"""
Shortest path algorithms for unweighted graphs.
"""
import networkx as nx
__all__ = [
"bidirectional_shortest_path",
"single_source_shortest_path",
"single_source_shortest_path_length",
"single_target_shortest_path",
"single_target_shortest_path_length",
"all_pairs_shortest_path",
"all_pairs_shortest_path_length",
"predecessor",
]
[docs]def single_source_shortest_path_length(G, source, cutoff=None):
"""Compute the shortest path lengths from source to all reachable nodes.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
lengths : dict
Dict keyed by node to shortest path length to source.
Examples
--------
>>> G = nx.path_graph(5)
>>> length = nx.single_source_shortest_path_length(G, 0)
>>> length[4]
4
>>> for node in length:
... print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 3
4: 4
See Also
--------
shortest_path_length
"""
if source not in G:
raise nx.NodeNotFound(f"Source {source} is not in G")
if cutoff is None:
cutoff = float("inf")
nextlevel = {source: 1}
return dict(_single_shortest_path_length(G.adj, nextlevel, cutoff))
def _single_shortest_path_length(adj, firstlevel, cutoff):
"""Yields (node, level) in a breadth first search
Shortest Path Length helper function
Parameters
----------
adj : dict
Adjacency dict or view
firstlevel : dict
starting nodes, e.g. {source: 1} or {target: 1}
cutoff : int or float
level at which we stop the process
"""
seen = {} # level (number of hops) when seen in BFS
level = 0 # the current level
nextlevel = set(firstlevel) # set of nodes to check at next level
n = len(adj)
while nextlevel and cutoff >= level:
thislevel = nextlevel # advance to next level
nextlevel = set() # and start a new set (fringe)
found = []
for v in thislevel:
if v not in seen:
seen[v] = level # set the level of vertex v
found.append(v)
yield (v, level)
if len(seen) == n:
return
for v in found:
nextlevel.update(adj[v])
level += 1
del seen
[docs]def single_target_shortest_path_length(G, target, cutoff=None):
"""Compute the shortest path lengths to target from all reachable nodes.
Parameters
----------
G : NetworkX graph
target : node
Target node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
lengths : iterator
(source, shortest path length) iterator
Examples
--------
>>> G = nx.path_graph(5, create_using=nx.DiGraph())
>>> length = dict(nx.single_target_shortest_path_length(G, 4))
>>> length[0]
4
>>> for node in range(5):
... print(f"{node}: {length[node]}")
0: 4
1: 3
2: 2
3: 1
4: 0
See Also
--------
single_source_shortest_path_length, shortest_path_length
"""
if target not in G:
raise nx.NodeNotFound(f"Target {target} is not in G")
if cutoff is None:
cutoff = float("inf")
# handle either directed or undirected
adj = G.pred if G.is_directed() else G.adj
nextlevel = {target: 1}
return _single_shortest_path_length(adj, nextlevel, cutoff)
[docs]def all_pairs_shortest_path_length(G, cutoff=None):
"""Computes the shortest path lengths between all nodes in `G`.
Parameters
----------
G : NetworkX graph
cutoff : integer, optional
Depth at which to stop the search. Only paths of length at most
`cutoff` are returned.
Returns
-------
lengths : iterator
(source, dictionary) iterator with dictionary keyed by target and
shortest path length as the key value.
Notes
-----
The iterator returned only has reachable node pairs.
Examples
--------
>>> G = nx.path_graph(5)
>>> length = dict(nx.all_pairs_shortest_path_length(G))
>>> for node in [0, 1, 2, 3, 4]:
... print(f"1 - {node}: {length[1][node]}")
1 - 0: 1
1 - 1: 0
1 - 2: 1
1 - 3: 2
1 - 4: 3
>>> length[3][2]
1
>>> length[2][2]
0
"""
length = single_source_shortest_path_length
# TODO This can be trivially parallelized.
for n in G:
yield (n, length(G, n, cutoff=cutoff))
[docs]def bidirectional_shortest_path(G, source, target):
"""Returns a list of nodes in a shortest path between source and target.
Parameters
----------
G : NetworkX graph
source : node label
starting node for path
target : node label
ending node for path
Returns
-------
path: list
List of nodes in a path from source to target.
Raises
------
NetworkXNoPath
If no path exists between source and target.
See Also
--------
shortest_path
Notes
-----
This algorithm is used by shortest_path(G, source, target).
"""
if source not in G or target not in G:
msg = f"Either source {source} or target {target} is not in G"
raise nx.NodeNotFound(msg)
# call helper to do the real work
results = _bidirectional_pred_succ(G, source, target)
pred, succ, w = results
# build path from pred+w+succ
path = []
# from source to w
while w is not None:
path.append(w)
w = pred[w]
path.reverse()
# from w to target
w = succ[path[-1]]
while w is not None:
path.append(w)
w = succ[w]
return path
def _bidirectional_pred_succ(G, source, target):
"""Bidirectional shortest path helper.
Returns (pred, succ, w) where
pred is a dictionary of predecessors from w to the source, and
succ is a dictionary of successors from w to the target.
"""
# does BFS from both source and target and meets in the middle
if target == source:
return ({target: None}, {source: None}, source)
# handle either directed or undirected
if G.is_directed():
Gpred = G.pred
Gsucc = G.succ
else:
Gpred = G.adj
Gsucc = G.adj
# predecesssor and successors in search
pred = {source: None}
succ = {target: None}
# initialize fringes, start with forward
forward_fringe = [source]
reverse_fringe = [target]
while forward_fringe and reverse_fringe:
if len(forward_fringe) <= len(reverse_fringe):
this_level = forward_fringe
forward_fringe = []
for v in this_level:
for w in Gsucc[v]:
if w not in pred:
forward_fringe.append(w)
pred[w] = v
if w in succ: # path found
return pred, succ, w
else:
this_level = reverse_fringe
reverse_fringe = []
for v in this_level:
for w in Gpred[v]:
if w not in succ:
succ[w] = v
reverse_fringe.append(w)
if w in pred: # found path
return pred, succ, w
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
[docs]def single_source_shortest_path(G, source, cutoff=None):
"""Compute shortest path between source
and all other nodes reachable from source.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
lengths : dictionary
Dictionary, keyed by target, of shortest paths.
Examples
--------
>>> G = nx.path_graph(5)
>>> path = nx.single_source_shortest_path(G, 0)
>>> path[4]
[0, 1, 2, 3, 4]
Notes
-----
The shortest path is not necessarily unique. So there can be multiple
paths between the source and each target node, all of which have the
same 'shortest' length. For each target node, this function returns
only one of those paths.
See Also
--------
shortest_path
"""
if source not in G:
raise nx.NodeNotFound(f"Source {source} not in G")
def join(p1, p2):
return p1 + p2
if cutoff is None:
cutoff = float("inf")
nextlevel = {source: 1} # list of nodes to check at next level
paths = {source: [source]} # paths dictionary (paths to key from source)
return dict(_single_shortest_path(G.adj, nextlevel, paths, cutoff, join))
def _single_shortest_path(adj, firstlevel, paths, cutoff, join):
"""Returns shortest paths
Shortest Path helper function
Parameters
----------
adj : dict
Adjacency dict or view
firstlevel : dict
starting nodes, e.g. {source: 1} or {target: 1}
paths : dict
paths for starting nodes, e.g. {source: [source]}
cutoff : int or float
level at which we stop the process
join : function
function to construct a path from two partial paths. Requires two
list inputs `p1` and `p2`, and returns a list. Usually returns
`p1 + p2` (forward from source) or `p2 + p1` (backward from target)
"""
level = 0 # the current level
nextlevel = firstlevel
while nextlevel and cutoff > level:
thislevel = nextlevel
nextlevel = {}
for v in thislevel:
for w in adj[v]:
if w not in paths:
paths[w] = join(paths[v], [w])
nextlevel[w] = 1
level += 1
return paths
[docs]def single_target_shortest_path(G, target, cutoff=None):
"""Compute shortest path to target from all nodes that reach target.
Parameters
----------
G : NetworkX graph
target : node label
Target node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
lengths : dictionary
Dictionary, keyed by target, of shortest paths.
Examples
--------
>>> G = nx.path_graph(5, create_using=nx.DiGraph())
>>> path = nx.single_target_shortest_path(G, 4)
>>> path[0]
[0, 1, 2, 3, 4]
Notes
-----
The shortest path is not necessarily unique. So there can be multiple
paths between the source and each target node, all of which have the
same 'shortest' length. For each target node, this function returns
only one of those paths.
See Also
--------
shortest_path, single_source_shortest_path
"""
if target not in G:
raise nx.NodeNotFound(f"Target {target} not in G")
def join(p1, p2):
return p2 + p1
# handle undirected graphs
adj = G.pred if G.is_directed() else G.adj
if cutoff is None:
cutoff = float("inf")
nextlevel = {target: 1} # list of nodes to check at next level
paths = {target: [target]} # paths dictionary (paths to key from source)
return dict(_single_shortest_path(adj, nextlevel, paths, cutoff, join))
[docs]def all_pairs_shortest_path(G, cutoff=None):
"""Compute shortest paths between all nodes.
Parameters
----------
G : NetworkX graph
cutoff : integer, optional
Depth at which to stop the search. Only paths of length at most
`cutoff` are returned.
Returns
-------
lengths : dictionary
Dictionary, keyed by source and target, of shortest paths.
Examples
--------
>>> G = nx.path_graph(5)
>>> path = dict(nx.all_pairs_shortest_path(G))
>>> print(path[0][4])
[0, 1, 2, 3, 4]
See Also
--------
floyd_warshall()
"""
# TODO This can be trivially parallelized.
for n in G:
yield (n, single_source_shortest_path(G, n, cutoff=cutoff))
[docs]def predecessor(G, source, target=None, cutoff=None, return_seen=None):
"""Returns dict of predecessors for the path from source to all nodes in G
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
target : node label, optional
Ending node for path. If provided only predecessors between
source and target are returned
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
pred : dictionary
Dictionary, keyed by node, of predecessors in the shortest path.
Examples
--------
>>> G = nx.path_graph(4)
>>> list(G)
[0, 1, 2, 3]
>>> nx.predecessor(G, 0)
{0: [], 1: [0], 2: [1], 3: [2]}
"""
if source not in G:
raise nx.NodeNotFound(f"Source {source} not in G")
level = 0 # the current level
nextlevel = [source] # list of nodes to check at next level
seen = {source: level} # level (number of hops) when seen in BFS
pred = {source: []} # predecessor dictionary
while nextlevel:
level = level + 1
thislevel = nextlevel
nextlevel = []
for v in thislevel:
for w in G[v]:
if w not in seen:
pred[w] = [v]
seen[w] = level
nextlevel.append(w)
elif seen[w] == level: # add v to predecessor list if it
pred[w].append(v) # is at the correct level
if cutoff and cutoff <= level:
break
if target is not None:
if return_seen:
if target not in pred:
return ([], -1) # No predecessor
return (pred[target], seen[target])
else:
if target not in pred:
return [] # No predecessor
return pred[target]
else:
if return_seen:
return (pred, seen)
else:
return pred