Question

In an algorithms course quite some years ago, I came across an interesting graph representation. It is basically a path matrix, but with extra information. Each cell Aij contains the (possibly empty) list of vertices adjacent to i that you can pass through to reach j.

For example, the directed graph informally represented as:

(Z → X)    (Z → Y)    (X → W)    (Y → W)

gets the following matrix:

When maintaining a matrix like this you have the advantage of knowing not only if there is a path from i to j, but what all possible paths are.

But I can’t for the life of me find any reference to this representation on the web. What’s it called?

Answer 1

After a lot of searching and a hint from an old teacher, I came upon the path reconstruction section of the Wikipedia article about the Floyd-Warshall algorithm. They store the ‘next node’ in the shortest path between i and j in the Aij cell of what they call a:

next matrix

In a set of slides by what appears to be a colleague of mine, it is called a:

distance-next table

in the same context. Granted, they are talking about storing the information of the shortest paths only. Of course, for that purpose it suffices to store only one node(-index) per cell. Neither of these sources is citing any scientific publication. But after searching for a long while, I strongly feel that the matrix representation from my original question was never officially named. So, I dub thee:

path-next matrix

If anyone can cite a scientific publication that says otherwise, I’d still be happy to accept their answer over my own.