More Compact Oracles for Approximate Distances in Planar Graphs

Distance oracles are data structures that provide fast (possibly approximate) answers to shortestpath and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS‘01, Thorup introduced approximate distance oracles for planar graphs. He proved that, for any > 0 and for any planar graph on n nodes, there exists a (1 + )–approximate distance oracle using space O(n −1 logn) such that approximate distance queries can be answered in time O( −1). Ten years later, we give the first improvements on the space–query time tradeoff for planar graphs. • We give the first oracle having a space–time product with subquadratic dependency on 1/ . For space Õ(n logn) we obtain query time Õ( −1) (assuming polynomial edge weights). We believe that the dependency on may be almost optimal. • For the case of moderate edge weights (average bounded by poly(logn), which appears to be the case for many real-world road networks), we hit a “sweet spot,” improving upon Thorup’s oracle both in terms of and n. Our oracle uses space Õ(n log log n) and it has query time Õ( −1 + log log log n). (Notation: Õ(·) hides low-degree polynomials in log(1/ ) and log∗(n).) ar X iv :1 10 9. 26 41 v2 [ cs .D S] 2 8 O ct 2 01 1