  Distance oracles are data structures that provide fast (possibly approximate)
answers to shortest-path 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 eps>0 and for any planar graph on n nodes, there exists
a (1+eps)-approximate distance oracle using space O(n eps^{-1} log n) such that
approximate distance queries can be answered in time O(1/eps).
  Ten years later, we give the first improvements on the space-querytime
tradeoff for planar graphs.
  * We give the first oracle having a space-time product with subquadratic
dependency on 1/eps. For space ~O(n log n) we obtain query time ~O(1/eps)
(assuming polynomial edge weights). The space shows a doubly logarithmic
dependency on 1/eps only. We believe that the dependency on eps may be almost
optimal.
  * For the case of moderate edge weights (average bounded by polylog(n), 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 eps and n. Our oracle
uses space ~O(n log log n) and it has query time ~O(log log log n + 1/eps).
  (Asymptotic notation in this abstract hides low-degree polynomials in
log(1/eps) and log*(n).)

Data Structures and Algorithms