Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

# networkx.algorithms.shortest_paths.weighted.bellman_ford_predecessor_and_distance¶

bellman_ford_predecessor_and_distance(G, source, target=None, weight='weight', heuristic=False)[source]

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.

• heuristic (bool) – Determines whether to use a heuristic to early detect negative cycles at a hopefully negligible cost.

Returns

pred, dist – Returns two dictionaries keyed by node to predecessor in the path and to the distance from the source respectively.

Return type

dictionaries

Raises
• NodeNotFound – If source is not in G.

• NetworkXUnbounded – If the (di)graph contains a negative cost (di)cycle, the algorithm raises an exception to indicate the presence of the negative cost (di)cycle. Note: any negative weight edge in an undirected graph is a negative cost 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 cost 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 cost (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 []