Fast and Exact Convex Hull Simplification

TL;DR

This paper introduces fast algorithms for convex hull simplification that efficiently approximate the hull with fewer points, providing exact solutions in special cases and near-linear approximations generally.

Contribution

It presents new algorithms for convex hull simplification with improved time complexity, including exact solutions for convex position and approximation methods for general point sets.

Findings

Exact algorithms for convex position in O(n log^2 n) and O(n log^3 n) time.

Reduction of the general problem to APSP with sub-quadratic algorithms.

Near-linear algorithms yield 2-approximations for arbitrary point sets.

Abstract

Given a point set $P$ in the plane, we seek a subset $Q\subseteq P$, whose convex hull gives a smaller and thus simpler representation of the convex hull of $P$. Specifically, let $cost(Q,P)$ denote the Hausdorff distance between the convex hulls $\mathcal{CH}(Q)$ and $\mathcal{CH}(P)$. Then given a value $\varepsilon>0$ we seek the smallest subset $Q\subseteq P$ such that $cost(Q,P)\leq \varepsilon$. We also consider the dual version, where given an integer $k$, we seek the subset $Q\subseteq P$ which minimizes $cost(Q,P)$, such that $|Q|\leq k$. For these problems, when $P$ is in convex position, we respectively give an $O(n\log^2 n)$ time algorithm and an $O(n\log^3 n)$ time algorithm, where the latter running time holds with high probability. When there is no restriction on $P$, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an $O(n^{2.5302})$ time algorithm when minimizing $k$ and an $O(\min\{n^{2.5302}, kn^{2.376}\})$ time algorithm when minimizing $\varepsilon$, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.