Editing Johnson's algorithm (section)

==Algorithm description==
Johnson's algorithm consists of the following steps:<ref name="clrs"/><ref name="black"/>
#First, a new [[Vertex (graph theory)|node]] {{mvar|q}} is added to the graph, connected by zero-weight [[Edge (graph theory)|edges]] to each of the other nodes.
#Second, the [[Bellman–Ford algorithm]] is used, starting from the new vertex {{mvar|q}}, to find for each vertex {{mvar|v}} the minimum weight {{math|''h''(''v'')}} of a path from {{mvar|q}} to {{mvar|v}}. If this step detects a negative cycle, the algorithm is terminated.
#Next the edges of the original graph are reweighted using the values computed by the Bellman–Ford algorithm: an edge from {{mvar|u}} to {{mvar|v}}, having length {{tmath|w(u, v)}}, is given the new length {{math|''w''(''u'',''v'') + ''h''(''u'') &minus; ''h''(''v'')}}.
#Finally, {{mvar|q}} is removed, and [[Dijkstra's algorithm]] is used to find the shortest paths from each node {{mvar|s}} to every other vertex in the reweighted graph. The distance in the original graph is then computed for each distance {{mvar|D}}({{mvar|u}}, {{mvar|v}}), by adding {{mvar|''h''(''v'') &minus; ''h''(''u'')}} to the distance returned by Dijkstra's algorithm.