__init__.py 64.7 KB
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#! /usr/bin/env python
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# -*- coding: utf-8 -*-
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#
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# graph_tool -- a general graph manipulation python module
#
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# Copyright (C) 2006-2013 Tiago de Paula Peixoto <tiago@skewed.de>
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#
# 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/>.

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"""
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``graph_tool.topology`` - Assessing graph topology
--------------------------------------------------
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Summary
+++++++

.. autosummary::
   :nosignatures:

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   shortest_distance
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   shortest_path
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   pseudo_diameter
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   similarity
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   isomorphism
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   subgraph_isomorphism
   mark_subgraph
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   max_cardinality_matching
   max_independent_vertex_set
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   min_spanning_tree
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   random_spanning_tree
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   dominator_tree
   topological_sort
   transitive_closure
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   tsp_tour
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   sequential_vertex_coloring
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   label_components
   label_biconnected_components
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   label_largest_component
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   label_out_component
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   kcore_decomposition
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   is_bipartite
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   is_DAG
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   is_planar
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   make_maximal_planar
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   edge_reciprocity
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Contents
++++++++
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"""

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from __future__ import division, absolute_import, print_function

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from .. dl_import import dl_import
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dl_import("from . import libgraph_tool_topology")
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from .. import _prop, Vector_int32_t, _check_prop_writable, \
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     _check_prop_scalar, _check_prop_vector, Graph, PropertyMap, GraphView,\
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     libcore, _get_rng, _degree
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import random, sys, numpy
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__all__ = ["isomorphism", "subgraph_isomorphism", "mark_subgraph",
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           "max_cardinality_matching", "max_independent_vertex_set",
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           "min_spanning_tree", "random_spanning_tree", "dominator_tree",
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           "topological_sort", "transitive_closure", "tsp_tour",
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           "sequential_vertex_coloring", "label_components",
           "label_largest_component", "label_biconnected_components",
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           "label_out_component", "kcore_decomposition", "shortest_distance",
           "shortest_path", "pseudo_diameter", "is_bipartite", "is_DAG",
           "is_planar", "make_maximal_planar", "similarity", "edge_reciprocity"]
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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`
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        Second graph to be compared.
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    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
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(100, lambda: (3,3))
    >>> u = g.copy()
    >>> gt.similarity(u, g)
    1.0
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    >>> gt.random_rewire(u)
    21
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    >>> gt.similarity(u, g)
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    0.03
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    """

    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
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def isomorphism(g1, g2, isomap=False):
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    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
    --------
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    .. testcode::
       :hide:

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

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    >>> 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

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    """
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    imap = g1.new_vertex_property("int32_t")
    iso = libgraph_tool_topology.\
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           check_isomorphism(g1._Graph__graph, g2._Graph__graph,
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                             _prop("v", g1, imap))
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    if isomap:
        return iso, imap
    else:
        return iso

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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`).
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    Parameters
    ----------
    sub : :class:`~graph_tool.Graph`
        Subgraph for which to be searched.
    g : :class:`~graph_tool.Graph`
        Graph in which the search is performed.
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    max_n : int (optional, default: `0`)
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        Maximum number of matches to find. If `max_n == 0`, all matches are
        found.
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    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.
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    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`.
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    Examples
    --------
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    .. testcode::
       :hide:

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

    >>> from numpy.random import poisson
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    >>> g = gt.random_graph(30, lambda: (poisson(6.1), poisson(6.1)))
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    >>> sub = gt.random_graph(10, lambda: (poisson(1.9), poisson(1.9)))
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    >>> vm, em = gt.subgraph_isomorphism(sub, g)
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    >>> print(len(vm))
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    >>> for i in range(len(vm)):
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    ...   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
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    >>> ewidth.a *= 2
    >>> gt.graph_draw(g, vertex_fill_color=vmask, edge_color=emask,
    ...               edge_pen_width=ewidth, output_size=(200, 200),
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    ...               output="subgraph-iso-embed.pdf")
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    <...>
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    >>> gt.graph_draw(sub, output_size=(200, 200), output="subgraph-iso.pdf")
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    <...>

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    .. image:: subgraph-iso.*
    .. image:: subgraph-iso-embed.*
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    **Left:** Subgraph searched, **Right:** One isomorphic subgraph found in main graph.
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    References
    ----------
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    .. [ullmann-algorithm-1976] Ullmann, J. R., "An algorithm for subgraph
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       isomorphism", Journal of the ACM 23 (1): 31-42, 1976, :doi:`10.1145/321921.321925`
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    .. [subgraph-isormophism-wikipedia] http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem
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    """
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    if sub.num_vertices() == 0:
        raise ValueError("Cannot search for an empty subgraph.")
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    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!")
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    vmaps = []
    emaps = []
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    if random:
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        rng = _get_rng()
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    else:
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        rng = libcore.rng_t()
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    libgraph_tool_topology.\
           subgraph_isomorphism(sub._Graph__graph, g._Graph__graph,
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                                _prop("v", sub, vertex_label[0]),
                                _prop("v", g, vertex_label[1]),
                                _prop("e", sub, edge_label[0]),
                                _prop("e", g, edge_label[1]),
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                                vmaps, emaps, max_n, rng)
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    for i in range(len(vmaps)):
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        vmaps[i] = PropertyMap(vmaps[i], sub, "v")
        emaps[i] = PropertyMap(emaps[i], sub, "e")
    return vmaps, emaps

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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
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    `sub`.
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    """
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    if vmask is None:
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        vmask = g.new_vertex_property("bool")
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    if emask is None:
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        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
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def min_spanning_tree(g, weights=None, root=None, tree_map=None):
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    """
    Return the minimum spanning tree of a given graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
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    weights : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
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        The edge weights. If provided, the minimum spanning tree will minimize
        the edge weights.
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    root : :class:`~graph_tool.Vertex` (optional, default: `None`)
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        Root of the minimum spanning tree. If this is provided, Prim's algorithm
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        is used. Otherwise, Kruskal's algorithm is used.
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    tree_map : :class:`~graph_tool.PropertyMap` (optional, default: `None`)
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        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
    --------
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    .. testcode::
       :hide:

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

    >>> from numpy.random import random
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    >>> g, pos = gt.triangulation(random((400, 2)) * 10, type="delaunay")
    >>> weight = g.new_edge_property("double")
    >>> for e in g.edges():
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    ...    weight[e] = linalg.norm(pos[e.target()].a - pos[e.source()].a)
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    >>> tree = gt.min_spanning_tree(g, weights=weight)
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    >>> gt.graph_draw(g, pos=pos, output="triang_orig.pdf")
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    <...>
    >>> g.set_edge_filter(tree)
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    >>> gt.graph_draw(g, pos=pos, output="triang_min_span_tree.pdf")
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    <...>


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    .. image:: triang_orig.*
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        :width: 400px
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    .. image:: triang_min_span_tree.*
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        :width: 400px
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    *Left:* Original graph, *Right:* The minimum spanning tree.
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    References
    ----------
    .. [kruskal-shortest-1956] J. B. Kruskal.  "On the shortest spanning subtree
       of a graph and the traveling salesman problem",  In Proceedings of the
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       American Mathematical Society, volume 7, pages 48-50, 1956.
       :doi:`10.1090/S0002-9939-1956-0078686-7`
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    .. [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
    """
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    if tree_map is None:
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        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.")

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    try:
        g.stash_filter(directed=True)
        g.set_directed(False)
        if root is None:
            libgraph_tool_topology.\
                   get_kruskal_spanning_tree(g._Graph__graph,
                                             _prop("e", g, weights),
                                             _prop("e", g, tree_map))
        else:
            libgraph_tool_topology.\
                   get_prim_spanning_tree(g._Graph__graph, int(root),
                                          _prop("e", g, weights),
                                          _prop("e", g, tree_map))
    finally:
        g.pop_filter(directed=True)
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    return tree_map
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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
    --------
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    .. testcode::
       :hide:

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

    >>> from numpy.random import random
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    >>> 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)
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    >>> gt.graph_draw(g, pos=pos, output="triang_random_span_tree.pdf")
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    <...>


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


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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
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(100, lambda: (2, 2))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
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    >>> root = [v for v in g.vertices() if v.in_degree() == 0]
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    >>> dom = gt.dominator_tree(g, root[0])
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    >>> print(dom.a)
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    [ 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
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      0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
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      0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
      0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
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    References
    ----------
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    .. [dominator-bgl] http://www.boost.org/libs/graph/doc/lengauer_tarjan_dominator.htm
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    """
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    if dom_map is None:
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        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" +
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                         " int32_t.")
    if not g.is_directed():
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        raise ValueError("dominator tree requires a directed graph.")
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    libgraph_tool_topology.\
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               dominator_tree(g._Graph__graph, int(root),
                              _prop("v", g, dom_map))
    return dom_map
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def topological_sort(g):
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    """
    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
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tree = gt.min_spanning_tree(g)
    >>> g.set_edge_filter(tree)
    >>> sort = gt.topological_sort(g)
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    >>> print(sort)
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    [ 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]
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    References
    ----------
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    .. [topological-boost] http://www.boost.org/libs/graph/doc/topological_sort.html
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    .. [topological-wiki] http://en.wikipedia.org/wiki/Topological_sorting

    """

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    topological_order = Vector_int32_t()
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    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()
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def transitive_closure(g):
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    """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
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(30, lambda: (3, 3))
    >>> tc = gt.transitive_closure(g)

    References
    ----------
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    .. [transitive-boost] http://www.boost.org/libs/graph/doc/transitive_closure.html
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    .. [transitive-wiki] http://en.wikipedia.org/wiki/Transitive_closure

    """

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    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

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

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

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    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    vprop : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
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        Vertex property to store the component labels. If none is supplied, one
        is created.
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    directed : bool (optional, default: ``None``)
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        Treat graph as directed or not, independently of its actual
        directionality.
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    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.
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    Returns
    -------
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    comp : :class:`~graph_tool.PropertyMap`
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        Vertex property map with component labels.
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    hist : :class:`~numpy.ndarray`
        Histogram of component labels.
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    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.
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    Notes
    -----
    The components are arbitrarily labeled from 0 to N-1, where N is the total
    number of components.

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

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

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    >>> g = gt.random_graph(100, lambda: (poisson(2), poisson(2)))
    >>> comp, hist, is_attractor = gt.label_components(g, attractors=True)
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    >>> print(comp.a)
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    [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]
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    >>> print(hist)
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    [ 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]
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    >>> print(is_attractor)
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    [ True  True  True False False False  True  True False False False False
      True  True False False False False False False False False  True False
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     False False False False False False False False False False False False
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     False False False False False]
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    """

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

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

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    if directed is not None:
        g = GraphView(g, directed=directed)
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    hist = libgraph_tool_topology.\
               label_components(g._Graph__graph, _prop("v", g, vprop))
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    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
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def label_largest_component(g, directed=None):
    """
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    Label the largest component in the graph. If the graph is directed, then the
    largest strongly connected component is labelled.
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    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`
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         Boolean vertex property map which labels the largest component.
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    Notes
    -----
    The algorithm runs in :math:`O(V + E)` time.

    Examples
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(100, lambda: poisson(1), directed=False)
    >>> l = gt.label_largest_component(g)
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    >>> print(l.a)
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    [1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
     0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
     0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]
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    >>> u = gt.GraphView(g, vfilt=l)   # extract the largest component as a graph
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    >>> print(u.num_vertices())
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    """

    label = g.new_vertex_property("bool")
    c, h = label_components(g, directed=directed)
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    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)
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    return label
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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
    --------
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    .. 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))
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    >>> print(l.a)
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    [1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0
     1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 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 1 1 1 1 0 1 1 1 0 1 1 0 0]
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    The in-component can be obtained by reversing the graph.

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    >>> l = gt.label_out_component(gt.GraphView(g, reversed=True, directed=True),
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    ...                            g.vertex(1))
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    >>> print(l.a)
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    [1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0
     0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1
     1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0]
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    """

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


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

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    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.

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    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
    --------
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    .. testcode::
       :hide:

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

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    >>> g = gt.random_graph(100, lambda: poisson(2), directed=False)
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    >>> comp, art, hist = gt.label_biconnected_components(g)
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    >>> print(comp.a)
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    [51 51 51 51 51 51 11 52 51 51 44 42 41 45 49 23 19 51 51 32 38 51 24 37 51
     51 51 10  8 51 20 43 51 51 51 51 51 47 46 51 51 13 14 51 51 51 51 33 30 51
      1 21 51 51 51 35 36  6 51 26 27  7 12  4  3 29 28 51 51 51 31 51 51  0 39
     51 51 51 34 40 51 51  9 17 51 51 18 15 22  2 16 50  5 48 51 51 53 51 51 25]
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    >>> print(art.a)
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    [1 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0
     0 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1
     1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0]
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    >>> print(hist)
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    [ 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  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 47  1  1]
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    """
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    if vprop is None:
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        vprop = g.new_vertex_property("bool")
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    if eprop is None:
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        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")

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    g = GraphView(g, directed=False)
    hist = libgraph_tool_topology.\
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             label_biconnected_components(g._Graph__graph, _prop("e", g, eprop),
                                          _prop("v", g, vprop))
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    return eprop, vprop, hist
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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
    .. [batagelk-algorithm] V. Batagelj, M. Zaversnik, "An O(m) Algorithm for
       Cores Decomposition of Networks", 2003, :arxiv:`cs/0310049`

    """

    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

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def shortest_distance(g, source=None, target=None, weights=None, max_dist=None,
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                      directed=None, dense=False, dist_map=None,
                      pred_map=False):
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    """
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    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.
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    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    source : :class:`~graph_tool.Vertex` (optional, default: None)
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        Source vertex of the search. If unspecified, the all pairs shortest
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        distances are computed.
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    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.
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    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
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        searched. This parameter has no effect if source is None.
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    directed : bool (optional, default:None)
        Treat graph as directed or not, independently of its actual
        directionality.
    dense : bool (optional, default: False)
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        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
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        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.
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    pred_map : bool (optional, default: False)
        If true, a vertex property map with the predecessors is returned.
        Ignored if source=None.
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    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
    --------
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    .. testcode::
       :hide:

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

    >>> from numpy.random import poisson
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    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
    >>> dist = gt.shortest_distance(g, source=g.vertex(0))
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    >>> print(dist.a)
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    [         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]
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    >>> dist = gt.shortest_distance(g)
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    >>> print(dist[g.vertex(0)].a)
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    [         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]
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    References
    ----------
    .. [bfs] Edward Moore, "The shortest path through a maze", International
       Symposium on the Theory of Switching (1959), Harvard University
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       Press;
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
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    .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with
       graphs." Numerische Mathematik, 1:269-271, 1959.
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    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
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    .. [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
    """

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    if weights is None:
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        dist_type = 'int32_t'
    else:
        dist_type = weights.value_type()

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    if dist_map is None:
        if source is not None:
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            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")
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    if source is not None:
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        _check_prop_scalar(dist_map, name="dist_map")
    else:
        _check_prop_vector(dist_map, name="dist_map")

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    if max_dist is None:
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        max_dist = 0

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    if directed is not None:
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        g.stash_filter(directed=True)
        g.set_directed(directed)

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    if target is None:
        target = -1

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    try:
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        if source is not None:
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            pmap = g.copy_property(g.vertex_index, value_type="int64_t")
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            libgraph_tool_topology.get_dists(g._Graph__graph,
                                             int(source),
                                             int(target),
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                                             _prop("v", g, dist_map),
                                             _prop("e", g, weights),
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                                             _prop("v", g, pmap),
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                                             float(max_dist))
        else:
            libgraph_tool_topology.get_all_dists(g._Graph__graph,
                                                 _prop("v", g, dist_map),
                                                 _prop("e", g, weights), dense)

    finally:
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        if directed is not None:
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            g.pop_filter(directed=True)
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    if source is not None and target != -1:
        dist_map = dist_map[target]

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    if source is not None and pred_map:
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        return dist_map, pmap
    else:
        return dist_map

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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.
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    target : :class:`~graph_tool.Vertex`
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        Target vertex of the search.
    weights : :class:`~graph_tool.PropertyMap` (optional, default: None)
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        The edge weights.
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    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
    --------
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    .. 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)))
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    >>> vlist, elist = gt.shortest_path(g, g.vertex(10), g.vertex(11))
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    >>> print([str(v) for v in vlist])
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    ['10', '131', '184', '265', '223', '11']
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    >>> print([str(e) for e in elist])
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    ['(10, 131)', '(131, 184)', '(184, 265)', '(265, 223)', '(223, 11)']
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    References
    ----------
    .. [bfs] Edward Moore, "The shortest path through a maze", International
       Symposium on the Theory of Switching (1959), Harvard University
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       Press
    .. [bfs-boost] http://www.boost.org/libs/graph/doc/breadth_first_search.html
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    .. [dijkstra] E. Dijkstra, "A note on two problems in connexion with
       graphs." Numerische Mathematik, 1:269-271, 1959.
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    .. [dijkstra-boost] http://www.boost.org/libs/graph/doc/dijkstra_shortest_paths.html
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    """

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    if pred_map is None:
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        pred_map = shortest_distance(g, source, target,
                                     weights=weights,
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                                     pred_map=True)[1]
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    if pred_map[target] == int(target):  # no path to target
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        return [], []

    vlist = [target]
    elist = []

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    if weights is not None:
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        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:
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                if weights is not None:
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                    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

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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
    --------
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    .. testcode::
       :hide:

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

    >>> from numpy.random import poisson
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    >>> g = gt.random_graph(300, lambda: (poisson(3), poisson(3)))
    >>> dist, ends = gt.pseudo_diameter(g)
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    >>> print(dist)
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    9.0
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    >>> print(int(ends[0]), int(ends[1]))
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    0 140
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    References
    ----------
    .. [pseudo-diameter] http://en.wikipedia.org/wiki/Distance_%28graph_theory%29
    """

    if source is None:
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        source = g.vertex(0, use_index=False)
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    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))


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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
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    >>> gt.graph_draw(g, vertex_fill_color=part, output_size=(300, 300), output="bipartite.pdf")
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    <...>

    .. 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


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def is_planar(g, embedding=False, kuratowski=False):
    """
    Test if the graph is planar.

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

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

    Notes
    -----

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

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

    Examples
    --------