__init__.py 76 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) 2006-2015 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
34
   all_shortest_paths
   all_predecessors
Tiago Peixoto's avatar
Tiago Peixoto committed
35
   pseudo_diameter
36
   similarity
37
   isomorphism
38
39
   subgraph_isomorphism
   mark_subgraph
40
41
   max_cardinality_matching
   max_independent_vertex_set
42
   min_spanning_tree
43
   random_spanning_tree
44
45
46
   dominator_tree
   topological_sort
   transitive_closure
Tiago Peixoto's avatar
Tiago Peixoto committed
47
   tsp_tour
48
   sequential_vertex_coloring
49
50
   label_components
   label_biconnected_components
51
   label_largest_component
52
   label_out_component
Tiago Peixoto's avatar
Tiago Peixoto committed
53
   kcore_decomposition
54
   is_bipartite
Tiago Peixoto's avatar
Tiago Peixoto committed
55
   is_DAG
56
   is_planar
57
   make_maximal_planar
Tiago Peixoto's avatar
Tiago Peixoto committed
58
   edge_reciprocity
59
60
61

Contents
++++++++
62

63
64
"""

65
66
from __future__ import division, absolute_import, print_function

Tiago Peixoto's avatar
Tiago Peixoto committed
67
from .. dl_import import dl_import
68
dl_import("from . import libgraph_tool_topology")
69

70
from .. import _prop, Vector_int32_t, _check_prop_writable, \
71
     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView,\
72
     libcore, _get_rng, _degree, perfect_prop_hash
73
from .. stats import label_self_loops
74
import random, sys, numpy, collections
75

76
__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
77
           "max_cardinality_matching", "max_independent_vertex_set",
78
           "min_spanning_tree", "random_spanning_tree", "dominator_tree",
Tiago Peixoto's avatar
Tiago Peixoto committed
79
           "topological_sort", "transitive_closure", "tsp_tour",
80
81
           "sequential_vertex_coloring", "label_components",
           "label_largest_component", "label_biconnected_components",
Tiago Peixoto's avatar
Tiago Peixoto committed
82
           "label_out_component", "kcore_decomposition", "shortest_distance",
Tiago Peixoto's avatar
Tiago Peixoto committed
83
84
85
           "shortest_path", "all_shortest_paths", "all_predecessors",
           "pseudo_diameter", "is_bipartite", "is_DAG", "is_planar",
           "make_maximal_planar", "similarity", "edge_reciprocity"]
86
87
88
89
90
91
92
93
94

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
95
        Second graph to be compared.
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
    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
    --------
122
123
124
125
126
127
128
    .. testcode::
       :hide:

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

129
130
131
132
    >>> g = gt.random_graph(100, lambda: (3,3))
    >>> u = g.copy()
    >>> gt.similarity(u, g)
    1.0
Tiago Peixoto's avatar
Tiago Peixoto committed
133
    >>> gt.random_rewire(u)
134
    24
135
    >>> gt.similarity(u, g)
136
    0.04666666666666667
137
138
139
140
141
142
143
    """

    if label1 is None:
        label1 = g1.vertex_index
    if label2 is None:
        label2 = g2.vertex_index
    if label1.value_type() != label2.value_type():
144
145
146
147
        try:
            label2 = label2.copy(label1.value_type())
        except ValueError:
            label1 = label1.copy(label2.value_type())
Tiago Peixoto's avatar
Tiago Peixoto committed
148
    if label1.is_writable() or label2.is_writable():
149
150
151
        s = libgraph_tool_topology.\
               similarity(g1._Graph__graph, g2._Graph__graph,
                          _prop("v", g1, label1), _prop("v", g2, label2))
Tiago Peixoto's avatar
Tiago Peixoto committed
152
153
154
155
    else:
        s = libgraph_tool_topology.\
               similarity_fast(g1._Graph__graph, g2._Graph__graph,
                               _prop("v", g1, label1), _prop("v", g2, label2))
156
157
158
159
160
    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
161

Tiago Peixoto's avatar
Tiago Peixoto committed
162

163
def isomorphism(g1, g2, vertex_inv1=None, vertex_inv2=None, isomap=False):
164
165
    r"""Check whether two graphs are isomorphic.

166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
    Parameters
    ----------
    g1 : :class:`~graph_tool.Graph`
        First graph.
    g2 : :class:`~graph_tool.Graph`
        Second graph.
    vertex_inv1 : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        Vertex invariant of the first graph. Only vertices with with the same
        invariants are considered in the isomorphism.
    vertex_inv2 : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
        Vertex invariant of the second graph. Only vertices with with the same
        invariants are considered in the isomorphism.
    isomap : ``bool`` (optional, default: ``False``)
        If ``True``, a vertex :class:`~graph_tool.PropertyMap` with the
        isomorphism mapping is returned as well.

    Returns
    -------
    is_isomorphism : ``bool``
        ``True`` if both graphs are isomorphic, otherwise ``False``.
    isomap : :class:`~graph_tool.PropertyMap`
         Isomorphism mapping corresponding to a property map belonging to the
         first graph which maps its vertices to their corresponding vertices of
         the second graph.
190
191
192

    Examples
    --------
193
194
195
196
197
198
199
    .. testcode::
       :hide:

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

200
201
202
203
204
205
206
207
208
    >>> 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

209
    """
210
    imap = g1.new_vertex_property("int32_t")
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
    if vertex_inv1 is None:
        vertex_inv1 = g1.degree_property_map("total").copy("int64_t")
    else:
        vertex_inv1 = vertex_inv1.copy("int64_t")
        d = g1.degree_property_map("total")
        vertex_inv1.fa += (vertex_inv1.fa.max() + 1) * d.a
    if vertex_inv2 is None:
        vertex_inv2 = g2.degree_property_map("total").copy("int64_t")
    else:
        vertex_inv2 = vertex_inv2.copy("int64_t")
        d = g2.degree_property_map("total")
        vertex_inv2.fa += (vertex_inv2.fa.max() + 1) * d.a

    inv_max = max(vertex_inv1.fa.max(),vertex_inv2.fa.max()) + 1

    l1 = label_self_loops(g1, mark_only=True)
    if l1.fa.max() > 0:
        g1 = GraphView(g1, efilt=1 - l1.fa)

    l2 = label_self_loops(g2, mark_only=True)
    if l2.fa.max() > 0:
        g2 = GraphView(g2, efilt=1 - l2.fa)

234
    iso = libgraph_tool_topology.\
235
           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
236
237
238
                             _prop("v", g1, vertex_inv1),
                             _prop("v", g2, vertex_inv2),
                             inv_max,
Tiago Peixoto's avatar
Tiago Peixoto committed
239
                             _prop("v", g1, imap))
240
241
242
243
244
    if isomap:
        return iso, imap
    else:
        return iso

Tiago Peixoto's avatar
Tiago Peixoto committed
245

246
def subgraph_isomorphism(sub, g, max_n=0, vertex_label=None, edge_label=None,
247
                         induced=False, subgraph=True, generator=False):
248
    r"""Obtain all subgraph isomorphisms of `sub` in `g` (or at most `max_n` subgraphs, if `max_n > 0`).
249

250

Tiago Peixoto's avatar
Tiago Peixoto committed
251
252
253
254
255
256
    Parameters
    ----------
    sub : :class:`~graph_tool.Graph`
        Subgraph for which to be searched.
    g : :class:`~graph_tool.Graph`
        Graph in which the search is performed.
257
    max_n : int (optional, default: ``0``)
Tiago Peixoto's avatar
Tiago Peixoto committed
258
259
        Maximum number of matches to find. If `max_n == 0`, all matches are
        found.
260
    vertex_label : pair of :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
261
        If provided, this should be a pair of :class:`~graph_tool.PropertyMap`
262
263
264
265
        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``)
266
        If provided, this should be a pair of :class:`~graph_tool.PropertyMap`
267
268
269
270
271
272
273
        objects, belonging to ``sub`` and ``g`` (in this order), which specify
        edge labels which should match, in addition to the topological
        isomorphism.
    induced : bool (optional, default: ``False``)
        If ``True``, only node-induced subgraphs are found.
    subgraph : bool (optional, default: ``True``)
        If ``False``, all non-subgraph isomorphisms between `sub` and `g` are
274
        found.
275
276
277
278
    generator : bool (optional, default: ``False``)
        If ``True``, a generator will be returned, instead of a list. This is
        useful if the number of isomorphisms is too large to store in memory. If
        ``generator == True``, the option ``max_n`` is ignored.
Tiago Peixoto's avatar
Tiago Peixoto committed
279
280
281

    Returns
    -------
282
283
284
285
    vertex_maps : list (or generator) of :class:`~graph_tool.PropertyMap` objects
        List (or generator) containing vertex property map objects which
        indicate different isomorphism mappings. The property maps vertices in
        `sub` to the corresponding vertex index in `g`.
Tiago Peixoto's avatar
Tiago Peixoto committed
286
287
288

    Notes
    -----
289
290
291
292
293
    The implementation is based on the VF2 algorithm, introduced by Cordella et al.
    [cordella-improved-2001]_ [cordella-subgraph-2004]_. The spatial complexity
    is of order :math:`O(V)`, where :math:`V` is the (maximum) number of vertices
    of the two graphs. Time complexity is :math:`O(V^2)` in the best case and
    :math:`O(V!\times V)` in the worst case.
294
295
296

    Examples
    --------
297
    >>> from numpy.random import poisson
298
299
300
    >>> g = gt.complete_graph(30)
    >>> sub = gt.complete_graph(10)
    >>> vm = gt.subgraph_isomorphism(sub, g, max_n=100)
301
    >>> print(len(vm))
302
    100
303
    >>> for i in range(len(vm)):
304
305
    ...   g.set_vertex_filter(None)
    ...   g.set_edge_filter(None)
306
    ...   vmask, emask = gt.mark_subgraph(g, sub, vm[i])
307
308
    ...   g.set_vertex_filter(vmask)
    ...   g.set_edge_filter(emask)
309
    ...   assert gt.isomorphism(g, sub)
310
311
312
313
    >>> 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
314
315
316
    >>> ewidth.a *= 2
    >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
    ...               edge_pen_width=ewidth, output_size=(200, 200),
317
    ...               output="subgraph-iso-embed.pdf")
318
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
319
    >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf")
320
321
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
322
323
324
325
326
327
328
329
    .. 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
330
331
    .. image:: subgraph-iso.*
    .. image:: subgraph-iso-embed.*
332

333

Tiago Peixoto's avatar
Tiago Peixoto committed
334
    **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph.
335
336
337

    References
    ----------
338
339
340
341
342
343
    .. [cordella-improved-2001] L. P. Cordella, P. Foggia, C. Sansone, and M. Vento,
       "An improved algorithm for matching large graphs.", 3rd IAPR-TC15 Workshop
       on Graph-based Representations in Pattern Recognition, pp. 149-159, Cuen, 2001.
       http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.101.5342
    .. [cordella-subgraph-2004] L. P. Cordella, P. Foggia, C. Sansone, and M. Vento,
       "A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs.",
Tiago Peixoto's avatar
Tiago Peixoto committed
344
       IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 10, pp. 1367-1372, 2004.
345
346
       :doi:`10.1109/TPAMI.2004.75`
    .. [boost-subgraph-iso] http://www.boost.org/libs/graph/doc/vf2_sub_graph_iso.html
347
    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
348
349

    """
350
351
    if sub.num_vertices() == 0:
        raise ValueError("Cannot search for an empty subgraph.")
352
353
354
355
    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!")
356
357
    elif vertex_label[0].value_type() != "int64_t":
        vertex_label = perfect_prop_hash(vertex_label, htype="int64_t")
358

359
360
361
362
    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!")
363
364
    elif edge_label[0].value_type() != "int64_t":
        edge_label = perfect_prop_hash(edge_label, htype="int64_t")
365

366
367
368
369
370
371
372
373
374
    vmaps = libgraph_tool_topology.\
            subgraph_isomorphism(sub._Graph__graph, g._Graph__graph,
                                 _prop("v", sub, vertex_label[0]),
                                 _prop("v", g, vertex_label[1]),
                                 _prop("e", sub, edge_label[0]),
                                 _prop("e", g, edge_label[1]),
                                 max_n, induced, not subgraph,
                                 generator)
    if generator:
375
        return (PropertyMap(vmap, sub, "v") for vmap in vmaps)
376
    else:
377
        return [PropertyMap(vmap, sub, "v") for vmap in vmaps]
378

Tiago Peixoto's avatar
Tiago Peixoto committed
379

380
def mark_subgraph(g, sub, vmap, vmask=None, emask=None):
381
382
383
384
385
386
387
388
389
    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
390
    `sub`.
391
    """
392
    if vmask is None:
393
        vmask = g.new_vertex_property("bool")
394
    if emask is None:
395
396
397
398
399
400
401
402
        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
403
404
        us = set([g.vertex(vmap[x]) for x in v.out_neighbours()])

405
        for ew in w.out_edges():
406
407
408
            if ew.target() in us:
                emask[ew] = True

409
    return vmask, emask
410

Tiago Peixoto's avatar
Tiago Peixoto committed
411

412
def min_spanning_tree(g, weights=None, root=None, tree_map=None):
413
414
415
416
417
418
419
    """
    Return the minimum spanning tree of a given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
420
    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
421
422
        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
423
    root : :class:`~graph_tool.Vertex` (optional, default: `None`)
424
        Root of the minimum spanning tree. If this is provided, Prim's algorithm
425
        is used. Otherwise, Kruskal's algorithm is used.
426
    tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
        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
    --------
442
443
444
445
446
447
448
449
    .. testcode::
       :hide:

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

    >>> from numpy.random import random
450
451
452
    >>> 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
453
    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
454
    >>> tree = gt.min_spanning_tree(g, weights=weight)
455
    >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf")
456
    <...>
457
458
    >>> u = gt.GraphView(g, efilt=tree)
    >>> gt.graph_draw(u, pos=pos, output="triang_min_span_tree.pdf")
459
460
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
461
462
463
464
    .. testcode::
       :hide:

       gt.graph_draw(g, pos=pos, output="triang_orig.png")
465
       gt.graph_draw(u, pos=pos, output="triang_min_span_tree.png")
466

467
    .. image:: triang_orig.*
Tiago Peixoto's avatar
Tiago Peixoto committed
468
        :width: 400px
469
    .. image:: triang_min_span_tree.*
Tiago Peixoto's avatar
Tiago Peixoto committed
470
        :width: 400px
471
472

    *Left:* Original graph, *Right:* The minimum spanning tree.
473
474
475
476
477

    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
478
479
       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
480
481
482
483
484
    .. [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
    """
485
    if tree_map is None:
486
487
488
489
        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.")

490
491
492
493
494
495
496
497
498
499
500
    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))
501
    return tree_map
502

Tiago Peixoto's avatar
Tiago Peixoto committed
503

504
def random_spanning_tree(g, weights=None, root=None, tree_map=None):
505
    r"""Return a random spanning tree of a given graph, which can be directed or
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
    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
    -----
528
529

    The running time for this algorithm is :math:`O(\tau)`, with :math:`\tau`
530
531
532
533
    being the mean hitting time of a random walk on the graph. In the worse case,
    we have :math:`\tau \sim O(V^3)`, with :math:`V` being the number of
    vertices in the graph. However, in much more typical cases (e.g. sparse
    random graphs) the running time is simply :math:`O(V)`.
534
535
536

    Examples
    --------
537
538
539
540
541
542
543
544
    .. testcode::
       :hide:

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

    >>> from numpy.random import random
545
    >>> g, pos = gt.triangulation(random((400, 2)), type="delaunay")
546
547
548
549
    >>> 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)
550
    >>> tree2 = gt.random_spanning_tree(g, weights=weight)
551
552
    >>> gt.graph_draw(g, pos=pos, output="rtriang_orig.pdf")
    <...>
553
554
555
556
557
    >>> u = gt.GraphView(g, efilt=tree)
    >>> gt.graph_draw(u, pos=pos, output="triang_random_span_tree.pdf")
    <...>
    >>> u2 = gt.GraphView(g, efilt=tree2)
    >>> gt.graph_draw(u2, pos=pos, output="triang_random_span_tree2.pdf")
558
559
    <...>

Tiago Peixoto's avatar
Tiago Peixoto committed
560
561
562
563
    .. testcode::
       :hide:

       gt.graph_draw(g, pos=pos, output="rtriang_orig.png")
564
565
       gt.graph_draw(u, pos=pos, output="triang_random_span_tree.png")
       gt.graph_draw(u2, pos=pos, output="triang_random_span_tree2.png")
566
567

    .. image:: rtriang_orig.*
568
        :width: 300px
569
    .. image:: triang_random_span_tree.*
570
571
572
        :width: 300px
    .. image:: triang_random_span_tree2.*
        :width: 300px
573

574
575
    *Left:* Original graph, *Middle:* A random spanning tree, *Right:* Another
    random spanning tree
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596

    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)
597
598
599
    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) )
600
601
602
603

    libgraph_tool_topology.\
        random_spanning_tree(g._Graph__graph, int(root),
                             _prop("e", g, weights),
604
                             _prop("e", g, tree_map), _get_rng())
605
606
607
    return tree_map


Tiago Peixoto's avatar
Tiago Peixoto committed
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
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
    --------
635
636
637
638
639
640
641
    .. testcode::
       :hide:

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

Tiago Peixoto's avatar
Tiago Peixoto committed
642
643
644
    >>> g = gt.random_graph(100, lambda: (2, 2))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
645
    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
Tiago Peixoto's avatar
Tiago Peixoto committed
646
    >>> dom = gt.dominator_tree(g, root[0])
647
    >>> print(dom.a)
648
649
650
    [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
     0 0 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
651
652
653

    References
    ----------
654
    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
Tiago Peixoto's avatar
Tiago Peixoto committed
655
656

    """
657
    if dom_map is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
658
659
660
        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" +
661
662
                         " int32_t.")
    if not g.is_directed():
Tiago Peixoto's avatar
Tiago Peixoto committed
663
        raise ValueError("dominator tree requires a directed graph.")
664
    libgraph_tool_topology.\
Tiago Peixoto's avatar
Tiago Peixoto committed
665
666
667
               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
668

Tiago Peixoto's avatar
Tiago Peixoto committed
669

670
def topological_sort(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
671
672
673
674
675
676
677
    """
    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
678
    such that if edge (u,v) appears in the graph, then u comes before v in the
Tiago Peixoto's avatar
Tiago Peixoto committed
679
680
681
682
683
684
    ordering. The graph must be a directed acyclic graph (DAG).

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

    Examples
    --------
685
686
687
688
689
690
691
    .. testcode::
       :hide:

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

Tiago Peixoto's avatar
Tiago Peixoto committed
692
693
694
695
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
    >>> sort = gt.topological_sort(g)
696
    >>> print(sort)
Tiago Peixoto's avatar
Tiago Peixoto committed
697
698
    [29 28 27 26 23 24 22 21 20 18 17 16 15 14 11 10  9  6  5  4 19 12 13  3  2
     25  1  0  7  8]
Tiago Peixoto's avatar
Tiago Peixoto committed
699
700
701

    References
    ----------
702
    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
Tiago Peixoto's avatar
Tiago Peixoto committed
703
704
705
706
    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

707
    topological_order = Vector_int32_t()
Tiago Peixoto's avatar
Tiago Peixoto committed
708
709
710
711
    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).");
712
    return topological_order.a[::-1].copy()
713

Tiago Peixoto's avatar
Tiago Peixoto committed
714

715
def transitive_closure(g):
Tiago Peixoto's avatar
Tiago Peixoto committed
716
717
718
719
720
721
722
723
724
725
726
727
728
    """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
    --------
729
730
731
732
733
734
735
    .. testcode::
       :hide:

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

Tiago Peixoto's avatar
Tiago Peixoto committed
736
737
738
739
740
    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tc = gt.transitive_closure(g)

    References
    ----------
741
    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
Tiago Peixoto's avatar
Tiago Peixoto committed
742
743
744
745
    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

746
747
748
749
750
751
752
    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
753

754
def label_components(g, vprop=None, directed=None, attractors=False):
755
    """
756
    Label the components to which each vertex in the graph belongs. If the
757
758
    graph is directed, it finds the strongly connected components.

759
760
761
    A property map with the component labels is returned, together with an
    histogram of component labels.

762
763
    Parameters
    ----------
764
    g : :class:`~graph_tool.Graph`
765
        Graph to be used.
766
    vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
767
768
        Vertex property to store the component labels. If none is supplied, one
        is created.
769
    directed : bool (optional, default: ``None``)
770
771
        Treat graph as directed or not, independently of its actual
        directionality.
772
773
774
775
    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.
776
777
778

    Returns
    -------
779
    comp : :class:`~graph_tool.PropertyMap`
780
        Vertex property map with component labels.
781
782
    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
783
784
785
786
    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.
787
788
789
790
791
792

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

793
    The algorithm runs in :math:`O(V + E)` time.
794
795
796

    Examples
    --------
797
798
799
800
801
802
    .. testcode::
       :hide:

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

803
804
    >>> g = gt.random_graph(100, lambda: (poisson(2), poisson(2)))
    >>> comp, hist, is_attractor = gt.label_components(g, attractors=True)
805
    >>> print(comp.a)
806
807
808
809
    [13 13 13 13 14 12 13 15 16 13 17 19 13 13 13 20 13 13 13 10 13 13 22 13 13
      4 13 13  2 23 13 13 24 13 13 26 27 13 13 13 13  0 13 13  3 13 13 13 28  1
      6 13 13 13 13  5 13 13 13 13 13 13 13  9 13 11 13 29 13 13 13 13 18 13 30
     31 13 13 32 13 33 34 35 13 13 21 13 25  8 36 13 13 13 13 13 37 13 13  7 13]
810
    >>> print(hist)
811
812
    [ 1  1  1  1  1  1  1  1  1  1  1  1  1 63  1  1  1  1  1  1  1  1  1  1  1
      1  1  1  1  1  1  1  1  1  1  1  1  1]
813
    >>> print(is_attractor)
814
815
816
817
    [ True False  True  True  True False False  True False  True  True  True
      True False  True False False False False False False False False False
     False False False False False False False False False  True False  True
     False False]
818
819
    """

820
    if vprop is None:
821
822
823
824
825
        vprop = g.new_vertex_property("int32_t")

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

826
827
    if directed is not None:
        g = GraphView(g, directed=directed)
828

829
830
    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
831
832
833
834
835
836
837
838
839

    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
840
841
842
843


def label_largest_component(g, directed=None):
    """
844
845
    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
846
847
848
849
850
851
852
853
854
855
856
857
858
859

    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`
860
         Boolean vertex property map which labels the largest component.
861
862
863
864
865
866
867

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

    Examples
    --------
868
869
870
871
872
873
874
    .. testcode::
       :hide:

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

875
876
    >>> g = gt.random_graph(100, lambda: poisson(1), directed=False)
    >>> l = gt.label_largest_component(g)
877
    >>> print(l.a)
878
879
880
    [0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0
     1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
     0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0]
881
    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
882
    >>> print(u.num_vertices())
883
    18
884
885
886
887
    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
888
    vfilt, inv = g.get_vertex_filter()
889
    label.fa = c.fa == h.argmax()
890
    return label
891

Tiago Peixoto's avatar
Tiago Peixoto committed
892

893
def label_out_component(g, root, label=None):
894
895
896
897
898
899
900
901
902
903
    """
    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.
904
905
906
    label : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        If provided, this must be an initialized Boolean vertex property map
        where the out-component will be labeled.
907
908
909

    Returns
    -------
910
    label : :class:`~graph_tool.PropertyMap`
911
912
913
914
915
916
917
918
         Boolean vertex property map which labels the out-component.

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

    Examples
    --------
919
920
921
922
923
924
925
926
927
    .. 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))
928
    >>> print(l.a)
929
930
931
    [1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1
     1 1 0 1 1 1 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 1 1 1 1 0 0
     1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0]
932
933
934

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

Tiago Peixoto's avatar
Tiago Peixoto committed
935
    >>> l = gt.label_out_component(gt.GraphView(g, reversed=True, directed=True),
936
    ...                            g.vertex(1))
937
    >>> print(l.a)
938
939
940
    [0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 1
     1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0
     1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 0]
941
942
    """

943
944
945
946
947
    if label is None:
        label = g.new_vertex_property("bool")
    elif label.value_type() != "bool":
        raise ValueError("value type of `label` must be `bool`, not %s" %
                         label.value_type())
948
949
950
951
952
953
    libgraph_tool_topology.\
             label_out_component(g._Graph__graph, int(root),
                                 _prop("v", g, label))
    return label


954
def label_biconnected_components(g, eprop=None, vprop=None):
955
956
957
958
    """
    Label the edges of biconnected components, and the vertices which are
    articulation points.

959
960
961
962
    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.

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
    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
    --------
1006
1007
1008
1009
1010
1011
1012
    .. testcode::
       :hide:

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

Tiago Peixoto's avatar
Tiago Peixoto committed
1013
    >>> g = gt.random_graph(100, lambda: poisson(2), directed=False)
1014
    >>> comp, art, hist = gt.label_biconnected_components(g)
1015
    >>> print(comp.a)
1016
1017
1018
1019
1020
    [31 42 41 41 41 21  2 41 41 19 41 33 41 41 12 41 40 41 41 41 41 41 41  8 41
     10 41 32 28 30 41 41 41  5 41 41 41 41 39 38 41 41 41 41 45 44 41 41 22 41
     41 41  0 41 41 41 41 41 41 41 41  7 13 41 20 41 41 41 41 34  9 41 41  4 43
     18 41 41 15 29  1 41 41 41 41  6 41 25 23 35 16 24 37 11  3 36 17 26 27 14
     41]
1021
    >>> print(art.a)
1022
1023
1024
    [1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0
     1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1
     1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0]
1025
    >>> print(hist)
Tiago Peixoto's avatar
Tiago Peixoto committed
1026
    [ 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
1027
      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 56  1  1  1  1]
1028
    """
1029

1030
    if vprop is None:
1031
        vprop = g.new_vertex_property("bool")
1032
    if eprop is None:
1033
1034
1035
1036
1037
1038
1039
        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")

1040
1041
    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
1042
1043
             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
1044
    return eprop, vprop, hist
1045

Tiago Peixoto's avatar
Tiago Peixoto committed
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
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
1098
1099
1100
1101
1102
    .. [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
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124

    """

    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
1125

1126
1127
1128
def shortest_distance(g, source=None, target=None, weights=None,
                      negative_weights=False, max_dist=None, directed=None,
                      dense=False, dist_map=None, pred_map=False):
1129
    """Calculate the distance from a source to a target vertex, or to of all
1130
1131
    vertices from a given source, or the all pairs shortest paths, if the source
    is not specified.
1132
1133
1134
1135
1136

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
1137
    source : :class:`~graph_tool.Vertex` (optional, default: ``None``)
1138
        Source vertex of the search. If unspecified, the all pairs shortest
1139
        distances are computed.
1140
    target : :class:`~graph_tool.Vertex` or iterable of such objects (optional, default: ``None``)
1141
1142
        Target vertex (or vertices) of the search. If unspecified, the distance
        to all vertices from the source will be computed.
1143
1144
1145
1146
    weights : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        The edge weights. If provided, the shortest path will correspond to the
        minimal sum of weights.
    negative_weights : ``bool`` (optional, default: ``False``)
Tiago Peixoto's avatar
Tiago Peixoto committed
1147
        If `True`, this will trigger the use of the Bellman-Ford algorithm.
1148
1149
        Ignored if ``source`` is ``None``.
    max_dist : scalar value (optional, default: ``None``)
1150
        If specified, this limits the maximum distance of the vertices
1151
1152
1153
        searched. This parameter has no effect if source is ``None``, or if
        `negative_weights=True`.
    directed : ``bool`` (optional, default:``None``)
1154
1155
        Treat graph as directed or not, independently of its actual
        directionality.
1156
1157
1158
    dense : ``bool`` (optional, default: ``False``)
        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
1159
        has no effect.
1160
    dist_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
1161
1162
        Vertex property to store the distances. If none is supplied, one
        is created.
1163
1164
1165
    pred_map : ``bool`` (optional, default: ``False``)
        If ``True``, a vertex property map with the predecessors is returned.
        Ignored if ``source`` is ``None``.
1166
1167
1168
1169

    Returns
    -------
    dist_map : :class:`~graph_tool.PropertyMap`
1170
        Vertex property map with the distances from source. If source is ``None``,
1171
        it will have a vector value type, with the distances to every vertex.
1172
1173
    pred_map : :class:`~graph_tool.PropertyMap` (optional, if ``pred_map == True``)
        Vertex property map with the predecessors in the search tree.
1174
1175
1176
1177
1178
1179

    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
1180
1181
1182
1183
1184
    ``negative_weights == True``, the Bellman-Ford algorithm is used
    [bellman_ford]_, which accepts negative weights, as long as there are no
    negative loops. 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.
1185
1186

    If source is specified, the algorithm runs in :math:`O(V + E)` time, or
1187
1188
1189
    :math:`O(V \log V)` if weights are given. If ``negative_weights == True``,
    the complexity is :math:`O(VE)`. If source is not specified, it runs in
    :math:`O(VE\log V)` time, or :math:`O(V^3)` if dense == True.
1190
1191
1192

    Examples
    --------
1193
1194
1195
1196
1197
1198
1199
1200
    .. testcode::
       :hide:

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

    >>> from numpy.random import poisson
1201
1202
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
    >>> dist = gt.shortest_distance(g, source=g.vertex(0))
1203
    >>> print(dist.a)
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
    [         0          1          5          4 2147483647          4
              9          5          8          5          7          6
              3          5          6          8          3          3
              5          6 2147483647          1          4          5
              5          2          5          7          4          5
              5          5          4          4          5          2
              5 2147483647          5          2 2147483647          6
              5          6          6          2          5          4
              3          6          5          4          4          5
              3          3          5          5          1          5
              4          6          3          4          3          3
              7          5          5          4 2147483647 2147483647
              2          5          3          5          5          6
              3          5          6          6          5          4
              5          3          6          3          4 2147483647
              4          6          4          4          4          4
              6          5          4          4]
    >>>
1222
    >>> dist = gt.shortest_distance(g)
1223
    >>> print(dist[g.vertex(0)].a)
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
    [         0          1          5          4 2147483647          4
              9          5          8          5          7          6
              3          5          6          8          3          3
              5          6 2147483647          1          4          5
              5          2          5          7          4          5
              5          5          4          4          5          2
              5 2147483647          5          2 2147483647          6
              5          6          6          2          5          4
              3          6          5          4          4          5
              3          3          5          5          1          5
              4          6          3          4          3          3
              7          5          5          4 2147483647 2147483647
              2          5          3          5          5          6
              3          5          6          6          5          4
              5          3          6          3          4 2147483647
              4          6          4          4          4          4
              6          5          4          4]
1241
    >>> dist = gt.shortest_distance(g, source=g.vertex(0), target=g.vertex(2))
1242
    >>> print(dist)
1243
1244
    5
    >>> dist = gt.shortest_distance(g, source=g.vertex(0), target=[g.vertex(2), g.vertex(6)])
1245
    >>> print(dist)
1246
    [5 9]
1247
1248
1249
1250

    References
    ----------
    .. [bfs] Edward Moore, "The shortest path through a maze", International
1251
       Symposium on the Theory of Switching (1959), Harvard University Press.
Tiago Peixoto's avatar
Tiago Peixoto committed
1252
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
1253
1254
    .. [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
1255
    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
1256
1257
    .. [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
1258
    .. [bellman-ford] http://www.boost.org/libs/graph/doc/bellman_ford_shortest.html
1259
1260
    """

1261
1262
1263
1264
1265
1266
    if isinstance(target, collections.Iterable):
        target = numpy.asarray(target, dtype="int64")
    elif target is None:
        target = numpy.array([], dtype="int64")
    else:
        target = numpy.asarray([int(target)], dtype="int64")
1267

1268
    if weights is None:
1269
1270
1271
1272
        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

1273
1274
    if dist_map is None:
        if source is not None:
1275
1276
1277
1278
1279
            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")
1280
    if source is not None:
1281
1282
1283
1284
        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

1285
    if max_dist is None:
1286
1287
        max_dist = 0

1288
    if directed is not None:
1289
1290
1291
        u = GraphView(g, directed=directed)
    else:
        u = g
1292