Revisiting Random Points: Combinatorial Complexity and Algorithms

TL;DR

This paper studies properties of random point sets in fixed dimensions, providing new concentration results, simple algorithms for geometric structures, and improved complexity bounds for certain problems.

Contribution

It introduces new concentration bounds, simple linear-time algorithms for Delaunay triangulation, MST, and convex hull, and offers a more straightforward proof of the linear complexity of Delaunay triangulation.

Findings

Number of pairs at distance at most r is tightly concentrated within O(n log n)

Linear-time algorithms for Delaunay triangulation, MST, and convex hull

Expected complexity of Delaunay triangulation is linear

Abstract

Consider a set $P$ of $n$ points picked uniformly and independently from $[0,1]^d$ for a constant dimension $d$ -- such a point set is extremely well behaved in many aspects. For example, for a fixed $r \in [0,1]$, we prove a new concentration result on the number of pairs of points of $P$ at a distance at most $r$ -- we show that this number lies in an interval that contains only $O(n \log n)$ numbers. We also present simple linear time algorithms to construct the Delaunay triangulation, Euclidean MST, and the convex hull of the points of $P$. The MST algorithm is an interesting divide-and-conquer algorithm which might be of independent interest. We also provide a new proof that the expected complexity of the Delaunay triangulation of $P$ is linear -- the new proof is simpler and more direct, and might be of independent interest. Finally, we present a simple $\tilde{O}(n^{4/3})$ time algorithm for the distance selection problem for $d=2$.