An efficient oracle for counting shortest paths in planar graphs

Abstract

We study the counting shortest paths problem: find the number ns(u,v) of shortest paths from a vertex u to a vertex v in a graph of n vertices. Bezáková and Searns [ISAAC 2018] give an oracle which, given u and v in a planar graph, answers ns(u,v) in O(n) time and O(n1.5) memory space. Applying this oracle directly, it takes O(n) time to answer whether there is a unique shortest path from u to v. We propose an oracle which answers ns(u,v) in O(n) time and O(n1.5) space. Our oracle answers whether there is a unique shortest path from u to v in O(log⁡n) time. Computational studies show that our oracle is faster than the oracle of Bezáková and Searns for large graphs. A key technique to speed up the query time of our oracle is applying Voronoi diagrams on planar graphs, an advanced data structure widely used in shortest distance oracles. Based on Voronoi diagrams on planar graphs, significant theoretical improvements have been made for distance oracles. Our studies confirm that Voronoi diagrams are efficient data structures for distance oracles in practice.