Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.shortest_paths.dense

# -*- coding: utf-8 -*-
"""Floyd-Warshall algorithm for shortest paths.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#
# Authors: Aric Hagberg <aric.hagberg@gmail.com>
#          Miguel Sozinho Ramalho <m.ramalho@fe.up.pt>
import networkx as nx

__all__ = ['floyd_warshall',
'floyd_warshall_predecessor_and_distance',
'reconstruct_path',
'floyd_warshall_numpy']

[docs]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/ ")

# To handle cases when an edge has weight=0, we must make sure that
# nonedges are not given the value 0 as well.
A = nx.to_numpy_matrix(G, nodelist=nodelist, multigraph_weight=min,
weight=weight, nonedge=np.inf)
n, m = A.shape
I = np.identity(n)
A[I == 1] = 0  # diagonal elements should be zero
for i in range(n):
A = np.minimum(A, A[i, :] + A[:, i])
return A

[docs]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.

Examples
--------
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from([('s', 'u', 10), ('s', 'x', 5),
...     ('u', 'v', 1), ('u', 'x', 2), ('v', 'y', 1), ('x', 'u', 3),
...     ('x', 'v', 5), ('x', 'y', 2), ('y', 's', 7), ('y', 'v', 6)])
>>> predecessors, _ = nx.floyd_warshall_predecessor_and_distance(G)
>>> print(nx.reconstruct_path('s', 'v', predecessors))
['s', 'x', 'u', 'v']

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 $O(n^2)$.

--------
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)

[docs]def reconstruct_path(source, target, predecessors):
"""Reconstruct a path from source to target using the predecessors
dict as returned by floyd_warshall_predecessor_and_distance

Parameters
----------
source : node
Starting node for path

target : node
Ending node for path

predecessors: dictionary
Dictionary, keyed by source and target, of predecessors in the
shortest path, as returned by floyd_warshall_predecessor_and_distance

Returns
-------
path : list
A list of nodes containing the shortest path from source to target

If source and target are the same, an empty list is returned

Notes
------
This function is meant to give more applicability to the
floyd_warshall_predecessor_and_distance function

--------
floyd_warshall_predecessor_and_distance
"""
if source == target:
return []
prev = predecessors[source]
curr = prev[target]
path = [target, curr]
while curr != source:
curr = prev[curr]
path.append(curr)
return list(reversed(path))

[docs]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 $O(n^2)$.

--------
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")