Johnson's algorithm


Definition: An algorithm to solve the all pairs shortest path problem in a sparse weighted, directed graph. First, it adds a new node with zero weight edges from it to all other nodes, and runs the Bellman-Ford algorithm to check for negative weight cycles and find h(v), the least weight of a path from the new node to node v. Next it reweights the edges using the nodes' h(v) values. Finally for each node, it runs Dijkstra's algorithm and stores the computed least weight to other nodes, reweighted using the nodes' h(v) values, as the final weight. The time complexity is O(V²log V + VE).

Aggregate child (... is a part of or used in me.)
Bellman-Ford algorithm, Dijkstra's algorithm.

See also Floyd-Warshall algorithm.

Note: After [CLR90, page 569].

Author: PEB

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Entry modified 17 December 2004.
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Cite this as:
Paul E. Black, "Johnson's algorithm", in Dictionary of Algorithms and Data Structures [online], Vreda Pieterse and Paul E. Black, eds. 17 December 2004. (accessed TODAY) Available from: