#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2006-2014 Tiago de Paula Peixoto
#
# 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 .
"""
``graph_tool.search`` - Search algorithms
-----------------------------------------
This module includes several search algorithms, which are customizable to
arbitrary purposes. It is mostly a wrapper around the Visitor interface of the
`Boost Graph Library `_,
and the respective search functions.
Summary
+++++++
.. autosummary::
:nosignatures:
bfs_search
dfs_search
dijkstra_search
bellman_ford_search
astar_search
BFSVisitor
DFSVisitor
DijkstraVisitor
BellmanFordVisitor
AStarVisitor
StopSearch
Examples
++++++++
In this module, most documentation examples will make use of the network
:download:`search_example.xml `, shown below.
>>> g = gt.load_graph("search_example.xml")
>>> name = g.vp["name"]
>>> weight = g.ep["weight"]
>>> pos = g.vp["pos"]
>>> gt.graph_draw(g, pos, vertex_text=name, vertex_font_size=12, vertex_shape="double_circle",
... vertex_fill_color="#729fcf", vertex_pen_width=3,
... edge_pen_width=weight, output="search_example.pdf")
<...>
.. testcode::
:hide:
gt.graph_draw(g, pos=pos, vertex_text=name, vertex_font_size=12, vertex_shape="double_circle",
vertex_fill_color="#729fcf", vertex_pen_width=3,
edge_pen_width=weight, output="search_example.png")
.. figure:: search_example.*
:alt: search example
:align: center
This is the network used in the examples below. The width of the edges
correspond to the values of the "weight" property map.
Contents
++++++++
"""
from __future__ import division, absolute_import, print_function
import sys
if sys.version_info < (3,):
range = xrange
from .. dl_import import dl_import
dl_import("from . import libgraph_tool_search")
from .. import _prop, _python_type
import weakref
__all__ = ["bfs_search", "BFSVisitor", "dfs_search", "DFSVisitor",
"dijkstra_search", "DijkstraVisitor", "bellman_ford_search",
"BellmanFordVisitor", "astar_search", "AStarVisitor",
"StopSearch"]
class VisitorWrapper(object):
def __init__(self, g, visitor, edge_members, vertex_members):
self.visitor = visitor
self.g = g
self.edge_members = set(edge_members)
self.vertex_members = set(vertex_members)
def __getattr__(self, attr):
try:
orig_attr = self.visitor.__getattribute__(attr)
except AttributeError:
return object.__getattribute__(self, attr)
if callable(orig_attr):
def wrapped_visitor_member(*args, **kwargs):
old_perms = dict(self.g._Graph__perms)
perms ={"del_vertex": False, "del_edge": False, "add_edge": False}
if attr in self.edge_members:
perms.update({"del_edge": True, "add_edge": True})
elif attr in self.vertex_members:
perms.update({"add_vertex": False})
self.g._Graph__perms.update(perms)
try:
ret = orig_attr(*args, **kwargs)
finally:
self.g._Graph__perms.update(old_perms)
return ret
return wrapped_visitor_member
else:
return orig_attr
class BFSVisitor(object):
r"""A visitor object that is invoked at the event-points inside the
:func:`~graph_tool.search.bfs_search` algorithm. By default, it performs no
action, and should be used as a base class in order to be useful."""
def initialize_vertex(self, u):
"""This is invoked on every vertex of the graph before the start of the
graph search. """
return
def discover_vertex(self, u):
"""This is invoked when a vertex is encountered for the first time."""
return
def examine_vertex(self, u):
"""This is invoked on a vertex as it is popped from the queue. This
happens immediately before examine_edge() is invoked on each of the
out-edges of vertex u."""
return
def examine_edge(self, e):
"""This is invoked on every out-edge of each vertex after it is
discovered."""
return
def tree_edge(self, e):
"""This is invoked on each edge as it becomes a member of the edges that
form the search tree."""
return
def non_tree_edge(self, e):
"""This is invoked on back or cross edges for directed graphs and cross
edges for undirected graphs. """
return
def gray_target(self, e):
"""This is invoked on the subset of non-tree edges whose target vertex
is colored gray at the time of examination. The color gray indicates
that the vertex is currently in the queue."""
return
def black_target(self, e):
"""This is invoked on the subset of non-tree edges whose target vertex
is colored black at the time of examination. The color black indicates
that the vertex has been removed from the queue."""
return
def finish_vertex(self, u):
"""This invoked on a vertex after all of its out edges have been added
to the search tree and all of the adjacent vertices have been discovered
(but before the out-edges of the adjacent vertices have been examined).
"""
return
def bfs_search(g, source, visitor=BFSVisitor()):
r"""Breadth-first traversal of a directed or undirected graph.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : :class:`~graph_tool.Vertex`
Source vertex.
visitor : :class:`~graph_tool.search.BFSVisitor` (optional, default: ``BFSVisitor()``)
A visitor object that is invoked at the event points inside the
algorithm. This should be a subclass of
:class:`~graph_tool.search.BFSVisitor`.
See Also
--------
dfs_search: Depth-first search
dijkstra_search: Dijkstra's search algorithm
astar_search: :math:`A^*` heuristic search algorithm
Notes
-----
A breadth-first traversal visits vertices that are closer to the source
before visiting vertices that are further away. In this context "distance"
is defined as the number of edges in the shortest path from the source
vertex.
The time complexity is :math:`O(V + E)`.
The pseudo-code for the BFS algorithm is listed below, with the annotated
event points, for which the given visitor object will be called with the
appropriate method.
::
BFS(G, source)
for each vertex u in V[G] initialize vertex u
color[u] := WHITE
d[u] := infinity
end for
color[source] := GRAY
d[source] := 0
ENQUEUE(Q, source) discover vertex source
while (Q != Ø)
u := DEQUEUE(Q) examine vertex u
for each vertex v in Adj[u] examine edge (u,v)
if (color[v] = WHITE) (u,v) is a tree edge
color[v] := GRAY
ENQUEUE(Q, v) discover vertex v
else (u,v) is a non-tree edge
if (color[v] = GRAY)
... (u,v) has a gray target
else
... (u,v) has a black target
end for
color[u] := BLACK finish vertex u
end while
Examples
--------
We must define what should be done during the search by subclassing
:class:`~graph_tool.search.BFSVisitor`, and specializing the appropriate
methods. In the following we will keep track of the distance from the root,
and the predecessor tree.
.. testcode::
class VisitorExample(gt.BFSVisitor):
def __init__(self, name, pred, dist):
self.name = name
self.pred = pred
self.dist = dist
def discover_vertex(self, u):
print("-->", self.name[u], "has been discovered!")
def examine_vertex(self, u):
print(self.name[u], "has been examined...")
def tree_edge(self, e):
self.pred[e.target()] = int(e.source())
self.dist[e.target()] = self.dist[e.source()] + 1
With the above class defined, we can perform the BFS search as follows.
>>> dist = g.new_vertex_property("int")
>>> pred = g.new_vertex_property("int")
>>> gt.bfs_search(g, g.vertex(0), VisitorExample(name, pred, dist))
--> Bob has been discovered!
Bob has been examined...
--> Eve has been discovered!
--> Chuck has been discovered!
--> Carlos has been discovered!
--> Isaac has been discovered!
Eve has been examined...
--> Imothep has been discovered!
--> Carol has been discovered!
Chuck has been examined...
Carlos has been examined...
--> Alice has been discovered!
Isaac has been examined...
Imothep has been examined...
Carol has been examined...
Alice has been examined...
--> Oscar has been discovered!
--> Dave has been discovered!
Oscar has been examined...
Dave has been examined...
>>> print(dist.a)
[0 2 2 1 1 3 1 1 3 2]
>>> print(pred.a)
[0 3 6 0 0 1 0 0 1 6]
References
----------
.. [bfs] Edward Moore, "The shortest path through a maze", International
Symposium on the Theory of Switching, 1959
.. [bfs-bgl] http://www.boost.org/doc/libs/release/libs/graph/doc/breadth_first_search.html
.. [bfs-wikipedia] http://en.wikipedia.org/wiki/Breadth-first_search
"""
visitor = VisitorWrapper(g, visitor,
["initialize_vertex", "examine_vertex", "finish_vertex"],
["initialize_vertex"])
try:
libgraph_tool_search.bfs_search(g._Graph__graph,
weakref.ref(g),
int(source), visitor)
except StopSearch:
pass
class DFSVisitor(object):
r"""
A visitor object that is invoked at the event-points inside the
:func:`~graph_tool.search.dfs_search` algorithm. By default, it performs no
action, and should be used as a base class in order to be useful.
"""
def initialize_vertex(self, u):
"""
This is invoked on every vertex of the graph before the start of the
graph search.
"""
return
def start_vertex(self, u):
"""
This is invoked on the source vertex once before the start of the
search.
"""
return
def discover_vertex(self, u):
"""This is invoked when a vertex is encountered for the first time."""
return
def examine_edge(self, e):
"""
This is invoked on every out-edge of each vertex after it is discovered.
"""
return
def tree_edge(self, e):
"""
This is invoked on each edge as it becomes a member of the edges that
form the search tree.
"""
return
def back_edge(self, e):
"""
This is invoked on the back edges in the graph. For an undirected graph
there is some ambiguity between tree edges and back edges since the edge
(u,v) and (v,u) are the same edge, but both the
:meth:`~graph_tool.search.DFSVisitor.tree_edge` and
:meth:`~graph_tool.search..DFSVisitor.back_edge` functions will be
invoked. One way to resolve this ambiguity is to record the tree edges,
and then disregard the back-edges that are already marked as tree
edges. An easy way to record tree edges is to record predecessors at the
tree_edge event point.
"""
return
def forward_or_cross_edge(self, e):
"""
This is invoked on forward or cross edges in the graph. In an undirected
graph this method is never called.
"""
return
def finish_vertex(self, e):
"""
This is invoked on vertex u after finish_vertex has been called for all
the vertices in the DFS-tree rooted at vertex u. If vertex u is a leaf
in the DFS-tree, then the
:meth:`~graph_tool..search.DFSVisitor.finish_vertex` function is called
on u after all the out-edges of u have been examined.
"""
return
def dfs_search(g, source, visitor=DFSVisitor()):
r"""Depth-first traversal of a directed or undirected graph.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : :class:`~graph_tool.Vertex`
Source vertex.
visitor : :class:`~graph_tool.search.DFSVisitor` (optional, default: ``DFSVisitor()``)
A visitor object that is invoked at the event points inside the
algorithm. This should be a subclass of
:class:`~graph_tool.search.DFSVisitor`.
See Also
--------
bfs_search: Breadth-first search
dijkstra_search: Dijkstra's search algorithm
astar_search: :math:`A^*` heuristic search algorithm
Notes
-----
When possible, a depth-first traversal chooses a vertex adjacent to the
current vertex to visit next. If all adjacent vertices have already been
discovered, or there are no adjacent vertices, then the algorithm backtracks
to the last vertex that had undiscovered neighbors. Once all reachable
vertices have been visited, the algorithm selects from any remaining
undiscovered vertices and continues the traversal. The algorithm finishes
when all vertices have been visited.
The time complexity is :math:`O(V + E)`.
The pseudo-code for the DFS algorithm is listed below, with the annotated
event points, for which the given visitor object will be called with the
appropriate method.
::
DFS(G)
for each vertex u in V
color[u] := WHITE initialize vertex u
end for
time := 0
call DFS-VISIT(G, source) start vertex s
DFS-VISIT(G, u)
color[u] := GRAY discover vertex u
for each v in Adj[u] examine edge (u,v)
if (color[v] = WHITE) (u,v) is a tree edge
call DFS-VISIT(G, v)
else if (color[v] = GRAY) (u,v) is a back edge
...
else if (color[v] = BLACK) (u,v) is a cross or forward edge
...
end for
color[u] := BLACK finish vertex u
Examples
--------
We must define what should be done during the search by subclassing
:class:`~graph_tool.search.DFSVisitor`, and specializing the appropriate
methods. In the following we will keep track of the discover time, and the
predecessor tree.
.. testcode::
class VisitorExample(gt.DFSVisitor):
def __init__(self, name, pred, time):
self.name = name
self.pred = pred
self.time = time
self.last_time = 0
def discover_vertex(self, u):
print("-->", self.name[u], "has been discovered!")
self.time[u] = self.last_time
self.last_time += 1
def examine_edge(self, e):
print("edge (%s, %s) has been examined..." % \
(self.name[e.source()], self.name[e.target()]))
def tree_edge(self, e):
self.pred[e.target()] = int(e.source())
With the above class defined, we can perform the DFS search as follows.
>>> time = g.new_vertex_property("int")
>>> pred = g.new_vertex_property("int")
>>> gt.dfs_search(g, g.vertex(0), VisitorExample(name, pred, time))
--> Bob has been discovered!
edge (Bob, Eve) has been examined...
--> Eve has been discovered!
edge (Eve, Isaac) has been examined...
--> Isaac has been discovered!
edge (Isaac, Bob) has been examined...
edge (Isaac, Chuck) has been examined...
--> Chuck has been discovered!
edge (Chuck, Eve) has been examined...
edge (Chuck, Isaac) has been examined...
edge (Chuck, Imothep) has been examined...
--> Imothep has been discovered!
edge (Imothep, Carol) has been examined...
--> Carol has been discovered!
edge (Carol, Eve) has been examined...
edge (Carol, Imothep) has been examined...
edge (Imothep, Carlos) has been examined...
--> Carlos has been discovered!
edge (Carlos, Eve) has been examined...
edge (Carlos, Imothep) has been examined...
edge (Carlos, Bob) has been examined...
edge (Carlos, Alice) has been examined...
--> Alice has been discovered!
edge (Alice, Oscar) has been examined...
--> Oscar has been discovered!
edge (Oscar, Alice) has been examined...
edge (Oscar, Dave) has been examined...
--> Dave has been discovered!
edge (Dave, Oscar) has been examined...
edge (Dave, Alice) has been examined...
edge (Alice, Dave) has been examined...
edge (Alice, Carlos) has been examined...
edge (Imothep, Chuck) has been examined...
edge (Imothep, Eve) has been examined...
edge (Chuck, Bob) has been examined...
edge (Isaac, Eve) has been examined...
edge (Eve, Imothep) has been examined...
edge (Eve, Bob) has been examined...
edge (Eve, Carol) has been examined...
edge (Eve, Carlos) has been examined...
edge (Eve, Chuck) has been examined...
edge (Bob, Chuck) has been examined...
edge (Bob, Carlos) has been examined...
edge (Bob, Isaac) has been examined...
>>> print(time.a)
[0 7 5 6 3 9 1 2 8 4]
>>> print(pred.a)
[0 3 9 9 7 8 0 6 1 4]
References
----------
.. [dfs-bgl] http://www.boost.org/doc/libs/release/libs/graph/doc/depth_first_search.html
.. [dfs-wikipedia] http://en.wikipedia.org/wiki/Depth-first_search
"""
visitor = VisitorWrapper(g, visitor,
["initialize_vertex", "discover_vertex", "finish_vertex",
"start_vertex"], ["initialize_vertex"])
try:
libgraph_tool_search.dfs_search(g._Graph__graph,
weakref.ref(g),
int(source), visitor)
except StopSearch:
pass
class DijkstraVisitor(object):
r"""A visitor object that is invoked at the event-points inside the
:func:`~graph_tool.search.dijkstra_search` algorithm. By default, it
performs no action, and should be used as a base class in order to be
useful.
"""
def initialize_vertex(self, u):
"""
This is invoked on every vertex of the graph before the start of the
graph search.
"""
return
def examine_vertex(self, u):
"""
This is invoked on a vertex as it is popped from the queue. This happens
immediately before :meth:`~graph_tool.DijsktraVisitor.examine_edge` is
invoked on each of the out-edges of vertex u.
"""
return
def examine_edge(self, e):
"""
This is invoked on every out-edge of each vertex after it is discovered.
"""
return
def discover_vertex(self, u):
"""This is invoked when a vertex is encountered for the first time."""
return
def edge_relaxed(self, e):
"""
Upon examination, if the following condition holds then the edge is
relaxed (its distance is reduced), and this method is invoked.
::
(u, v) = tuple(e)
assert(compare(combine(d[u], weight[e]), d[v]))
"""
return
def edge_not_relaxed(self, e):
"""
Upon examination, if the edge is not relaxed (see
:meth:`~graph_tool.search.DijsktraVisitor.edge_relaxed`) then this
method is invoked.
"""
return
def finish_vertex(self, u):
"""
This invoked on a vertex after all of its out edges have been added to
the search tree and all of the adjacent vertices have been discovered
(but before their out-edges have been examined).
"""
return
def dijkstra_search(g, source, weight, visitor=DijkstraVisitor(), dist_map=None,
pred_map=None, combine=lambda a, b: a + b,
compare=lambda a, b: a < b, zero=0, infinity=float('inf')):
r"""Dijsktra traversal of a directed or undirected graph, with non-negative weights.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : :class:`~graph_tool.Vertex`
Source vertex.
weight : :class:`~graph_tool.PropertyMap`
Edge property map with weight values.
visitor : :class:`~graph_tool.search.DijkstraVisitor` (optional, default: ``DijkstraVisitor()``)
A visitor object that is invoked at the event points inside the
algorithm. This should be a subclass of
:class:`~graph_tool.search.DijkstraVisitor`.
dist_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the distances from the source will be
stored.
pred_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the predecessor map will be
stored (must have value type "int").
combine : binary function (optional, default: ``lambda a, b: a + b``)
This function is used to combine distances to compute the distance of a
path.
compare : binary function (optional, default: ``lambda a, b: a < b``)
This function is use to compare distances to determine which vertex is
closer to the source vertex.
zero : int or float (optional, default: ``0``)
Value assumed to correspond to a distance of zero by the combine and
compare functions.
infinity : int or float (optional, default: ``float('inf')``)
Value assumed to correspond to a distance of infinity by the combine and
compare functions.
Returns
-------
dist_map : :class:`~graph_tool.PropertyMap`
A vertex property map with the computed distances from the source.
pred_map : :class:`~graph_tool.PropertyMap`
A vertex property map with the predecessor tree.
See Also
--------
bfs_search: Breadth-first search
dfs_search: Depth-first search
astar_search: :math:`A^*` heuristic search algorithm
Notes
-----
Dijkstra's algorithm finds all the shortest paths from the source vertex to
every other vertex by iteratively "growing" the set of vertices S to which
it knows the shortest path. At each step of the algorithm, the next vertex
added to S is determined by a priority queue. The queue contains the
vertices in V - S prioritized by their distance label, which is the length
of the shortest path seen so far for each vertex. The vertex u at the top of
the priority queue is then added to S, and each of its out-edges is relaxed:
if the distance to u plus the weight of the out-edge (u,v) is less than the
distance label for v then the estimated distance for vertex v is
reduced. The algorithm then loops back, processing the next vertex at the
top of the priority queue. The algorithm finishes when the priority queue is
empty.
The time complexity is :math:`O(V \log V)`.
The pseudo-code for Dijkstra's algorithm is listed below, with the annotated
event points, for which the given visitor object will be called with the
appropriate method.
::
DIJKSTRA(G, source, weight)
for each vertex u in V initialize vertex u
d[u] := infinity
p[u] := u
end for
d[source] := 0
INSERT(Q, source) discover vertex s
while (Q != Ø)
u := EXTRACT-MIN(Q) examine vertex u
for each vertex v in Adj[u] examine edge (u,v)
if (weight[(u,v)] + d[u] < d[v]) edge (u,v) relaxed
d[v] := weight[(u,v)] + d[u]
p[v] := u
DECREASE-KEY(Q, v)
else edge (u,v) not relaxed
...
if (d[v] was originally infinity)
INSERT(Q, v) discover vertex v
end for finish vertex u
end while
return d
Examples
--------
We must define what should be done during the search by subclassing
:class:`~graph_tool.search.DijkstraVisitor`, and specializing the
appropriate methods. In the following we will keep track of the discover
time, and the predecessor tree.
.. testcode::
class VisitorExample(gt.DijkstraVisitor):
def __init__(self, name, time):
self.name = name
self.time = time
self.last_time = 0
def discover_vertex(self, u):
print("-->", self.name[u], "has been discovered!")
self.time[u] = self.last_time
self.last_time += 1
def examine_edge(self, e):
print("edge (%s, %s) has been examined..." % \
(self.name[e.source()], self.name[e.target()]))
def edge_relaxed(self, e):
print("edge (%s, %s) has been relaxed..." % \
(self.name[e.source()], self.name[e.target()]))
With the above class defined, we can perform the Dijkstra search as follows.
>>> time = g.new_vertex_property("int")
>>> dist, pred = gt.dijkstra_search(g, g.vertex(0), weight, VisitorExample(name, time))
--> Bob has been discovered!
edge (Bob, Eve) has been examined...
edge (Bob, Eve) has been relaxed...
--> Eve has been discovered!
edge (Bob, Chuck) has been examined...
edge (Bob, Chuck) has been relaxed...
--> Chuck has been discovered!
edge (Bob, Carlos) has been examined...
edge (Bob, Carlos) has been relaxed...
--> Carlos has been discovered!
edge (Bob, Isaac) has been examined...
edge (Bob, Isaac) has been relaxed...
--> Isaac has been discovered!
edge (Eve, Isaac) has been examined...
edge (Eve, Imothep) has been examined...
edge (Eve, Imothep) has been relaxed...
--> Imothep has been discovered!
edge (Eve, Bob) has been examined...
edge (Eve, Carol) has been examined...
edge (Eve, Carol) has been relaxed...
--> Carol has been discovered!
edge (Eve, Carlos) has been examined...
edge (Eve, Chuck) has been examined...
edge (Isaac, Bob) has been examined...
edge (Isaac, Chuck) has been examined...
edge (Isaac, Eve) has been examined...
edge (Chuck, Eve) has been examined...
edge (Chuck, Isaac) has been examined...
edge (Chuck, Imothep) has been examined...
edge (Chuck, Bob) has been examined...
edge (Carlos, Eve) has been examined...
edge (Carlos, Imothep) has been examined...
edge (Carlos, Bob) has been examined...
edge (Carlos, Alice) has been examined...
edge (Carlos, Alice) has been relaxed...
--> Alice has been discovered!
edge (Imothep, Carol) has been examined...
edge (Imothep, Carlos) has been examined...
edge (Imothep, Chuck) has been examined...
edge (Imothep, Eve) has been examined...
edge (Alice, Oscar) has been examined...
edge (Alice, Oscar) has been relaxed...
--> Oscar has been discovered!
edge (Alice, Dave) has been examined...
edge (Alice, Dave) has been relaxed...
--> Dave has been discovered!
edge (Alice, Carlos) has been examined...
edge (Carol, Eve) has been examined...
edge (Carol, Imothep) has been examined...
edge (Oscar, Alice) has been examined...
edge (Oscar, Dave) has been examined...
edge (Dave, Oscar) has been examined...
edge (Dave, Alice) has been examined...
>>> print(time.a)
[0 7 6 3 2 9 1 4 8 5]
>>> print(pred.a)
[0 3 6 0 0 1 0 0 1 6]
>>> print(dist.a)
[ 0. 8.91915887 9.27141329 4.29277116 4.02118246
12.23513866 3.23790211 3.45487436 11.04391549 7.74858396]
References
----------
.. [dijkstra] E. Dijkstra, "A note on two problems in connexion with
graphs", Numerische Mathematik, 1:269-271, 1959.
.. [dijkstra-bgl] http://www.boost.org/doc/libs/release/libs/graph/doc/dijkstra_shortest_paths_no_color_map.html
.. [dijkstra-wikipedia] http://en.wikipedia.org/wiki/Dijkstra's_algorithm
"""
visitor = VisitorWrapper(g, visitor,
["initialize_vertex", "examine_vertex", "finish_vertex"],
["initialize_vertex"])
if visitor is None:
visitor = DijkstraVisitor()
if dist_map is None:
dist_map = g.new_vertex_property(weight.value_type())
if pred_map is None:
pred_map = g.new_vertex_property("int")
if pred_map.value_type() != "int32_t":
raise ValueError("pred_map must be of value type 'int32_t', not '%s'." % \
pred_map.value_type())
try:
if dist_map.value_type() != "python::object":
zero = _python_type(dist_map.value_type())(zero)
except OverflowError:
zero = (weight.a.max() + 1) * g.num_vertices()
zero = _python_type(dist_map.value_type())(zero)
try:
if dist_map.value_type() != "python::object":
infinity = _python_type(dist_map.value_type())(infinity)
except OverflowError:
infinity = (weight.a.max() + 1) * g.num_vertices()
infinity = _python_type(dist_map.value_type())(infinity)
try:
libgraph_tool_search.dijkstra_search(g._Graph__graph,
weakref.ref(g),
int(source),
_prop("v", g, dist_map),
_prop("v", g, pred_map),
_prop("e", g, weight), visitor,
compare, combine, zero, infinity)
except StopSearch:
pass
return dist_map, pred_map
class BellmanFordVisitor(object):
r"""A visitor object that is invoked at the event-points inside the
:func:`~graph_tool.search.bellman_ford_search` algorithm. By default, it
performs no action, and should be used as a base class in order to be
useful.
"""
def examine_edge(self, e):
"""
This is invoked on every edge in the graph ``|V|`` times.
"""
return
def edge_relaxed(self, e):
"""
This is invoked when the distance label for the target vertex is
decreased. The edge (u,v) that participated in the last relaxation for
vertex v is an edge in the shortest paths tree.
"""
return
def edge_not_relaxed(self, e):
"""
This is invoked if the distance label for the target vertex is not
decreased.
"""
return
def edge_minimized(self, e):
"""
This is invoked during the second stage of the algorithm, during the
test of whether each edge was minimized. If the edge is minimized then
this function is invoked.
"""
return
def edge_not_minimized(self, e):
"""
This is invoked during the second stage of the algorithm, during the
test of whether each edge was minimized. If the edge was not minimized,
this function is invoked. This happens when there is a negative cycle in
the graph.
"""
return
def bellman_ford_search(g, source, weight, visitor=BellmanFordVisitor(),
dist_map=None, pred_map=None,
combine=lambda a, b: a + b,
compare=lambda a, b: a < b, zero=0,
infinity=float('inf')):
r"""Bellman-Ford traversal of a directed or undirected graph, with negative weights.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : :class:`~graph_tool.Vertex`
Source vertex.
weight : :class:`~graph_tool.PropertyMap`
Edge property map with weight values.
visitor : :class:`~graph_tool.search.DijkstraVisitor` (optional, default: ``DijkstraVisitor()``)
A visitor object that is invoked at the event points inside the
algorithm. This should be a subclass of
:class:`~graph_tool.search.DijkstraVisitor`.
dist_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the distances from the source will be
stored.
pred_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the predecessor map will be
stored (must have value type "int").
combine : binary function (optional, default: ``lambda a, b: a + b``)
This function is used to combine distances to compute the distance of a
path.
compare : binary function (optional, default: ``lambda a, b: a < b``)
This function is use to compare distances to determine which vertex is
closer to the source vertex.
zero : int or float (optional, default: ``0``)
Value assumed to correspond to a distance of zero by the combine and
compare functions.
infinity : int or float (optional, default: ``float('inf')``)
Value assumed to correspond to a distance of infinity by the combine and
compare functions.
Returns
-------
minimized : bool
True if all edges were successfully minimized, or False if there is a
negative loop in the graph.
dist_map : :class:`~graph_tool.PropertyMap`
A vertex property map with the computed distances from the source.
pred_map : :class:`~graph_tool.PropertyMap`
A vertex property map with the predecessor tree.
See Also
--------
bfs_search: Breadth-first search
dfs_search: Depth-first search
dijsktra_search: Dijkstra search
astar_search: :math:`A^*` heuristic search
Notes
-----
The Bellman-Ford algorithm [bellman-ford]_ solves the single-source shortest
paths problem for a graph with both positive and negative edge weights. If
you only need to solve the shortest paths problem for positive edge weights,
:func:`~graph_tool.search.dijkstra_search` provides a more efficient
alternative. If all the edge weights are all equal, then
:func:`~graph_tool.search.bfs_search` provides an even more efficient
alternative.
The Bellman-Ford algorithm proceeds by looping through all of the edges in
the graph, applying the relaxation operation to each edge. In the following
pseudo-code, ``v`` is a vertex adjacent to ``u``, ``w`` maps edges to their
weight, and ``d`` is a distance map that records the length of the shortest
path to each vertex seen so far. ``p`` is a predecessor map which records
the parent of each vertex, which will ultimately be the parent in the
shortest paths tree
::
RELAX(u, v, w, d, p)
if (w(u,v) + d[u] < d[v])
d[v] := w(u,v) + d[u] relax edge (u,v)
p[v] := u
else
... edge (u,v) is not relaxed
The algorithm repeats this loop ``|V|`` times after which it is guaranteed
that the distances to each vertex have been reduced to the minimum possible
unless there is a negative cycle in the graph. If there is a negative cycle,
then there will be edges in the graph that were not properly minimized. That
is, there will be edges ``(u,v)`` such that ``w(u,v) + d[u] < d[v]``. The
algorithm loops over the edges in the graph one final time to check if all
the edges were minimized, returning true if they were and returning false
otherwise.
::
BELLMAN-FORD(G)
for each vertex u in V
d[u] := infinity
p[u] := u
end for
for i := 1 to |V|-1
for each edge (u,v) in E examine edge (u,v)
RELAX(u, v, w, d, p)
end for
end for
for each edge (u,v) in E
if (w(u,v) + d[u] < d[v])
return (false, , ) edge (u,v) was not minimized
else
... edge (u,v) was minimized
end for
return (true, p, d)
The time complexity is :math:`O(V E)`.
Examples
--------
We must define what should be done during the search by subclassing
:class:`~graph_tool.search.BellmanFordVisitor`, and specializing the
appropriate methods. In the following we will keep track of the edge
minimizations.
.. testcode::
class VisitorExample(gt.BellmanFordVisitor):
def __init__(self, name):
self.name = name
def edge_minimized(self, e):
print("edge (%s, %s) has been minimized..." % \
(self.name[e.source()], self.name[e.target()]))
def edge_not_minimized(self, e):
print("edge (%s, %s) has not been minimized..." % \
(self.name[e.source()], self.name[e.target()]))
With the above class defined, we can perform the Bellman-Ford search as
follows.
>>> nweight = g.copy_property(weight)
>>> nweight.a[6] *= -1 # include negative weight in edge (Carlos, Alice)
>>> minimized, dist, pred = gt.bellman_ford_search(g, g.vertex(0), nweight, VisitorExample(name))
edge (Bob, Eve) has been minimized...
edge (Bob, Chuck) has been minimized...
edge (Bob, Carlos) has been minimized...
edge (Bob, Isaac) has been minimized...
edge (Alice, Oscar) has been minimized...
edge (Alice, Dave) has been minimized...
edge (Alice, Carlos) has been minimized...
edge (Carol, Eve) has been minimized...
edge (Carol, Imothep) has been minimized...
edge (Carlos, Eve) has been minimized...
edge (Carlos, Imothep) has been minimized...
edge (Chuck, Eve) has been minimized...
edge (Chuck, Isaac) has been minimized...
edge (Chuck, Imothep) has been minimized...
edge (Dave, Oscar) has been minimized...
edge (Eve, Isaac) has been minimized...
edge (Eve, Imothep) has been minimized...
>>> print(minimized)
True
>>> print(pred.a)
[3 3 9 1 6 1 3 6 1 3]
>>> print(dist.a)
[-28.42555934 -37.34471821 -25.20438243 -41.97110592 -35.20316571
-34.02873843 -36.58860946 -33.55645565 -35.2199616 -36.0029274 ]
References
----------
.. [bellman-ford] R. Bellman, "On a routing problem", Quarterly of Applied
Mathematics, 16(1):87-90, 1958.
.. [bellman-ford-bgl] http://www.boost.org/doc/libs/release/libs/graph/doc/bellman_ford_shortest.html
.. [bellman-ford-wikipedia] http://en.wikipedia.org/wiki/Bellman-Ford_algorithm
"""
visitor = VisitorWrapper(g, visitor, [], [])
if dist_map is None:
dist_map = g.new_vertex_property(weight.value_type())
if pred_map is None:
pred_map = g.new_vertex_property("int")
if pred_map.value_type() != "int32_t":
raise ValueError("pred_map must be of value type 'int32_t', not '%s'." % \
pred_map.value_type())
try:
if dist_map.value_type() != "python::object":
zero = _python_type(dist_map.value_type())(zero)
except OverflowError:
zero = (weight.a.max() + 1) * g.num_vertices()
zero = _python_type(dist_map.value_type())(zero)
try:
if dist_map.value_type() != "python::object":
infinity = _python_type(dist_map.value_type())(infinity)
except OverflowError:
infinity = (weight.a.max() + 1) * g.num_vertices()
infinity = _python_type(dist_map.value_type())(infinity)
minimized = False
try:
minimized = \
libgraph_tool_search.bellman_ford_search(g._Graph__graph,
weakref.ref(g),
int(source),
_prop("v", g, dist_map),
_prop("v", g, pred_map),
_prop("e", g, weight),
visitor, compare, combine,
zero, infinity)
except StopSearch:
pass
return minimized, dist_map, pred_map
class AStarVisitor(object):
r"""A visitor object that is invoked at the event-points inside the
:func:`~graph_tool.search.astar_search` algorithm. By default, it
performs no action, and should be used as a base class in order to be
useful.
"""
def initialize_vertex(self, u):
"""
This is invoked on every vertex of the graph before the start of the
graph search.
"""
return
def examine_vertex(self, u):
""" This is invoked on a vertex as it is popped from the queue (i.e. it
has the lowest cost on the ``OPEN`` list). This happens immediately
before examine_edge() is invoked on each of the out-edges of vertex u.
"""
return
def examine_edge(self, e):
"""
This is invoked on every out-edge of each vertex after it is examined.
"""
return
def discover_vertex(self, u):
"""
This is invoked when a vertex is first discovered and is added to the
``OPEN`` list.
"""
return
def edge_relaxed(self, e):
"""
Upon examination, if the following condition holds then the edge is
relaxed (its distance is reduced), and this method is invoked.
::
(u, v) = tuple(e)
assert(compare(combine(d[u], weight[e]), d[v]))
"""
return
def edge_not_relaxed(self, e):
"""
Upon examination, if the edge is not relaxed (see
:meth:`~graph_tool.search.AStarVisitor.edge_relaxed`) then this
method is invoked.
"""
return
def black_target(self, e):
""" This is invoked when a vertex that is on the ``CLOSED`` list is
"rediscovered" via a more efficient path, and is re-added to the
``OPEN`` list.
"""
return
def finish_vertex(self, u):
"""
This is invoked on a vertex when it is added to the CLOSED list,
which happens after all of its out edges have been examined.
"""
return
def astar_search(g, source, weight, visitor=AStarVisitor(),
heuristic=lambda v: 1, dist_map=None, pred_map=None,
cost_map=None, combine=lambda a, b: a + b,
compare=lambda a, b: a < b, zero=0,
infinity=float('inf'), implicit=False):
r"""
Heuristic :math:`A^*` search on a weighted, directed or undirected graph for the case where all edge weights are non-negative.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be used.
source : :class:`~graph_tool.Vertex`
Source vertex.
weight : :class:`~graph_tool.PropertyMap`
Edge property map with weight values.
visitor : :class:`~graph_tool.search.AStarVisitor` (optional, default: ``AStarVisitor()``)
A visitor object that is invoked at the event points inside the
algorithm. This should be a subclass of
:class:`~graph_tool.search.AStarVisitor`.
heuristic : unary function (optional, default: ``lambda v: 1``)
The heuristic function that guides the search. It should take a single
argument which is a :class:`~graph_tool.Vertex`, and output an estimated
distance from the source vertex.
dist_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the distances from the source will be
stored.
pred_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the predecessor map will be
stored (must have value type "int").
cost_map : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
A vertex property map where the vertex costs will be stored. It must
have the same value type as ``dist_map``. This parameter is only used if
``implicit`` is True.
combine : binary function (optional, default: ``lambda a, b: a + b``)
This function is used to combine distances to compute the distance of a
path.
compare : binary function (optional, default: ``lambda a, b: a < b``)
This function is use to compare distances to determine which vertex is
closer to the source vertex.
implicit : bool (optional, default: ``False``)
If true, the underlying graph will be assumed to be implicit
(i.e. constructed during the search).
zero : int or float (optional, default: ``0``)
Value assumed to correspond to a distance of zero by the combine and
compare functions.
infinity : int or float (optional, default: ``float('inf')``)
Value assumed to correspond to a distance of infinity by the combine and
compare functions.
Returns
-------
dist_map : :class:`~graph_tool.PropertyMap`
A vertex property map with the computed distances from the source.
See Also
--------
bfs_search: Breadth-first search
dfs_search: Depth-first search
dijkstra_search: Dijkstra's search algorithm
Notes
-----
The :math:`A^*` algorithm is a heuristic graph search algorithm: an
:math:`A^*` search is "guided" by a heuristic function. A heuristic function
:math:`h(v)` is one which estimates the cost from a non-goal state (v) in
the graph to some goal state, t. Intuitively, :math:`A^*` follows paths
(through the graph) to the goal that are estimated by the heuristic function
to be the best paths. Unlike best-first search, :math:`A^*` takes into
account the known cost from the start of the search to v; the paths
:math:`A^*` takes are guided by a function :math:`f(v) = g(v) + h(v)`, where
:math:`h(v)` is the heuristic function, and :math:`g(v)` (sometimes denoted
:math:`c(s, v)`) is the known cost from the start to v. Clearly, the
efficiency of :math:`A^*` is highly dependent on the heuristic function with
which it is used.
The time complexity is :math:`O((E + V) \log V)`.
The pseudo-code for the :math:`A^*` algorithm is listed below, with the
annotated event points, for which the given visitor object will be called
with the appropriate method.
::
A*(G, source, h)
for each vertex u in V initialize vertex u
d[u] := f[u] := infinity
color[u] := WHITE
end for
color[s] := GRAY
d[s] := 0
f[s] := h(source)
INSERT(Q, source) discover vertex source
while (Q != Ø)
u := EXTRACT-MIN(Q) examine vertex u
for each vertex v in Adj[u] examine edge (u,v)
if (w(u,v) + d[u] < d[v])
d[v] := w(u,v) + d[u] edge (u,v) relaxed
f[v] := d[v] + h(v)
if (color[v] = WHITE)
color[v] := GRAY
INSERT(Q, v) discover vertex v
else if (color[v] = BLACK)
color[v] := GRAY
INSERT(Q, v) reopen vertex v
end if
else
... edge (u,v) not relaxed
end for
color[u] := BLACK finish vertex u
end while
Examples
--------
We will use an irregular two-dimensional lattice as an example, where the
heuristic function will be based on the euclidean distance to the target.
The heuristic function will be defined as:
.. testcode::
def h(v, target, pos):
return sqrt(sum((pos[v].a - pos[target].a) ** 2))
where ``pos`` is the vertex position in the plane.
We must define what should be done during the search by subclassing
:class:`~graph_tool.search.AStarVisitor`, and specializing the appropriate
methods. In the following we will keep track of the discovered vertices, and
which edges were examined, as well as the predecessor tree. We will also
abort the search when a given target vertex is found, by raising the
:class:`~graph_tool.search.StopSearch` exception.
.. testcode::
class VisitorExample(gt.AStarVisitor):
def __init__(self, touched_v, touched_e, target):
self.touched_v = touched_v
self.touched_e = touched_e
self.target = target
def discover_vertex(self, u):
self.touched_v[u] = True
def examine_edge(self, e):
self.touched_e[e] = True
def edge_relaxed(self, e):
if e.target() == self.target:
raise gt.StopSearch()
With the above class defined, we can perform the :math:`A^*` search as
follows.
.. testsetup::
from numpy.random import seed, random
import matplotlib.cm
seed(42)
>>> points = random((500, 2)) * 4
>>> points[0] = [-0.01, 0.01]
>>> points[1] = [4.01, 4.01]
>>> g, pos = gt.triangulation(points, type="delaunay")
>>> weight = g.new_edge_property("double") # Edge weights corresponding to
... # Euclidean distances
>>> for e in g.edges():
... weight[e] = sqrt(sum((pos[e.source()].a -
... pos[e.target()].a) ** 2))
>>> touch_v = g.new_vertex_property("bool")
>>> touch_e = g.new_edge_property("bool")
>>> target = g.vertex(1)
>>> dist, pred = gt.astar_search(g, g.vertex(0), weight,
... VisitorExample(touch_v, touch_e, target),
... heuristic=lambda v: h(v, target, pos))
We can now observe the best path found, and how many vertices and edges were
visited in the process.
>>> ecolor = g.new_edge_property("string")
>>> ewidth = g.new_edge_property("double")
>>> ewidth.a = 1
>>> for e in g.edges():
... ecolor[e] = "blue" if touch_e[e] else "black"
>>> v = target
>>> while v != g.vertex(0):
... p = g.vertex(pred[v])
... for e in v.out_edges():
... if e.target() == p:
... ecolor[e] = "#a40000"
... ewidth[e] = 3
... v = p
>>> gt.graph_draw(g, pos=pos, output_size=(300, 300), vertex_fill_color=touch_v,
... vcmap=matplotlib.cm.binary, edge_color=ecolor,
... edge_pen_width=ewidth, output="astar-delaunay.pdf")
<...>
.. testcode::
:hide:
gt.graph_draw(g, pos=pos, output_size=(300, 300), vertex_fill_color=touch_v,
vcmap=matplotlib.cm.binary, edge_color=ecolor,
edge_pen_width=ewidth, output="astar-delaunay.png")
.. figure:: astar-delaunay.*
:align: center
The shortest path is shown in red. The visited edges are shown in blue,
and the visited vertices in black.
The :math:`A^*` algorithm is very useful for searching *implicit* graphs,
i.e. graphs which are not entirely stored in memory and are generated
"on-the-fly" during the search. In the following example we will carry out a
search in a hamming hypercube of 10 bits witch has random weights on its
edges in the range :math:`[0,1]`. The vertices of the hypercube will be
created during the search.
The heuristic function will use the Hamming distance between vertices:
.. testcode::
def h(v, target, state):
return sum(abs(state[v].a - target)) / 2
In the following visitor we will keep growing the graph on-the-fly, and
abort the search when a given target vertex is found, by raising the
:class:`~graph_tool.search.StopSearch` exception.
.. testcode::
from numpy.random import random
class HammingVisitor(gt.AStarVisitor):
def __init__(self, g, target, state, weight, dist, cost):
self.g = g
self.state = state
self.target = target
self.weight = weight
self.dist = dist
self.cost = cost
self.visited = {}
def examine_vertex(self, u):
for i in range(len(self.state[u])):
nstate = list(self.state[u])
nstate[i] ^= 1
if tuple(nstate) in self.visited:
v = self.visited[tuple(nstate)]
else:
v = self.g.add_vertex()
self.visited[tuple(nstate)] = v
self.state[v] = nstate
self.dist[v] = self.cost[v] = float('inf')
for e in u.out_edges():
if e.target() == v:
break
else:
e = self.g.add_edge(u, v)
self.weight[e] = random()
self.visited[tuple(self.state[u])] = u
def edge_relaxed(self, e):
if self.state[e.target()] == self.target:
self.visited[tuple(self.target)] = e.target()
raise gt.StopSearch()
With the above class defined, we can perform the :math:`A^*` search as
follows.
.. testsetup::
from numpy.random import seed, random
seed(42)
gt.seed_rng(42)
>>> g = gt.Graph(directed=False)
>>> state = g.new_vertex_property("vector")
>>> v = g.add_vertex()
>>> state[v] = [0] * 10
>>> target = [1] * 10
>>> weight = g.new_edge_property("double")
>>> dist = g.new_vertex_property("double")
>>> cost = g.new_vertex_property("double")
>>> visitor = HammingVisitor(g, target, state, weight, dist, cost)
>>> dist, pred = gt.astar_search(g, g.vertex(0), weight, visitor, dist_map=dist,
... cost_map=cost, heuristic=lambda v: h(v, array(target), state),
... implicit=True)
We can now observe the best path found, and how many vertices and edges were
visited in the process.
>>> ecolor = g.new_edge_property("string")
>>> vcolor = g.new_vertex_property("string")
>>> ewidth = g.new_edge_property("double")
>>> ewidth.a = 1
>>> for e in g.edges():
... ecolor[e] = "black"
>>> for v in g.vertices():
... vcolor[v] = "white"
>>> v = visitor.visited[tuple(target)]
>>> while v != g.vertex(0):
... vcolor[v] = "black"
... p = g.vertex(pred[v])
... for e in v.out_edges():
... if e.target() == p:
... ecolor[e] = "#a40000"
... ewidth[e] = 3
... v = p
>>> vcolor[v] = "black"
>>> pos = gt.graph_draw(g, output_size=(300, 300), vertex_fill_color=vcolor, edge_color=ecolor,
... edge_pen_width=ewidth, output="astar-implicit.pdf")
.. testcode::
:hide:
gt.graph_draw(g, pos=pos, output_size=(300, 300), vertex_fill_color=vcolor,
edge_color=ecolor, edge_pen_width=ewidth,
output="astar-implicit.png")
.. figure:: astar-implicit.*
:align: center
The shortest path is shown in red, and the vertices which belong to it
are in black. Note that the number of vertices visited is much smaller
than the total number :math:`2^{10} = 1024`.
References
----------
.. [astar] Hart, P. E.; Nilsson, N. J.; Raphael, B. "A Formal Basis for the
Heuristic Determination of Minimum Cost Paths". IEEE Transactions on
Systems Science and Cybernetics SSC4 4 (2): 100-107, 1968.
:doi:`10.1109/TSSC.1968.300136`
.. [astar-bgl] http://www.boost.org/doc/libs/release/libs/graph/doc/astar_search.html
.. [astar-wikipedia] http://en.wikipedia.org/wiki/A*_search_algorithm
"""
visitor = VisitorWrapper(g, visitor,
["initialize_vertex", "examine_vertex", "finish_vertex"],
["initialize_vertex"])
if dist_map is None:
dist_map = g.new_vertex_property(weight.value_type())
if pred_map is None:
pred_map = g.new_vertex_property("int")
if pred_map.value_type() != "int32_t":
raise ValueError("pred_map must be of value type 'int32_t', not '%s'." % \
pred_map.value_type())
dist_type = dist_map.python_value_type()
if dist_type is not object:
h = lambda v: dist_type(heuristic(v))
else:
h = heuristic
try:
if dist_map.value_type() != "python::object":
zero = _python_type(dist_map.value_type())(zero)
except OverflowError:
zero = (weight.a.max() + 1) * g.num_vertices()
zero = _python_type(dist_map.value_type())(zero)
try:
if dist_map.value_type() != "python::object":
infinity = _python_type(dist_map.value_type())(infinity)
except OverflowError:
infinity = (weight.a.max() + 1) * g.num_vertices()
infinity = _python_type(dist_map.value_type())(infinity)
try:
if not implicit:
g._Graph__perms.update({"del_vertex": False, "del_edge": False,
"add_edge": False})
libgraph_tool_search.astar_search(g._Graph__graph,
weakref.ref(g),
int(source), _prop("v", g, dist_map),
_prop("v", g, pred_map),
_prop("e", g, weight), visitor,
compare, combine, zero, infinity,
h)
else:
if cost_map is None:
cost_map = g.new_vertex_property(dist_map.value_type())
elif cost_map.value_type() != dist_map.value_type():
raise ValueError("The cost_map value type must be the same as" +
" dist_map.")
g._Graph__perms.update({"del_vertex": False})
libgraph_tool_search.astar_search_implicit\
(g._Graph__graph, weakref.ref(g), int(source),
_prop("v", g, dist_map), _prop("v", g, pred_map),
_prop("v", g, cost_map), _prop("e", g, weight), visitor,
compare, combine, zero, infinity, h)
except StopSearch:
g._Graph__perms.update({"del_vertex": True, "del_edge": True,
"add_edge": True})
finally:
g._Graph__perms.update({"del_vertex": True, "del_edge": True,
"add_edge": True})
return dist_map, pred_map
class StopSearch(Exception):
"""If this exception is raised from inside any search visitor object, the search is aborted."""
pass