[Solved]: Efficiently enumerating all paths from i to j of given length in a graph

Problem Detail: 

I've been trying to efficiently solve this problem : given a integer p > 0 and a directed graph whose nodes are 0, ..., N-1, enumerate (not simply count) all the paths (not necessarily elementary) from a node i to a node j of length p. For instance, i->j is of length 2 if the edge i->j exists, i->k->j is of length 3, etc...). The graph is either given as an adjacency matrix or adjacency lists. In my very special case, I'm searching for cycles 0->...->0 of given length, but I'm pretty sure I'm gonna need the more general case i->...->j.

Right now, I'm dynamically storing paths from k to j of length q in a matrix Z[k][q], and since I need Z[i][p], I first compute all the Z[j][p-1] for i->j, then append i to them.

The problem is that this approach uses an awful lot of memory and doesn't scale very nicely for large graphs. Any advice on this ?

Asked By : Mrktn

Answered By : Tom van der Zanden

It's not possible to solve this problem "efficiently". There can be exponentially many paths of a given length, so enumerating all of them is not feasible in large graphs.

Your dynamic programming approach is a pretty good one, though you might be able to improve it by not storing the paths explicitly but instead storing the paths as linked lists or using pointers, so that you only use $O(1)$ storage per path, letting paths share their common suffixes (if you do not do this already).

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Question Source : http://cs.stackexchange.com/questions/42942