Question

I have a graph which consists of nodes and I need a fast algorithm that generates a random path between two nodes. I designed several algorithms from scratch for this but can’t seem to get it right.

Either the algorithm gets stuck in loops, or when I keep record of the visited nodes it sometimes gets stuck between visited nodes. Another problem I encountered is that my algorithm was too unstable in performance.

So my question is; does anyone know a fast and stable algorithm for a random path between two reachable nodes in an undirected graph?

Answer 1

Let your graph be G=(V,E). Create a subset U of V such that U = { u | there is a path from u to the target }.

• Note that finding this subset U is easy – and can be done in linear
  time using DFS or BFS on the reversed edges from the
  target.

Using this subset, create a graph G'=(U,E') where U is defined above and E' = E [intersection] UxU [The same edges, but applied only on vertices in U].

Run randomized (choosing which vertex to explore next on random) DFS on G' until you reach the target.

• Note – the idea is to find a path – not necesseraly simple, and thus you shouldn’t maintain a visited set.
• You might add a “break rule” that if you reach a certain depth, you will seek the target – unrandomised, to avoid infinite loops in circles.
• Perofrmance is expected to be varying, since it is linear in the length of the found path, which might be very long or very short…