Entropy, Triangulation, and Point Location in Planar Subdivisions

TL;DR

This paper introduces an efficient point location data structure for planar subdivisions that adapts to known query distributions, achieving near-optimal expected query times with linear size and near-minimum entropy triangulation.

Contribution

It presents a new data structure for point location in planar subdivisions that is optimized for known query distributions, with provably near-minimum expected comparison counts.

Findings

Expected query time within a constant factor of the lower bound

Preprocessing runs in O(n log n) time with O(n) space

Uses Steiner triangulation with near-minimum entropy

Abstract

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.