Transdichotomous Results in Computational Geometry, II: Offline Search

TL;DR

This paper introduces a new offline algorithm for point location in computational geometry, achieving significantly faster bounds for various problems in the word RAM model, surpassing classic methods.

Contribution

The paper presents a novel offline point location algorithm that improves the bounds for multiple geometric problems in the word RAM model, outperforming previous online solutions.

Findings

Convex hull in 3D can be constructed in n 2^O(sqrt{lglg n}) time.

Improved bounds for planar Voronoi diagrams and triangulation.

Significant speedup over classic O(n log n) algorithms.

Abstract

We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a two-dimensional analog of sorting where one needs to order points with respect to segments. This result implies, for example, that the convex hull of n points in three dimensions can be constructed in (randomized) time n 2^O(sqrt{lglg n}). Similar bounds hold for numerous other geometric problems, such as planar Voronoi diagrams, planar off-line nearest neighbor search, line segment intersection, and triangulation of non-simple polygons. In FOCS'06, we developed a data structure for online point location, which implied a bound of O(n lg n/lglg n) for three-dimensional convex hulls and the other problems. Our current bounds are dramatically better, and a convincing improvement over the classic O(nlg n) algorithms. As in the field of integer sorting, the main challenge is to find ways to manipulate information, while avoiding the online problem (in that case, predecessor search).