto's avatar
Tiago Peixoto committed
1282
                                     pred_map=True)[1]
1283

1284
    if pred_map[target] == int(target):  # no path to target
1285
1286
1287
1288
1289
        return [], []

    vlist = [target]
    elist = []

1290
    if weights is not None:
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
        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:
1304
                if weights is not None:
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
                    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

1316

Tiago Peixoto's avatar
Tiago Peixoto committed
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
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
    --------
1358
1359
1360
1361
1362
1363
1364
1365
    .. testcode::
       :hide:

       import numpy.random
       numpy.random.seed(42)
       gt.seed_rng(42)

    >>> from numpy.random import poisson
Tiago Peixoto's avatar
Tiago Peixoto committed
1366
1367
    >>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3)))
    >>> dist, ends = gt.pseudo_diameter(g)
1368
    >>> print(dist)
Tiago Peixoto's avatar
Tiago Peixoto committed
1369
    9.0
1370
    >>> print(int(ends[0]), int(ends[1]))
Tiago Peixoto's avatar
Tiago Peixoto committed
1371
    0 140
Tiago Peixoto's avatar
Tiago Peixoto committed
1372
1373
1374
1375
1376
1377
1378

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

    if source is None:
1379
        source = g.vertex(0, use_index=False)
Tiago Peixoto's avatar
Tiago Peixoto committed
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
    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))


1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
def is_bipartite(g, partition=False):
    """
    Test if the graph is bipartite.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    partition : bool (optional, default: ``False``)
        If ``True``, return the two partitions in case the graph is bipartite.

    Returns
    -------
    is_bipartite : bool
        Whether or not the graph is bipartite.
    partition : :class:`~graph_tool.PropertyMap` (only if `partition=True`)
        A vertex property map with the graph partitioning (or `None`) if the
        graph is not bipartite.

    Notes
    -----

    An undirected graph is bipartite if one can partition its set of vertices
    into two sets, such that all edges go from one set to the other.

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

    Examples
    --------
    >>> g = gt.lattice([10, 10])
    >>> is_bi, part = gt.is_bipartite(g, partition=True)
    >>> print(is_bi)
    True
Tiago Peixoto's avatar
Tiago Peixoto committed
1428
    >>> gt.graph_draw(g, vertex_fill_color=part, output_size=(300, 300), output="bipartite.pdf")
1429
1430
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
1431
1432
1433
1434
1435
    .. testcode::
       :hide:

       gt.graph_draw(g, vertex_fill_color=part, output_size=(300, 300), output="bipartite.png")

1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
    .. figure:: bipartite.*
        :align: center

        Bipartition of a 2D lattice.

    References
    ----------
    .. [boost-bipartite] http://www.boost.org/libs/graph/doc/is_bipartite.html
    """

    if partition:
        part = g.new_vertex_property("bool")
    else:
        part = None
    g = GraphView(g, directed=False)
    is_bi = libgraph_tool_topology.is_bipartite(g._Graph__graph,
                                                _prop("v", g, part))
    if partition:
        return is_bi, part
    else:
        return is_bi