""" Shortest path algorithms for weighted graphs. """ from collections import deque from heapq import heappop, heappush from itertools import count import networkx as nx from networkx.algorithms.shortest_paths.generic import _build_paths_from_predecessors __all__ = [ "dijkstra_path", "dijkstra_path_length", "bidirectional_dijkstra", "single_source_dijkstra", "single_source_dijkstra_path", "single_source_dijkstra_path_length", "multi_source_dijkstra", "multi_source_dijkstra_path", "multi_source_dijkstra_path_length", "all_pairs_dijkstra", "all_pairs_dijkstra_path", "all_pairs_dijkstra_path_length", "dijkstra_predecessor_and_distance", "bellman_ford_path", "bellman_ford_path_length", "single_source_bellman_ford", "single_source_bellman_ford_path", "single_source_bellman_ford_path_length", "all_pairs_bellman_ford_path", "all_pairs_bellman_ford_path_length", "bellman_ford_predecessor_and_distance", "negative_edge_cycle", "find_negative_cycle", "goldberg_radzik", "johnson", ] def _weight_function(G, weight): """Returns a function that returns the weight of an edge. The returned function is specifically suitable for input to functions :func:`_dijkstra` and :func:`_bellman_ford_relaxation`. Parameters ---------- G : NetworkX graph. weight : string or function If it is callable, `weight` itself is returned. If it is a string, it is assumed to be the name of the edge attribute that represents the weight of an edge. In that case, a function is returned that gets the edge weight according to the specified edge attribute. Returns ------- function This function returns a callable that accepts exactly three inputs: a node, an node adjacent to the first one, and the edge attribute dictionary for the eedge joining those nodes. That function returns a number representing the weight of an edge. If `G` is a multigraph, and `weight` is not callable, the minimum edge weight over all parallel edges is returned. If any edge does not have an attribute with key `weight`, it is assumed to have weight one. """ if callable(weight): return weight # If the weight keyword argument is not callable, we assume it is a # string representing the edge attribute containing the weight of # the edge. if G.is_multigraph(): return lambda u, v, d: min(attr.get(weight, 1) for attr in d.values()) return lambda u, v, data: data.get(weight, 1) @nx._dispatchable(edge_attrs="weight") def dijkstra_path(G, source, target, weight="weight"): """Returns the shortest weighted path from source to target in G. Uses Dijkstra's Method to compute the shortest weighted path between two nodes in a graph. Parameters ---------- G : NetworkX graph source : node Starting node target : node Ending node weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- path : list List of nodes in a shortest path. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXNoPath If no path exists between source and target. Examples -------- >>> G = nx.path_graph(5) >>> print(nx.dijkstra_path(G, 0, 4)) [0, 1, 2, 3, 4] Find edges of shortest path in Multigraph >>> G = nx.MultiDiGraph() >>> G.add_weighted_edges_from([(1, 2, 0.75), (1, 2, 0.5), (2, 3, 0.5), (1, 3, 1.5)]) >>> nodes = nx.dijkstra_path(G, 1, 3) >>> edges = nx.utils.pairwise(nodes) >>> list( ... (u, v, min(G[u][v], key=lambda k: G[u][v][k].get("weight", 1))) ... for u, v in edges ... ) [(1, 2, 1), (2, 3, 0)] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. The weight function can be used to include node weights. >>> def func(u, v, d): ... node_u_wt = G.nodes[u].get("node_weight", 1) ... node_v_wt = G.nodes[v].get("node_weight", 1) ... edge_wt = d.get("weight", 1) ... return node_u_wt / 2 + node_v_wt / 2 + edge_wt In this example we take the average of start and end node weights of an edge and add it to the weight of the edge. The function :func:`single_source_dijkstra` computes both path and length-of-path if you need both, use that. See Also -------- bidirectional_dijkstra bellman_ford_path single_source_dijkstra """ (length, path) = single_source_dijkstra(G, source, target=target, weight=weight) return path @nx._dispatchable(edge_attrs="weight") def dijkstra_path_length(G, source, target, weight="weight"): """Returns the shortest weighted path length in G from source to target. Uses Dijkstra's Method to compute the shortest weighted path length between two nodes in a graph. Parameters ---------- G : NetworkX graph source : node label starting node for path target : node label ending node for path weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- length : number Shortest path length. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXNoPath If no path exists between source and target. Examples -------- >>> G = nx.path_graph(5) >>> nx.dijkstra_path_length(G, 0, 4) 4 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. The function :func:`single_source_dijkstra` computes both path and length-of-path if you need both, use that. See Also -------- bidirectional_dijkstra bellman_ford_path_length single_source_dijkstra """ if source not in G: raise nx.NodeNotFound(f"Node {source} not found in graph") if source == target: return 0 weight = _weight_function(G, weight) length = _dijkstra(G, source, weight, target=target) try: return length[target] except KeyError as err: raise nx.NetworkXNoPath(f"Node {target} not reachable from {source}") from err @nx._dispatchable(edge_attrs="weight") def single_source_dijkstra_path(G, source, cutoff=None, weight="weight"): """Find shortest weighted paths in G from a source node. Compute shortest path between source and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph source : node Starting node for path. cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- paths : dictionary Dictionary of shortest path lengths keyed by target. Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> path = nx.single_source_dijkstra_path(G, 0) >>> path[4] [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. See Also -------- single_source_dijkstra, single_source_bellman_ford """ return multi_source_dijkstra_path(G, {source}, cutoff=cutoff, weight=weight) @nx._dispatchable(edge_attrs="weight") def single_source_dijkstra_path_length(G, source, cutoff=None, weight="weight"): """Find shortest weighted path lengths in G from a source node. Compute the shortest path length between source and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph source : node label Starting node for path cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- length : dict Dict keyed by node to shortest path length from source. Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> length = nx.single_source_dijkstra_path_length(G, 0) >>> length[4] 4 >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 3 4: 4 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. See Also -------- single_source_dijkstra, single_source_bellman_ford_path_length """ return multi_source_dijkstra_path_length(G, {source}, cutoff=cutoff, weight=weight) @nx._dispatchable(edge_attrs="weight") def single_source_dijkstra(G, source, target=None, cutoff=None, weight="weight"): """Find shortest weighted paths and lengths from a source node. Compute the shortest path length between source and all other reachable nodes for a weighted graph. Uses Dijkstra's algorithm to compute shortest paths and lengths between a source and all other reachable nodes in a weighted graph. Parameters ---------- G : NetworkX graph source : node label Starting node for path target : node label, optional Ending node for path cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- distance, path : pair of dictionaries, or numeric and list. If target is None, paths and lengths to all nodes are computed. The return value is a tuple of two dictionaries keyed by target nodes. The first dictionary stores distance to each target node. The second stores the path to each target node. If target is not None, returns a tuple (distance, path), where distance is the distance from source to target and path is a list representing the path from source to target. Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> length, path = nx.single_source_dijkstra(G, 0) >>> length[4] 4 >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 3 4: 4 >>> path[4] [0, 1, 2, 3, 4] >>> length, path = nx.single_source_dijkstra(G, 0, 1) >>> length 1 >>> path [0, 1] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. Based on the Python cookbook recipe (119466) at https://code.activestate.com/recipes/119466/ This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows and roundoff errors can cause problems). See Also -------- single_source_dijkstra_path single_source_dijkstra_path_length single_source_bellman_ford """ return multi_source_dijkstra( G, {source}, cutoff=cutoff, target=target, weight=weight ) @nx._dispatchable(edge_attrs="weight") def multi_source_dijkstra_path(G, sources, cutoff=None, weight="weight"): """Find shortest weighted paths in G from a given set of source nodes. Compute shortest path between any of the source nodes and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph sources : non-empty set of nodes Starting nodes for paths. If this is just a set containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in the set, the computed paths may begin from any one of the start nodes. cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- paths : dictionary Dictionary of shortest paths keyed by target. Examples -------- >>> G = nx.path_graph(5) >>> path = nx.multi_source_dijkstra_path(G, {0, 4}) >>> path[1] [0, 1] >>> path[3] [4, 3] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. Raises ------ ValueError If `sources` is empty. NodeNotFound If any of `sources` is not in `G`. See Also -------- multi_source_dijkstra, multi_source_bellman_ford """ length, path = multi_source_dijkstra(G, sources, cutoff=cutoff, weight=weight) return path @nx._dispatchable(edge_attrs="weight") def multi_source_dijkstra_path_length(G, sources, cutoff=None, weight="weight"): """Find shortest weighted path lengths in G from a given set of source nodes. Compute the shortest path length between any of the source nodes and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph sources : non-empty set of nodes Starting nodes for paths. If this is just a set containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in the set, the computed paths may begin from any one of the start nodes. cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- length : dict Dict keyed by node to shortest path length to nearest source. Examples -------- >>> G = nx.path_graph(5) >>> length = nx.multi_source_dijkstra_path_length(G, {0, 4}) >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 1 4: 0 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. Raises ------ ValueError If `sources` is empty. NodeNotFound If any of `sources` is not in `G`. See Also -------- multi_source_dijkstra """ if not sources: raise ValueError("sources must not be empty") for s in sources: if s not in G: raise nx.NodeNotFound(f"Node {s} not found in graph") weight = _weight_function(G, weight) return _dijkstra_multisource(G, sources, weight, cutoff=cutoff) @nx._dispatchable(edge_attrs="weight") def multi_source_dijkstra(G, sources, target=None, cutoff=None, weight="weight"): """Find shortest weighted paths and lengths from a given set of source nodes. Uses Dijkstra's algorithm to compute the shortest paths and lengths between one of the source nodes and the given `target`, or all other reachable nodes if not specified, for a weighted graph. Parameters ---------- G : NetworkX graph sources : non-empty set of nodes Starting nodes for paths. If this is just a set containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in the set, the computed paths may begin from any one of the start nodes. target : node label, optional Ending node for path cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- distance, path : pair of dictionaries, or numeric and list If target is None, returns a tuple of two dictionaries keyed by node. The first dictionary stores distance from one of the source nodes. The second stores the path from one of the sources to that node. If target is not None, returns a tuple of (distance, path) where distance is the distance from source to target and path is a list representing the path from source to target. Examples -------- >>> G = nx.path_graph(5) >>> length, path = nx.multi_source_dijkstra(G, {0, 4}) >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 1 4: 0 >>> path[1] [0, 1] >>> path[3] [4, 3] >>> length, path = nx.multi_source_dijkstra(G, {0, 4}, 1) >>> length 1 >>> path [0, 1] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. Based on the Python cookbook recipe (119466) at https://code.activestate.com/recipes/119466/ This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows and roundoff errors can cause problems). Raises ------ ValueError If `sources` is empty. NodeNotFound If any of `sources` is not in `G`. See Also -------- multi_source_dijkstra_path multi_source_dijkstra_path_length """ if not sources: raise ValueError("sources must not be empty") for s in sources: if s not in G: raise nx.NodeNotFound(f"Node {s} not found in graph") if target in sources: return (0, [target]) weight = _weight_function(G, weight) paths = {source: [source] for source in sources} # dictionary of paths dist = _dijkstra_multisource( G, sources, weight, paths=paths, cutoff=cutoff, target=target ) if target is None: return (dist, paths) try: return (dist[target], paths[target]) except KeyError as err: raise nx.NetworkXNoPath(f"No path to {target}.") from err def _dijkstra(G, source, weight, pred=None, paths=None, cutoff=None, target=None): """Uses Dijkstra's algorithm to find shortest weighted paths from a single source. This is a convenience function for :func:`_dijkstra_multisource` with all the arguments the same, except the keyword argument `sources` set to ``[source]``. """ return _dijkstra_multisource( G, [source], weight, pred=pred, paths=paths, cutoff=cutoff, target=target ) def _dijkstra_multisource( G, sources, weight, pred=None, paths=None, cutoff=None, target=None ): """Uses Dijkstra's algorithm to find shortest weighted paths Parameters ---------- G : NetworkX graph sources : non-empty iterable of nodes Starting nodes for paths. If this is just an iterable containing a single node, then all paths computed by this function will start from that node. If there are two or more nodes in this iterable, the computed paths may begin from any one of the start nodes. weight: function Function with (u, v, data) input that returns that edge's weight or None to indicate a hidden edge pred: dict of lists, optional(default=None) dict to store a list of predecessors keyed by that node If None, predecessors are not stored. paths: dict, optional (default=None) dict to store the path list from source to each node, keyed by node. If None, paths are not stored. target : node label, optional Ending node for path. Search is halted when target is found. cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. Returns ------- distance : dictionary A mapping from node to shortest distance to that node from one of the source nodes. Raises ------ NodeNotFound If any of `sources` is not in `G`. Notes ----- The optional predecessor and path dictionaries can be accessed by the caller through the original pred and paths objects passed as arguments. No need to explicitly return pred or paths. """ G_succ = G._adj # For speed-up (and works for both directed and undirected graphs) push = heappush pop = heappop dist = {} # dictionary of final distances seen = {} # fringe is heapq with 3-tuples (distance,c,node) # use the count c to avoid comparing nodes (may not be able to) c = count() fringe = [] for source in sources: seen[source] = 0 push(fringe, (0, next(c), source)) while fringe: (d, _, v) = pop(fringe) if v in dist: continue # already searched this node. dist[v] = d if v == target: break for u, e in G_succ[v].items(): cost = weight(v, u, e) if cost is None: continue vu_dist = dist[v] + cost if cutoff is not None: if vu_dist > cutoff: continue if u in dist: u_dist = dist[u] if vu_dist < u_dist: raise ValueError("Contradictory paths found:", "negative weights?") elif pred is not None and vu_dist == u_dist: pred[u].append(v) elif u not in seen or vu_dist < seen[u]: seen[u] = vu_dist push(fringe, (vu_dist, next(c), u)) if paths is not None: paths[u] = paths[v] + [u] if pred is not None: pred[u] = [v] elif vu_dist == seen[u]: if pred is not None: pred[u].append(v) # The optional predecessor and path dictionaries can be accessed # by the caller via the pred and paths objects passed as arguments. return dist @nx._dispatchable(edge_attrs="weight") def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight="weight"): """Compute weighted shortest path length and predecessors. Uses Dijkstra's Method to obtain the shortest weighted paths and return dictionaries of predecessors for each node and distance for each node from the `source`. Parameters ---------- G : NetworkX graph source : node label Starting node for path cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- pred, distance : dictionaries Returns two dictionaries representing a list of predecessors of a node and the distance to each node. Raises ------ NodeNotFound If `source` is not in `G`. Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The list of predecessors contains more than one element only when there are more than one shortest paths to the key node. Examples -------- >>> G = nx.path_graph(5, create_using=nx.DiGraph()) >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0) >>> sorted(pred.items()) [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] >>> sorted(dist.items()) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0, 1) >>> sorted(pred.items()) [(0, []), (1, [0])] >>> sorted(dist.items()) [(0, 0), (1, 1)] """ if source not in G: raise nx.NodeNotFound(f"Node {source} is not found in the graph") weight = _weight_function(G, weight) pred = {source: []} # dictionary of predecessors return (pred, _dijkstra(G, source, weight, pred=pred, cutoff=cutoff)) @nx._dispatchable(edge_attrs="weight") def all_pairs_dijkstra(G, cutoff=None, weight="weight"): """Find shortest weighted paths and lengths between all nodes. Parameters ---------- G : NetworkX graph cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edge[u][v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Yields ------ (node, (distance, path)) : (node obj, (dict, dict)) Each source node has two associated dicts. The first holds distance keyed by target and the second holds paths keyed by target. (See single_source_dijkstra for the source/target node terminology.) If desired you can apply `dict()` to this function to create a dict keyed by source node to the two dicts. Examples -------- >>> G = nx.path_graph(5) >>> len_path = dict(nx.all_pairs_dijkstra(G)) >>> len_path[3][0][1] 2 >>> for node in [0, 1, 2, 3, 4]: ... print(f"3 - {node}: {len_path[3][0][node]}") 3 - 0: 3 3 - 1: 2 3 - 2: 1 3 - 3: 0 3 - 4: 1 >>> len_path[3][1][1] [3, 2, 1] >>> for n, (dist, path) in nx.all_pairs_dijkstra(G): ... print(path[1]) [0, 1] [1] [2, 1] [3, 2, 1] [4, 3, 2, 1] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The yielded dicts only have keys for reachable nodes. """ for n in G: dist, path = single_source_dijkstra(G, n, cutoff=cutoff, weight=weight) yield (n, (dist, path)) @nx._dispatchable(edge_attrs="weight") def all_pairs_dijkstra_path_length(G, cutoff=None, weight="weight"): """Compute shortest path lengths between all nodes in a weighted graph. Parameters ---------- G : NetworkX graph cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- distance : iterator (source, dictionary) iterator with dictionary keyed by target and shortest path length as the key value. Examples -------- >>> G = nx.path_graph(5) >>> length = dict(nx.all_pairs_dijkstra_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 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionary returned only has keys for reachable node pairs. """ length = single_source_dijkstra_path_length for n in G: yield (n, length(G, n, cutoff=cutoff, weight=weight)) @nx._dispatchable(edge_attrs="weight") def all_pairs_dijkstra_path(G, cutoff=None, weight="weight"): """Compute shortest paths between all nodes in a weighted graph. Parameters ---------- G : NetworkX graph cutoff : integer or float, optional Length (sum of edge weights) at which the search is stopped. If cutoff is provided, only return paths with summed weight <= cutoff. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- paths : iterator (source, dictionary) iterator with dictionary keyed by target and shortest path as the key value. Examples -------- >>> G = nx.path_graph(5) >>> path = dict(nx.all_pairs_dijkstra_path(G)) >>> path[0][4] [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- floyd_warshall, all_pairs_bellman_ford_path """ path = single_source_dijkstra_path # TODO This can be trivially parallelized. for n in G: yield (n, path(G, n, cutoff=cutoff, weight=weight)) @nx._dispatchable(edge_attrs="weight") def bellman_ford_predecessor_and_distance( G, source, target=None, weight="weight", heuristic=False ): """Compute shortest path lengths and predecessors on shortest paths in weighted graphs. The algorithm has a running time of $O(mn)$ where $n$ is the number of nodes and $m$ is the number of edges. It is slower than Dijkstra but can handle negative edge weights. If a negative cycle is detected, you can use :func:`find_negative_cycle` to return the cycle and examine it. Shortest paths are not defined when a negative cycle exists because once reached, the path can cycle forever to build up arbitrarily low weights. Parameters ---------- G : NetworkX graph The algorithm works for all types of graphs, including directed graphs and multigraphs. source: node label Starting node for path target : node label, optional Ending node for path weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. heuristic : bool Determines whether to use a heuristic to early detect negative cycles at a hopefully negligible cost. Returns ------- pred, dist : dictionaries Returns two dictionaries keyed by node to predecessor in the path and to the distance from the source respectively. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXUnbounded If the (di)graph contains a negative (di)cycle, the algorithm raises an exception to indicate the presence of the negative (di)cycle. Note: any negative weight edge in an undirected graph is a negative cycle. Examples -------- >>> G = nx.path_graph(5, create_using=nx.DiGraph()) >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0) >>> sorted(pred.items()) [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] >>> sorted(dist.items()) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0, 1) >>> sorted(pred.items()) [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] >>> sorted(dist.items()) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) >>> G[1][2]["weight"] = -7 >>> nx.bellman_ford_predecessor_and_distance(G, 0) Traceback (most recent call last): ... networkx.exception.NetworkXUnbounded: Negative cycle detected. See Also -------- find_negative_cycle Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionaries returned only have keys for nodes reachable from the source. In the case where the (di)graph is not connected, if a component not containing the source contains a negative (di)cycle, it will not be detected. In NetworkX v2.1 and prior, the source node had predecessor `[None]`. In NetworkX v2.2 this changed to the source node having predecessor `[]` """ if source not in G: raise nx.NodeNotFound(f"Node {source} is not found in the graph") weight = _weight_function(G, weight) if G.is_multigraph(): if any( weight(u, v, {k: d}) < 0 for u, v, k, d in nx.selfloop_edges(G, keys=True, data=True) ): raise nx.NetworkXUnbounded("Negative cycle detected.") else: if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)): raise nx.NetworkXUnbounded("Negative cycle detected.") dist = {source: 0} pred = {source: []} if len(G) == 1: return pred, dist weight = _weight_function(G, weight) dist = _bellman_ford( G, [source], weight, pred=pred, dist=dist, target=target, heuristic=heuristic ) return (pred, dist) def _bellman_ford( G, source, weight, pred=None, paths=None, dist=None, target=None, heuristic=True, ): """Calls relaxation loop for Bellman–Ford algorithm and builds paths This is an implementation of the SPFA variant. See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm Parameters ---------- G : NetworkX graph source: list List of source nodes. The shortest path from any of the source nodes will be found if multiple sources are provided. weight : function The weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. pred: dict of lists, optional (default=None) dict to store a list of predecessors keyed by that node If None, predecessors are not stored paths: dict, optional (default=None) dict to store the path list from source to each node, keyed by node If None, paths are not stored dist: dict, optional (default=None) dict to store distance from source to the keyed node If None, returned dist dict contents default to 0 for every node in the source list target: node label, optional Ending node for path. Path lengths to other destinations may (and probably will) be incorrect. heuristic : bool Determines whether to use a heuristic to early detect negative cycles at a hopefully negligible cost. Returns ------- dist : dict Returns a dict keyed by node to the distance from the source. Dicts for paths and pred are in the mutated input dicts by those names. Raises ------ NodeNotFound If any of `source` is not in `G`. NetworkXUnbounded If the (di)graph contains a negative (di)cycle, the algorithm raises an exception to indicate the presence of the negative (di)cycle. Note: any negative weight edge in an undirected graph is a negative cycle """ if pred is None: pred = {v: [] for v in source} if dist is None: dist = {v: 0 for v in source} negative_cycle_found = _inner_bellman_ford( G, source, weight, pred, dist, heuristic, ) if negative_cycle_found is not None: raise nx.NetworkXUnbounded("Negative cycle detected.") if paths is not None: sources = set(source) dsts = [target] if target is not None else pred for dst in dsts: gen = _build_paths_from_predecessors(sources, dst, pred) paths[dst] = next(gen) return dist def _inner_bellman_ford( G, sources, weight, pred, dist=None, heuristic=True, ): """Inner Relaxation loop for Bellman–Ford algorithm. This is an implementation of the SPFA variant. See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm Parameters ---------- G : NetworkX graph source: list List of source nodes. The shortest path from any of the source nodes will be found if multiple sources are provided. weight : function The weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. pred: dict of lists dict to store a list of predecessors keyed by that node dist: dict, optional (default=None) dict to store distance from source to the keyed node If None, returned dist dict contents default to 0 for every node in the source list heuristic : bool Determines whether to use a heuristic to early detect negative cycles at a hopefully negligible cost. Returns ------- node or None Return a node `v` where processing discovered a negative cycle. If no negative cycle found, return None. Raises ------ NodeNotFound If any of `source` is not in `G`. """ for s in sources: if s not in G: raise nx.NodeNotFound(f"Source {s} not in G") if pred is None: pred = {v: [] for v in sources} if dist is None: dist = {v: 0 for v in sources} # Heuristic Storage setup. Note: use None because nodes cannot be None nonexistent_edge = (None, None) pred_edge = {v: None for v in sources} recent_update = {v: nonexistent_edge for v in sources} G_succ = G._adj # For speed-up (and works for both directed and undirected graphs) inf = float("inf") n = len(G) count = {} q = deque(sources) in_q = set(sources) while q: u = q.popleft() in_q.remove(u) # Skip relaxations if any of the predecessors of u is in the queue. if all(pred_u not in in_q for pred_u in pred[u]): dist_u = dist[u] for v, e in G_succ[u].items(): dist_v = dist_u + weight(u, v, e) if dist_v < dist.get(v, inf): # In this conditional branch we are updating the path with v. # If it happens that some earlier update also added node v # that implies the existence of a negative cycle since # after the update node v would lie on the update path twice. # The update path is stored up to one of the source nodes, # therefore u is always in the dict recent_update if heuristic: if v in recent_update[u]: # Negative cycle found! pred[v].append(u) return v # Transfer the recent update info from u to v if the # same source node is the head of the update path. # If the source node is responsible for the cost update, # then clear the history and use it instead. if v in pred_edge and pred_edge[v] == u: recent_update[v] = recent_update[u] else: recent_update[v] = (u, v) if v not in in_q: q.append(v) in_q.add(v) count_v = count.get(v, 0) + 1 if count_v == n: # Negative cycle found! return v count[v] = count_v dist[v] = dist_v pred[v] = [u] pred_edge[v] = u elif dist.get(v) is not None and dist_v == dist.get(v): pred[v].append(u) # successfully found shortest_path. No negative cycles found. return None @nx._dispatchable(edge_attrs="weight") def bellman_ford_path(G, source, target, weight="weight"): """Returns the shortest path from source to target in a weighted graph G. Parameters ---------- G : NetworkX graph source : node Starting node target : node Ending node weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- path : list List of nodes in a shortest path. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXNoPath If no path exists between source and target. Examples -------- >>> G = nx.path_graph(5) >>> nx.bellman_ford_path(G, 0, 4) [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- dijkstra_path, bellman_ford_path_length """ length, path = single_source_bellman_ford(G, source, target=target, weight=weight) return path @nx._dispatchable(edge_attrs="weight") def bellman_ford_path_length(G, source, target, weight="weight"): """Returns the shortest path length from source to target in a weighted graph. Parameters ---------- G : NetworkX graph source : node label starting node for path target : node label ending node for path weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- length : number Shortest path length. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXNoPath If no path exists between source and target. Examples -------- >>> G = nx.path_graph(5) >>> nx.bellman_ford_path_length(G, 0, 4) 4 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- dijkstra_path_length, bellman_ford_path """ if source == target: if source not in G: raise nx.NodeNotFound(f"Node {source} not found in graph") return 0 weight = _weight_function(G, weight) length = _bellman_ford(G, [source], weight, target=target) try: return length[target] except KeyError as err: raise nx.NetworkXNoPath(f"node {target} not reachable from {source}") from err @nx._dispatchable(edge_attrs="weight") def single_source_bellman_ford_path(G, source, weight="weight"): """Compute shortest path between source and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph source : node Starting node for path. weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- paths : dictionary Dictionary of shortest path lengths keyed by target. Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> path = nx.single_source_bellman_ford_path(G, 0) >>> path[4] [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- single_source_dijkstra, single_source_bellman_ford """ (length, path) = single_source_bellman_ford(G, source, weight=weight) return path @nx._dispatchable(edge_attrs="weight") def single_source_bellman_ford_path_length(G, source, weight="weight"): """Compute the shortest path length between source and all other reachable nodes for a weighted graph. Parameters ---------- G : NetworkX graph source : node label Starting node for path weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- length : dictionary Dictionary of shortest path length keyed by target Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> length = nx.single_source_bellman_ford_path_length(G, 0) >>> length[4] 4 >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 3 4: 4 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- single_source_dijkstra, single_source_bellman_ford """ weight = _weight_function(G, weight) return _bellman_ford(G, [source], weight) @nx._dispatchable(edge_attrs="weight") def single_source_bellman_ford(G, source, target=None, weight="weight"): """Compute shortest paths and lengths in a weighted graph G. Uses Bellman-Ford algorithm for shortest paths. Parameters ---------- G : NetworkX graph source : node label Starting node for path target : node label, optional Ending node for path weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- distance, path : pair of dictionaries, or numeric and list If target is None, returns a tuple of two dictionaries keyed by node. The first dictionary stores distance from one of the source nodes. The second stores the path from one of the sources to that node. If target is not None, returns a tuple of (distance, path) where distance is the distance from source to target and path is a list representing the path from source to target. Raises ------ NodeNotFound If `source` is not in `G`. Examples -------- >>> G = nx.path_graph(5) >>> length, path = nx.single_source_bellman_ford(G, 0) >>> length[4] 4 >>> for node in [0, 1, 2, 3, 4]: ... print(f"{node}: {length[node]}") 0: 0 1: 1 2: 2 3: 3 4: 4 >>> path[4] [0, 1, 2, 3, 4] >>> length, path = nx.single_source_bellman_ford(G, 0, 1) >>> length 1 >>> path [0, 1] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- single_source_dijkstra single_source_bellman_ford_path single_source_bellman_ford_path_length """ if source == target: if source not in G: raise nx.NodeNotFound(f"Node {source} is not found in the graph") return (0, [source]) weight = _weight_function(G, weight) paths = {source: [source]} # dictionary of paths dist = _bellman_ford(G, [source], weight, paths=paths, target=target) if target is None: return (dist, paths) try: return (dist[target], paths[target]) except KeyError as err: msg = f"Node {target} not reachable from {source}" raise nx.NetworkXNoPath(msg) from err @nx._dispatchable(edge_attrs="weight") def all_pairs_bellman_ford_path_length(G, weight="weight"): """Compute shortest path lengths between all nodes in a weighted graph. Parameters ---------- G : NetworkX graph weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- distance : iterator (source, dictionary) iterator with dictionary keyed by target and shortest path length as the key value. Examples -------- >>> G = nx.path_graph(5) >>> length = dict(nx.all_pairs_bellman_ford_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 Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionary returned only has keys for reachable node pairs. """ length = single_source_bellman_ford_path_length for n in G: yield (n, dict(length(G, n, weight=weight))) @nx._dispatchable(edge_attrs="weight") def all_pairs_bellman_ford_path(G, weight="weight"): """Compute shortest paths between all nodes in a weighted graph. Parameters ---------- G : NetworkX graph weight : string or function (default="weight") If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- paths : iterator (source, dictionary) iterator with dictionary keyed by target and shortest path as the key value. Examples -------- >>> G = nx.path_graph(5) >>> path = dict(nx.all_pairs_bellman_ford_path(G)) >>> path[0][4] [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. See Also -------- floyd_warshall, all_pairs_dijkstra_path """ path = single_source_bellman_ford_path for n in G: yield (n, path(G, n, weight=weight)) @nx._dispatchable(edge_attrs="weight") def goldberg_radzik(G, source, weight="weight"): """Compute shortest path lengths and predecessors on shortest paths in weighted graphs. The algorithm has a running time of $O(mn)$ where $n$ is the number of nodes and $m$ is the number of edges. It is slower than Dijkstra but can handle negative edge weights. Parameters ---------- G : NetworkX graph The algorithm works for all types of graphs, including directed graphs and multigraphs. source: node label Starting node for path weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- pred, dist : dictionaries Returns two dictionaries keyed by node to predecessor in the path and to the distance from the source respectively. Raises ------ NodeNotFound If `source` is not in `G`. NetworkXUnbounded If the (di)graph contains a negative (di)cycle, the algorithm raises an exception to indicate the presence of the negative (di)cycle. Note: any negative weight edge in an undirected graph is a negative cycle. As of NetworkX v3.2, a zero weight cycle is no longer incorrectly reported as a negative weight cycle. Examples -------- >>> G = nx.path_graph(5, create_using=nx.DiGraph()) >>> pred, dist = nx.goldberg_radzik(G, 0) >>> sorted(pred.items()) [(0, None), (1, 0), (2, 1), (3, 2), (4, 3)] >>> sorted(dist.items()) [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) >>> G[1][2]["weight"] = -7 >>> nx.goldberg_radzik(G, 0) Traceback (most recent call last): ... networkx.exception.NetworkXUnbounded: Negative cycle detected. Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionaries returned only have keys for nodes reachable from the source. In the case where the (di)graph is not connected, if a component not containing the source contains a negative (di)cycle, it will not be detected. """ if source not in G: raise nx.NodeNotFound(f"Node {source} is not found in the graph") weight = _weight_function(G, weight) if G.is_multigraph(): if any( weight(u, v, {k: d}) < 0 for u, v, k, d in nx.selfloop_edges(G, keys=True, data=True) ): raise nx.NetworkXUnbounded("Negative cycle detected.") else: if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)): raise nx.NetworkXUnbounded("Negative cycle detected.") if len(G) == 1: return {source: None}, {source: 0} G_succ = G._adj # For speed-up (and works for both directed and undirected graphs) inf = float("inf") d = {u: inf for u in G} d[source] = 0 pred = {source: None} def topo_sort(relabeled): """Topologically sort nodes relabeled in the previous round and detect negative cycles. """ # List of nodes to scan in this round. Denoted by A in Goldberg and # Radzik's paper. to_scan = [] # In the DFS in the loop below, neg_count records for each node the # number of edges of negative reduced costs on the path from a DFS root # to the node in the DFS forest. The reduced cost of an edge (u, v) is # defined as d[u] + weight[u][v] - d[v]. # # neg_count also doubles as the DFS visit marker array. neg_count = {} for u in relabeled: # Skip visited nodes. if u in neg_count: continue d_u = d[u] # Skip nodes without out-edges of negative reduced costs. if all(d_u + weight(u, v, e) >= d[v] for v, e in G_succ[u].items()): continue # Nonrecursive DFS that inserts nodes reachable from u via edges of # nonpositive reduced costs into to_scan in (reverse) topological # order. stack = [(u, iter(G_succ[u].items()))] in_stack = {u} neg_count[u] = 0 while stack: u, it = stack[-1] try: v, e = next(it) except StopIteration: to_scan.append(u) stack.pop() in_stack.remove(u) continue t = d[u] + weight(u, v, e) d_v = d[v] if t < d_v: is_neg = t < d_v d[v] = t pred[v] = u if v not in neg_count: neg_count[v] = neg_count[u] + int(is_neg) stack.append((v, iter(G_succ[v].items()))) in_stack.add(v) elif v in in_stack and neg_count[u] + int(is_neg) > neg_count[v]: # (u, v) is a back edge, and the cycle formed by the # path v to u and (u, v) contains at least one edge of # negative reduced cost. The cycle must be of negative # cost. raise nx.NetworkXUnbounded("Negative cycle detected.") to_scan.reverse() return to_scan def relax(to_scan): """Relax out-edges of relabeled nodes.""" relabeled = set() # Scan nodes in to_scan in topological order and relax incident # out-edges. Add the relabled nodes to labeled. for u in to_scan: d_u = d[u] for v, e in G_succ[u].items(): w_e = weight(u, v, e) if d_u + w_e < d[v]: d[v] = d_u + w_e pred[v] = u relabeled.add(v) return relabeled # Set of nodes relabled in the last round of scan operations. Denoted by B # in Goldberg and Radzik's paper. relabeled = {source} while relabeled: to_scan = topo_sort(relabeled) relabeled = relax(to_scan) d = {u: d[u] for u in pred} return pred, d @nx._dispatchable(edge_attrs="weight") def negative_edge_cycle(G, weight="weight", heuristic=True): """Returns True if there exists a negative edge cycle anywhere in G. Parameters ---------- G : NetworkX graph weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. heuristic : bool Determines whether to use a heuristic to early detect negative cycles at a negligible cost. In case of graphs with a negative cycle, the performance of detection increases by at least an order of magnitude. Returns ------- negative_cycle : bool True if a negative edge cycle exists, otherwise False. Examples -------- >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) >>> print(nx.negative_edge_cycle(G)) False >>> G[1][2]["weight"] = -7 >>> print(nx.negative_edge_cycle(G)) True Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. This algorithm uses bellman_ford_predecessor_and_distance() but finds negative cycles on any component by first adding a new node connected to every node, and starting bellman_ford_predecessor_and_distance on that node. It then removes that extra node. """ if G.size() == 0: return False # find unused node to use temporarily newnode = -1 while newnode in G: newnode -= 1 # connect it to all nodes G.add_edges_from([(newnode, n) for n in G]) try: bellman_ford_predecessor_and_distance( G, newnode, weight=weight, heuristic=heuristic ) except nx.NetworkXUnbounded: return True finally: G.remove_node(newnode) return False @nx._dispatchable(edge_attrs="weight") def find_negative_cycle(G, source, weight="weight"): """Returns a cycle with negative total weight if it exists. Bellman-Ford is used to find shortest_paths. That algorithm stops if there exists a negative cycle. This algorithm picks up from there and returns the found negative cycle. The cycle consists of a list of nodes in the cycle order. The last node equals the first to make it a cycle. You can look up the edge weights in the original graph. In the case of multigraphs the relevant edge is the minimal weight edge between the nodes in the 2-tuple. If the graph has no negative cycle, a NetworkXError is raised. Parameters ---------- G : NetworkX graph source: node label The search for the negative cycle will start from this node. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Examples -------- >>> G = nx.DiGraph() >>> G.add_weighted_edges_from( ... [(0, 1, 2), (1, 2, 2), (2, 0, 1), (1, 4, 2), (4, 0, -5)] ... ) >>> nx.find_negative_cycle(G, 0) [4, 0, 1, 4] Returns ------- cycle : list A list of nodes in the order of the cycle found. The last node equals the first to indicate a cycle. Raises ------ NetworkXError If no negative cycle is found. """ weight = _weight_function(G, weight) pred = {source: []} v = _inner_bellman_ford(G, [source], weight, pred=pred) if v is None: raise nx.NetworkXError("No negative cycles detected.") # negative cycle detected... find it neg_cycle = [] stack = [(v, list(pred[v]))] seen = {v} while stack: node, preds = stack[-1] if v in preds: # found the cycle neg_cycle.extend([node, v]) neg_cycle = list(reversed(neg_cycle)) return neg_cycle if preds: nbr = preds.pop() if nbr not in seen: stack.append((nbr, list(pred[nbr]))) neg_cycle.append(node) seen.add(nbr) else: stack.pop() if neg_cycle: neg_cycle.pop() else: if v in G[v] and weight(G, v, v) < 0: return [v, v] # should not reach here raise nx.NetworkXError("Negative cycle is detected but not found") # should not get here... msg = "negative cycle detected but not identified" raise nx.NetworkXUnbounded(msg) @nx._dispatchable(edge_attrs="weight") def bidirectional_dijkstra(G, source, target, weight="weight"): r"""Dijkstra's algorithm for shortest paths using bidirectional search. Parameters ---------- G : NetworkX graph source : node Starting node. target : node Ending node. weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. Returns ------- length, path : number and list length is the distance from source to target. path is a list of nodes on a path from source to target. Raises ------ NodeNotFound If `source` or `target` is not in `G`. NetworkXNoPath If no path exists between source and target. Examples -------- >>> G = nx.path_graph(5) >>> length, path = nx.bidirectional_dijkstra(G, 0, 4) >>> print(length) 4 >>> print(path) [0, 1, 2, 3, 4] Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The weight function can be used to hide edges by returning None. So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` will find the shortest red path. In practice bidirectional Dijkstra is much more than twice as fast as ordinary Dijkstra. Ordinary Dijkstra expands nodes in a sphere-like manner from the source. The radius of this sphere will eventually be the length of the shortest path. Bidirectional Dijkstra will expand nodes from both the source and the target, making two spheres of half this radius. Volume of the first sphere is `\pi*r*r` while the others are `2*\pi*r/2*r/2`, making up half the volume. This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows and roundoff errors can cause problems). See Also -------- shortest_path shortest_path_length """ if source not in G: raise nx.NodeNotFound(f"Source {source} is not in G") if target not in G: raise nx.NodeNotFound(f"Target {target} is not in G") if source == target: return (0, [source]) weight = _weight_function(G, weight) push = heappush pop = heappop # Init: [Forward, Backward] dists = [{}, {}] # dictionary of final distances paths = [{source: [source]}, {target: [target]}] # dictionary of paths fringe = [[], []] # heap of (distance, node) for choosing node to expand seen = [{source: 0}, {target: 0}] # dict of distances to seen nodes c = count() # initialize fringe heap push(fringe[0], (0, next(c), source)) push(fringe[1], (0, next(c), target)) # neighs for extracting correct neighbor information if G.is_directed(): neighs = [G._succ, G._pred] else: neighs = [G._adj, G._adj] # variables to hold shortest discovered path # finaldist = 1e30000 finalpath = [] dir = 1 while fringe[0] and fringe[1]: # choose direction # dir == 0 is forward direction and dir == 1 is back dir = 1 - dir # extract closest to expand (dist, _, v) = pop(fringe[dir]) if v in dists[dir]: # Shortest path to v has already been found continue # update distance dists[dir][v] = dist # equal to seen[dir][v] if v in dists[1 - dir]: # if we have scanned v in both directions we are done # we have now discovered the shortest path return (finaldist, finalpath) for w, d in neighs[dir][v].items(): # weight(v, w, d) for forward and weight(w, v, d) for back direction cost = weight(v, w, d) if dir == 0 else weight(w, v, d) if cost is None: continue vwLength = dists[dir][v] + cost if w in dists[dir]: if vwLength < dists[dir][w]: raise ValueError("Contradictory paths found: negative weights?") elif w not in seen[dir] or vwLength < seen[dir][w]: # relaxing seen[dir][w] = vwLength push(fringe[dir], (vwLength, next(c), w)) paths[dir][w] = paths[dir][v] + [w] if w in seen[0] and w in seen[1]: # see if this path is better than the already # discovered shortest path totaldist = seen[0][w] + seen[1][w] if finalpath == [] or finaldist > totaldist: finaldist = totaldist revpath = paths[1][w][:] revpath.reverse() finalpath = paths[0][w] + revpath[1:] raise nx.NetworkXNoPath(f"No path between {source} and {target}.") @nx._dispatchable(edge_attrs="weight") def johnson(G, weight="weight"): r"""Uses Johnson's Algorithm to compute shortest paths. Johnson's Algorithm finds a shortest path between each pair of nodes in a weighted graph even if negative weights are present. Parameters ---------- G : NetworkX graph weight : string or function If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be ``G.edges[u, v][weight]``). If no such edge attribute exists, the weight of the edge is assumed to be one. If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number. Returns ------- distance : dictionary Dictionary, keyed by source and target, of shortest paths. Examples -------- >>> graph = nx.DiGraph() >>> graph.add_weighted_edges_from( ... [("0", "3", 3), ("0", "1", -5), ("0", "2", 2), ("1", "2", 4), ("2", "3", 1)] ... ) >>> paths = nx.johnson(graph, weight="weight") >>> paths["0"]["2"] ['0', '1', '2'] Notes ----- Johnson's algorithm is suitable even for graphs with negative weights. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph. The time complexity of this algorithm is $O(n^2 \log n + n m)$, where $n$ is the number of nodes and $m$ the number of edges in the graph. For dense graphs, this may be faster than the Floyd–Warshall algorithm. See Also -------- floyd_warshall_predecessor_and_distance floyd_warshall_numpy all_pairs_shortest_path all_pairs_shortest_path_length all_pairs_dijkstra_path bellman_ford_predecessor_and_distance all_pairs_bellman_ford_path all_pairs_bellman_ford_path_length """ dist = {v: 0 for v in G} pred = {v: [] for v in G} weight = _weight_function(G, weight) # Calculate distance of shortest paths dist_bellman = _bellman_ford(G, list(G), weight, pred=pred, dist=dist) # Update the weight function to take into account the Bellman--Ford # relaxation distances. def new_weight(u, v, d): return weight(u, v, d) + dist_bellman[u] - dist_bellman[v] def dist_path(v): paths = {v: [v]} _dijkstra(G, v, new_weight, paths=paths) return paths return {v: dist_path(v) for v in G}