Succinct Geometric Indexes Supporting Point Location Queries

TL;DR

This paper introduces space-efficient geometric indexes for point location queries that operate in optimal or near-optimal time, significantly reducing space usage compared to previous structures while maintaining query efficiency.

Contribution

The paper presents the first succinct geometric index supporting point location in planar triangulations with optimal or near-optimal query times, using o(n) bits of space.

Findings

Supports point location in O(lg n) time on planar triangulations.

Achieves point location in o(lg n) time for integer coordinates bounded by U.

Provides an implicit data structure with O(lg^2 n) query time.

Abstract

We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence, to support geometric queries in optimal time. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time. We also design three variants of this index. The first supports point location using $\lg n + 2\sqrt{\lg n} + O(\lg^{1/4} n)$ point-line comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location in O(H+1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that use O(n) words or O(n lg n) bits, while saving drastic amounts of space. We then generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in $O(\lg^2 n)$ time.