Constant Query Time $$(1+\epsilon )$$ -Approximate Distance Oracle for Planar Graphs

Abstract

We give a (1 + ǫ)-approximate distance oracle with O(1) query time for an undirected planar graph G with n vertices and non-negative edge lengths. For ǫ > 0 and any two vertices u and v in G, our oracle gives a distanced(u, v) with stretch (1 + ǫ) in O(1) time. The oracle has size O(n log n((log n)/ǫ + f (ǫ))) and pre-processing time O(n log n((log 3 n)/ǫ 2 + f (ǫ))), where f (ǫ) = 2 O(1/ǫ) . This is the first (1 + ǫ)-approximate distance oracle with O(1) query time independent of ǫ and the size and pre-processing time nearly linear in n, and improves the query time O(1/ǫ) of previous (1 + ǫ)-approximate distance oracle with size nearly linear in n.

Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

“…All of our oracles have Ω(log n) query and updates since we handle root-to-leaf paths in the decomposition tree. It would be interesting to study whether this can be avoided, as done in the vertex-tovertex case, where approximate distance oracles with faster query times exist (see e.g., [11,14,2] and references therein). Another interesting question that arises is that of faster dynamic prefix minimum data structures.…”

Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

“…Many results reporting (1 + ε)approximate distances with various tradeoffs exist, all with (nearly) linear size and polylogarithmic, or even O(1/ε) query-time [23,15,14,26]. Gu and Xu [13] presented a size O(n polylog n) distance oracle capable of reporting (1 + ε)-approximate distances in time O(1). While their query time is a constant independent of ε, the preprocessing time and space are nearly linear, but with an exponential dependency on (1/ε).…”

Constant query time (1 + ϵ)-approximate distance oracle for planar graphs

“…A preliminary version of the paper appeared in the Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC 2015)[8] …”