Randomized Triangle Algorithms for Convex Hull Membership

TL;DR

This paper introduces randomized modifications to the triangle algorithm for convex hull membership testing, leveraging chaos game concepts to improve efficiency while maintaining similar complexity bounds.

Contribution

It proposes two novel randomized variants of the triangle algorithm that incorporate relaxed steps inspired by the Sierpinski triangle, enhancing the effectiveness of convex hull membership testing.

Findings

Expected complexity bounds match the deterministic version.

Randomized iterates can lead to more effective pivots.

The approach integrates chaos game principles into convex hull algorithms.

Abstract

We present randomized versions of the {\it triangle algorithm} introduced in \cite{kal14}. The triangle algorithm tests membership of a distinguished point $p \in \mathbb{R} ^m$ in the convex hull of a given set $S$ of $n$ points in $\mathbb{R}^m$. Given any {\it iterate} $p' \in conv(S)$, it searches for a {\it pivot}, a point $v \in S$ so that $d(p',v) \geq d(p,v)$. It replaces $p'$ with the point on the line segment $p'v$ closest to $p$ and repeats this process. If a pivot does not exist, $p'$ certifies that $p \not \in conv(S)$. Here we propose two random variations of the triangle algorithm that allow relaxed steps so as to take more effective steps possible in subsequent iterations. One is inspired by the {\it chaos game} known to result in the Sierpinski triangle. The incentive is that randomized iterates together with a property of Sierpinski triangle would result in effective pivots. Bounds on their expected complexity coincides with those of the deterministic version derived in \cite{kal14}.