Johnson's algorithm

The first three stages of Johnson's algorithm are depicted in the illustration below.

Johnson's algorithm consists of the following steps:

Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in a sparse directed graph.

On a sparse graph, Johnson's algorithm may be more efficient, taking Big O notation time.

Compared to the Floyd-Warshall algorithm, Johnson's algorithm is more efficient for sparse graphs.

Johnson's algorithm solves all pairs' shortest paths, and may be faster than Floyd-Warshall on sparse graphs.

Johnson's algorithm, an algorithm for solving the same problem as the Floyd-Warshall algorithm, all pairs shortest paths in graphs with some edge weights negative.

For weighted graphs, one may instead use the Floyd-Warshall algorithm or Johnson's algorithm, with running time O(n) or O(nm + n log n) respectively.

The modification to the weights is similar to the weight modification in Johnson's algorithm, and preserves the non-negativity of the weights while allowing the second instance of Dijkstra's algorithm to find the correct second path.