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.
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
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.
distance – Dictionary, keyed by source and target, of shortest paths.
- Return type
NetworkXError – If given graph is not weighted.
>>> import networkx as nx >>> 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']
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.