Delaunay Triangulations in Linear Time? (Part I)

TL;DR

This paper introduces a simple randomized algorithm for constructing Delaunay triangulations in linear expected time for certain point sets, leveraging nearest neighbor graphs and applicable in higher dimensions under specific conditions.

Contribution

It presents a novel, efficient randomized algorithm for Delaunay triangulation construction that operates in linear expected time under polynomial spread assumptions.

Findings

Runs in linear expected time for points with polynomially bounded spread

Applicable to higher dimensions with bounded spread and linear sample complexity

Uses nearest neighbor graphs for efficient point location

Abstract

We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearest neighbor graphs for point location. Under suitable assumptions, it runs in linear expected time for points in the plane with polynomially bounded spread, i.e., if the ratio between the largest and smallest pointwise distance is polynomially bounded. This also holds for point sets with bounded spread in higher dimensions as long as the expected complexity of the Delaunay triangulation of a sample of the points is linear in the sample size.