## The story behind this post

Recently I’ve received `+10` karma on StackOverflow. I was curious for what question or answer and clicked to check this. It appeared to be a seven-year-old answer about a Floyd-Warshall algorithm. I was surprised both of my bad English back those days and the very small value the answer had. So I’ve revised it and here it is – the brand-new version!

## The definitions

Let us have a graph, described by matrix `D`, where `D[i][j]` is the length of the edge `(i -> j)` (from graph’s vertex with index `i` to the vertex with index `j`).

Matrix `D` has the size of `N * N`, where `N` is a total number of vertices in a graph because we can reach the maximum of paths by connecting each graph’s vertex to each other.

Also, we’ll need matrix `R`, where we will store shortest paths (`R[i][j]` contains the index of a next vertex in the shortest path, starting at vertex `i` and ending at vertex `j`).

Matrix `R` has the same size as `D`.

The Floyd-Warshall algorithm performs these steps:

1. initialize the matrix of all the paths between any two pairs of vertices in a graph with the edge’s end vertex (this is important since this value will be used for path reconstruction)

2. for each pair of connected vertices (read: for each edge `(u -> v)`), `u` and `v`, find the vertex, which forms shortest path between them: if the vertex `k` defines two valid edges `(u -> k)` and `(k -> v)` (if they are present in the graph), which are together shorter than path `(u -> v)`, then assume the shortest path between `u` and `v` lies through `k`; set the shortest pivot point in matrix `R` for edge `(u -> v)` to be the corresponding pivot point for edge `(u -> k)`

But how do we read the matrix `D`?

## Inputs

Let us have a graph:

In GraphViz it would be described as follows:

We first create a two-dimensional array of size `4` (since there are exactly `4` vertices in our graph).

We initialize its main diagonal (the items, whose indices are equal, for ex. `G[0, 0]`, `G[1, 1]`, etc.) with zeros, because the shortest path from vertex to itself has the length `0` and the other elements with a very large number (to indicate there is no edge or an infinitely long edge between them). The defined elements, corresponding to graph’s edges, we fill with edges’ lengths:

## The algorithm itself

Now that we are on a same page with definitions, algorithm can be implemented like this:

After the algorithm run, the matrix `R` will be filled with vertices’ indices, describing shortest paths between them.

## Path reconstruction

In order to reconstruct the path from vertex `u` to vertex `v`, you need follow the elements of matrix `R`:

## Summary

The whole code could be wrapped in a couple of methods:

You can read ‘bout this algorithm on wikipedia and get some data structures generated automatically here