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:
GNetworkX graph

The algorithm works for all types of graphs, including directed graphs and multigraphs.

source: node label

Starting node for path

weightstring 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, distdictionaries

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.

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.

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