johnson#

johnson(G, weight='weight')[source]#

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:

GNetworkX graph

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:

distancedictionary: Dictionary, keyed by source and target, of shortest paths.

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

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.

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']

Additional backends implement this function

parallelParallel backend for NetworkX algorithms

The parallel computation is implemented by dividing the nodes into chunks and computing the shortest paths using Johnson’s Algorithm for each chunk in parallel.

Additional parameters:

get_chunksstr, function (default = “chunks”): A function that takes in an iterable of all the nodes as input and returns an iterable node_chunks. The default chunking is done by slicing the G.nodes into n chunks, where n is the number of CPU cores.

[Source]