Expected Size of Random Tukey Layers and Convex Layers

TL;DR

This paper analyzes the expected number of vertices in the first few Tukey and convex layers of a random planar point set, providing bounds and implications for computational geometry algorithms.

Contribution

It derives bounds on the expected size of Tukey and convex layers for random points in a convex polygon, advancing understanding of their average-case complexity.

Findings

Expected vertices on Tukey layers: O(kt log(n/k))

Expected vertices on convex layers: O(kt^3 log(n/(kt^2)))

Lower bounds of Ω(t log n) for specific cases

Abstract

We study the Tukey layers and convex layers of a planar point set, which consists of $n$ points independently and uniformly sampled from a convex polygon with $k$ vertices. We show that the expected number of vertices on the first $t$ Tukey layers is $O\left(kt\log(n/k)\right)$ and the expected number of vertices on the first $t$ convex layers is $O\left(kt^{3}\log(n/(kt^2))\right)$. We also show a lower bound of $\Omega(t\log n)$ for both quantities in the special cases where $k=3,4$. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.