scipy.sparse.csgraph.johnson(csgraph, directed=True, indices=None, return_predecessors=False, unweighted=False)

Compute the shortest path lengths using Johnson’s algorithm.

Johnson’s algorithm combines the Bellman-Ford algorithm and Dijkstra’s algorithm to quickly find shortest paths in a way that is robust to the presence of negative cycles. If a negative cycle is detected, an error is raised. For graphs without negative edge weights, dijkstra() may be faster.

New in version 0.11.0.


csgraph : array, matrix, or sparse matrix, 2 dimensions

The N x N array of distances representing the input graph.

directed : bool, optional

If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i]

indices : array_like or int, optional

if specified, only compute the paths for the points at the given indices.

return_predecessors : bool, optional

If True, return the size (N, N) predecesor matrix

unweighted : bool, optional

If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized.


dist_matrix : ndarray

The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph.

predecessors : ndarray

Returned only if return_predecessors == True. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999



if there are negative cycles in the graph


This routine is specially designed for graphs with negative edge weights. If all edge weights are positive, then Dijkstra’s algorithm is a better choice.