# Johnson’s algorithm for All-pairs shortest paths

The problem is to find the shortest paths between every pair of vertices in a given weighted directed Graph and weights may be negative. We have discussed Floyd Warshall Algorithm for this problem. The time complexity of the Floyd Warshall Algorithm is Θ(V^{3}).

*Using Johnson’s algorithm, we can find all pair shortest paths in O(V ^{2}log V + VE) time.* Johnson’s algorithm uses both Dijkstra and Bellman-Ford as subroutines. If we apply Dijkstra’s Single Source shortest path algorithm for every vertex, considering every vertex as the source, we can find all pair shortest paths in O(V*VLogV) time.

So using Dijkstra’s single-source shortest path seems to be a better option than Floyd Warshall’s Algorithm(https://www.geeksforgeeks.org/floyd-warshall-algorithm-dp-16/?ref=lbp) , but the problem with Dijkstra’s algorithm is, that it doesn’t work for negative weight edge. *The idea of Johnson’s algorithm is to re-weight all edges and make them all positive, then apply Dijkstra’s algorithm for every vertex.*

**How to transform a given graph into a graph with all non-negative weight edges?**

One may think of a simple approach of finding the minimum weight edge and adding this weight to all edges. Unfortunately, this doesn’t work as there may be a different number of edges in different paths (See this for an example). If there are multiple paths from a vertex u to v, then all paths must be increased by the same amount, so that the shortest path remains the shortest in the transformed graph. The idea of Johnson’s algorithm is to assign a weight to every vertex. Let the weight assigned to vertex u be h[u].

We reweight edges using vertex weights. For example, for an edge (u, v) of weight w(u, v), the new weight becomes w(u, v) + h[u] – h[v]. The great thing about this reweighting is, that all set of paths between any two vertices is increased by the same amount and all negative weights become non-negative. Consider any path between two vertices s and t, the weight of every path is increased by h[s] – h[t], and all h[] values of vertices on the path from s to t cancel each other.

How do we calculate h[] values?

Bellman-Ford algorithm is used for this purpose. Following is the complete algorithm. A new vertex is added to the graph and connected to all existing vertices. The shortest distance values from the new vertex to all existing vertices are h[] values.

Algorithm:

- Let the given graph be G. Add a new vertex s to the graph, add edges from the new vertex to all vertices of G. Let the modified graph be G’.
- Run the Bellman-Ford algorithm on G’ with s as the source. Let the distances calculated by Bellman-Ford be h[0], h[1], .. h[V-1]. If we find a negative weight cycle, then return. Note that the negative weight cycle cannot be created by new vertex s as there is no edge to s. All edges are from s.
- Reweight the edges of the original graph. For each edge (u, v), assign the new weight as “original weight + h[u] – h[v]”.
- Remove the added vertex s and run Dijkstra’s algorithm for every vertex.

**How does the transformation ensure nonnegative weight edges?**

The following property is always true about h[] values as they are the shortest distances.

h[v] <= h[u] + w(u, v)

The property simply means that the shortest distance from s to v must be smaller than or equal to the shortest distance from s to u plus the weight of the edge (u, v). The new weights are w(u, v) + h[u] – h[v]. The value of the new weights must be greater than or equal to zero because of the inequality “h[v] <= h[u] + w(u, v)”.

**Example:** Let us consider the following graph.

We add a source s and add edges from s to all vertices of the original graph. In the following diagram s is 4.

We calculate the shortest distances from 4 to all other vertices using Bellman-Ford algorithm. The shortest distances from 4 to 0, 1, 2 and 3 are 0, -5, -1 and 0 respectively, i.e., h[] = {0, -5, -1, 0}. Once we get these distances, we remove the source vertex 4 and reweight the edges using following formula. w(u, v) = w(u, v) + h[u] – h[v].

Since all weights are positive now, we can run Dijkstra’s shortest path algorithm for every vertex as the source.

**Time Complexity: **The main steps in the algorithm are Bellman-Ford Algorithm called once and Dijkstra called V times. Time complexity of Bellman Ford is O(VE) and time complexity of Dijkstra is O(VLogV). So overall time complexity is O(V^{2}log V + VE).

The time complexity of Johnson’s algorithm becomes the same as Floyd Warshall’s Algorithm (https://www.geeksforgeeks.org/floyd-warshall-algorithm-dp-16/?ref=lbp)

when the graph is complete (For a complete graph E = O(V^{2}). But for sparse graphs, the algorithm performs much better than Floyd Warshall’s Algorithm( https://www.geeksforgeeks.org/floyd-warshall-algorithm-dp-16/?ref=lbp ).

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