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# -*- coding: utf-8 -*-
"""Floyd-Warshall algorithm for shortest paths.
"""
# Copyright (C) 2004-2012 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
import networkx as nx
__author__ = """Aric Hagberg <aric.hagberg@gmail.com>"""
__all__ = ['floyd_warshall',
'floyd_warshall_predecessor_and_distance',
'floyd_warshall_numpy']
def floyd_warshall_numpy(G, nodelist=None, weight='weight'):
"""Find all-pairs shortest path lengths using Floyd's algorithm.
Parameters
----------
G : NetworkX graph
nodelist : list, optional
The rows and columns are ordered by the nodes in nodelist.
If nodelist is None then the ordering is produced by G.nodes().
weight: string, optional (default= 'weight')
Edge data key corresponding to the edge weight.
Returns
-------
distance : NumPy matrix
A matrix of shortest path distances between nodes.
If there is no path between to nodes the corresponding matrix entry
will be Inf.
Notes
------
Floyd's algorithm is appropriate for finding shortest paths in
dense graphs or graphs with negative weights when Dijkstra's
algorithm fails. This algorithm can still fail if there are
negative cycles. It has running time O(n^3) with running space of O(n^2).
"""
try:
import numpy as np
except ImportError:
raise ImportError(\
"to_numpy_matrix() requires numpy: http://scipy.org/ ")
A = nx.to_numpy_matrix(G, nodelist=nodelist, multigraph_weight=min,
weight=weight)
n,m = A.shape
I = np.identity(n)
A[A==0] = np.inf # set zero entries to inf
A[I==1] = 0 # except diagonal which should be zero
for i in range(n):
A = np.minimum(A, A[i,:] + A[:,i])
return A
def floyd_warshall_predecessor_and_distance(G, weight='weight'):
"""Find all-pairs shortest path lengths using Floyd's algorithm.
Parameters
----------
G : NetworkX graph
weight: string, optional (default= 'weight')
Edge data key corresponding to the edge weight.
Returns
-------
predecessor,distance : dictionaries
Dictionaries, keyed by source and target, of predecessors and distances
in the shortest path.
Notes
------
Floyd's algorithm is appropriate for finding shortest paths
in dense graphs or graphs with negative weights when Dijkstra's algorithm
fails. This algorithm can still fail if there are negative cycles.
It has running time O(n^3) with running space of O(n^2).
See Also
--------
floyd_warshall
floyd_warshall_numpy
all_pairs_shortest_path
all_pairs_shortest_path_length
"""
from collections import defaultdict
# dictionary-of-dictionaries representation for dist and pred
# use some defaultdict magick here
# for dist the default is the floating point inf value
dist = defaultdict(lambda : defaultdict(lambda: float('inf')))
for u in G:
dist[u][u] = 0
pred = defaultdict(dict)
# initialize path distance dictionary to be the adjacency matrix
# also set the distance to self to 0 (zero diagonal)
undirected = not G.is_directed()
for u,v,d in G.edges(data=True):
e_weight = d.get(weight, 1.0)
dist[u][v] = min(e_weight, dist[u][v])
pred[u][v] = u
if undirected:
dist[v][u] = min(e_weight, dist[v][u])
pred[v][u] = v
for w in G:
for u in G:
for v in G:
if dist[u][v] > dist[u][w] + dist[w][v]:
dist[u][v] = dist[u][w] + dist[w][v]
pred[u][v] = pred[w][v]
return dict(pred),dict(dist)
def floyd_warshall(G, weight='weight'):
"""Find all-pairs shortest path lengths using Floyd's algorithm.
Parameters
----------
G : NetworkX graph
weight: string, optional (default= 'weight')
Edge data key corresponding to the edge weight.
Returns
-------
distance : dict
A dictionary, keyed by source and target, of shortest paths distances
between nodes.
Notes
------
Floyd's algorithm is appropriate for finding shortest paths
in dense graphs or graphs with negative weights when Dijkstra's algorithm
fails. This algorithm can still fail if there are negative cycles.
It has running time O(n^3) with running space of O(n^2).
See Also
--------
floyd_warshall_predecessor_and_distance
floyd_warshall_numpy
all_pairs_shortest_path
all_pairs_shortest_path_length
"""
# could make this its own function to reduce memory costs
return floyd_warshall_predecessor_and_distance(G, weight=weight)[1]
# fixture for nose tests
def setup_module(module):
from nose import SkipTest
try:
import numpy
except:
raise SkipTest("NumPy not available")
|
