"""Floyd-Warshall algorithm for shortest paths."""
import networkx as nx
__all__ = [
"floyd_warshall",
"floyd_warshall_predecessor_and_distance",
"reconstruct_path",
"floyd_warshall_numpy",
"floyd_warshall_tree",
]
[docs]
@nx._dispatchable(edge_attrs="weight")
def floyd_warshall_numpy(G, nodelist=None, weight="weight"):
"""Find all-pairs shortest path lengths using Floyd's algorithm.
This algorithm for finding shortest paths takes advantage of
matrix representations of a graph and works well for dense
graphs where all-pairs shortest path lengths are desired.
The results are returned as a NumPy array, distance[i, j],
where i and j are the indexes of two nodes in nodelist.
The entry distance[i, j] is the distance along a shortest
path from i to j. If no path exists the distance is Inf.
Parameters
----------
G : NetworkX graph
nodelist : list, optional (default=G.nodes)
The rows and columns are ordered by the nodes in nodelist.
If nodelist is None then the ordering is produced by G.nodes.
Nodelist should include all nodes in G.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
Returns
-------
distance : 2D numpy.ndarray
A numpy array of shortest path distances between nodes.
If there is no path between two nodes the value is Inf.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from(
... [(0, 1, 5), (1, 2, 2), (2, 3, -3), (1, 3, 10), (3, 2, 8)]
... )
>>> nx.floyd_warshall_numpy(G)
array([[ 0., 5., 7., 4.],
[inf, 0., 2., -1.],
[inf, inf, 0., -3.],
[inf, inf, 8., 0.]])
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 $O(n^2)$.
Raises
------
NetworkXError
If nodelist is not a list of the nodes in G.
"""
import numpy as np
if nodelist is not None:
if not (len(nodelist) == len(G) == len(set(nodelist))):
raise nx.NetworkXError(
"nodelist must contain every node in G with no repeats."
"If you wanted a subgraph of G use G.subgraph(nodelist)"
)
# 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_array(
G, nodelist, multigraph_weight=min, weight=weight, nonedge=np.inf
)
n, m = A.shape
np.fill_diagonal(A, 0) # diagonal elements should be zero
for i in range(n):
# The second term has the same shape as A due to broadcasting
A = np.minimum(A, A[i, :][np.newaxis, :] + A[:, i][:, np.newaxis])
return A
[docs]
@nx._dispatchable(edge_attrs="weight")
def floyd_warshall_tree(G, weight="weight"):
r"""Find all-pairs shortest path lengths using a Tree-based
modification of Floyd's algorithm.
This variant implements the Tree algorithm of Brodnik, Grgurovic and Pozar.
It differs from the classical Floyd Warshall algorithm by using a shortest
path tree rooted at each intermediate vertex w. For every w, the algorithm
builds a tree ``OUT_w`` of shortest paths for the current iteration and
scans the tree in depth first order. If an update at a vertex v cannot
improve any distance, the algorithm skips the entire subtree below v,
since none of its nodes can produce a shorter path, as proved in [1]_.
Parameters
----------
G : NetworkX graph
weight : string, optional (default= 'weight')
Edge data key corresponding to the edge weight.
Returns
-------
predecessor, distance : dict and dict-of-dict
Predecessor is a dict keyed by node to the predecessor node in
the shortest path. The distance output is a dict keyed by source
node to a dict keyed by target node to the distance value of the
shortest path between the source and target.
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, distances = nx.floyd_warshall_tree(G)
>>> 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 worst case running time $O(|V|^3)$ with running space of $O(|V|^2)$.
For complete directed graphs with independent edge weights drawn from the
uniform distribution on [0, 1], the expected running time is
$O(|V|^2 (\log |V|)^2)$, as shown in [1]_.
See Also
--------
floyd_warshall_predecessor_and_distance
floyd_warshall
floyd_warshall_numpy
References
----------
.. [1] Brodnik, Andrej, Marko Grgurovic, and Rok Pozar.
"Modifications of the Floyd-Warshall algorithm with
nearly quadratic expected-time."
Ars Math. Contemp. 22, no. 1 (2022).
https://doi.org/10.26493/1855-3974.2467.497
"""
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
inf = float("inf")
dist = defaultdict(lambda: defaultdict(lambda: 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
# dont check for those w, `from` which `no` path exists
for w, pred_w in pred.items():
# out_w will store the adjacency list of the OUT_W tree of the paper,
# it is a tree, parent --> list of children
out_w = {parent: [] for parent in G}
for v, parent in pred_w.items(): # w to v path exist
if v == w:
continue
out_w[parent].append(v)
# dfs order dict and skip dict in practical improvements in the paper
stack = [w]
dfs_dict = {} # {node: next node in dfs order}
skip_dict = {} # {node: next node after subtree is skipped}
node = None
while stack:
next_node = stack.pop()
dfs_dict[node] = next_node
if stack:
skip_dict[next_node] = stack[-1]
stack.extend(out_w[next_node])
node = next_node
dist_w = dist[w] # for speed
# main inner loop starts here
for u in G:
if u == w: # small optimization
continue
dist_u = dist[u]
dist_uw = dist_u[w]
if dist_uw == inf: # small optimization
continue
# note: we skip v=w as relaxation would always fail
v = dfs_dict[w]
while v is not None:
dist_uwv = dist_uw + dist_w[v]
if dist_u[v] > dist_uwv:
dist_u[v] = dist_uwv
# update/new entries to be created in pred[u]
pred[u][v] = pred_w[v] # v must be in pred_w as checked above
v = dfs_dict.get(v, None)
else:
v = skip_dict.get(v, None)
return dict(pred), dict(dist)
[docs]
@nx._dispatchable(edge_attrs="weight")
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 $O(n^2)$.
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:
dist_w = dist[w] # save recomputation
for u in G:
dist_u = dist[u] # save recomputation
for v in G:
d = dist_u[w] + dist_w[v]
if dist_u[v] > d:
dist_u[v] = d
pred[u][v] = pred[w][v]
return dict(pred), dict(dist)
[docs]
@nx._dispatchable(graphs=None)
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
See Also
--------
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]
@nx._dispatchable(edge_attrs="weight")
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.
Examples
--------
>>> from pprint import pprint
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from(
... [(0, 1, 5), (1, 2, 2), (2, 3, -3), (1, 3, 10), (3, 2, 8)]
... )
>>> fw = nx.floyd_warshall(G, weight="weight")
>>> results = {a: dict(b) for a, b in fw.items()}
>>> pprint(results)
{0: {0: 0, 1: 5, 2: 7, 3: 4},
1: {0: inf, 1: 0, 2: 2, 3: -1},
2: {0: inf, 1: inf, 2: 0, 3: -3},
3: {0: inf, 1: inf, 2: 8, 3: 0}}
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]
