On Counting Oracles for Path Problems

We initiate the study of counting oracles for various path problems in graphs. Distance oracles have gained a lot of attention in recent years, with studies of the underlying space and time tradeoffs. For a given graph G, a distance oracle is a data structure which can be used to answer distance queries for pairs of vertices s,t in V(G). In this work, we extend the set up to answering counting queries: for a pair of vertices s,t, the oracle needs to provide the number of (shortest or all) paths from s to t. We present O(n^{1.5}) preprocessing time, O(n^{1.5}) space, and O(sqrt{n}) query time algorithms for oracles counting shortest paths in planar graphs and for counting all paths in planar directed acyclic graphs. We extend our results to other graphs which admit small balanced separators and present applications where our oracle improves the currently best known running times.