Johnson's Algorithm - Example

Prerequisites

To understand the workings of Johnson’s algorithm, it’s necessary to be familiar with two other shortest path algorithms:

1. Dijkstra’s Algorithm - Example
2. Bellman-Ford Algorithm - Example

Idea of the Algorithm

1. Adding a Vertex: We add a vertex $w$ to the graph $G$ along with $|V|$ edges connecting $w$ to each vertex of $G$.
2. We use Bellman-Ford’s algorithm from $w$ towards all other vertices of $G$. $d(s)$ gives us the distance between $w$ and $s$. If Bellman-Ford detects a negative cycle, the algorithm stops.
3. Removal of Vertex $w$: This vertex is no longer useful. Remove this vertex and all the edges added in step 1.
4. Re-weighting the Edges of the Graph: For each edge $(u,v)$ in the graph, do: $p((u,v)) := p((u,v)) + d(u) - d(v)$.
5. Calculating the Shortest Path: For each vertex in $G$, apply Dijkstra’s algorithm to determine the shortest path between all vertices.

It is important to note that the distances are not preserved. If for an edge, we had a weight of $-2$, the new weight will be $\geq 0$. However, this is not the point. Johnson’s algorithm gives us the shortest path from point $A$ to point $B$ “only”. If you need to retrieve the sum of the weights of this path, just return to the original graph and follow the path given by Johnson.

Algorithm / Solution / Complexity

This article only presents an example of the application of Johnson’s algorithm. However, you can easily find this information on the internet:

• Johnson’s Algorithm on Stringfixer
• Johnson’s Algorithm on Wikiwand

Example:

Let $G$ be a graph with negative weight edges.

Step 1 - Adding Vertex w

Step 2 - Shortest Paths from w to Other Vertices Using Bellman-Ford

In purple are the results of Bellman-Ford’s algorithm.

Steps 3 and 4 - Re-weighting the Edges

Step 5 - Dijkstra’s Algorithm on All Edges

The final graph $G'$ is as follows:

Step 5 is simply an application of Dijkstra’s algorithm. An example of its use is already detailed on this site: Dijkstra’s Algorithm - Example.

To take multiple results from Dijkstra’s algorithm:

• To go from a to d, the path to take is $a \rightarrow b \rightarrow c \rightarrow d$ with a weight of -9 (not 0, the weights from $G$ and not $G'$ must be used).
• To go from f to d, the path to take is $f \rightarrow e \rightarrow c \rightarrow d$ with a weight of -2.
• There is no possible path from d to e.

The rest is left as an exercise for the reader.