Practical Insertion-Only Convex Hull

TL;DR

This paper introduces a vector-based convex hull data structure optimized for insertion-only scenarios, balancing theoretical bounds with practical performance, and demonstrating superior empirical results over traditional tree-based methods.

Contribution

We develop a novel vector-based convex hull data structure that improves cache performance and practical efficiency, outperforming existing tree-based approaches in real-world and synthetic tests.

Findings

Naive O(h) Graham scan outperforms state-of-the-art methods in practice.

Tree-based methods with O(log h) updates are theoretically sound but less practical.

Vector-based logarithmic method is highly competitive and optimal for large convex hulls.

Abstract

Convex hull data structures are fundamental in computational geometry. We study insertion-only data structures, supporting various containment and intersection queries. When $P$ is sorted by $x$- or $y$-coordinate, convex hulls can be constructed in linear time using classical algorithms such as Graham scan. We investigate a variety of methods tailored to the insertion-only setting. We explore a broad selection of trade-offs involving robustness, memory access patterns, and space usage, providing an extensive evaluation of both existing and novel techniques. Logarithmic-time methods rely on pointer-based tree structures, which suffer in practice due to poor memory locality. Motivated by this, we develop a vector-based solution inspired by Overmars' logarithmic method. Our structure has worse asymptotic bounds, supporting queries in $O(\log^2 n)$ time, but stores data in $O(\log n)$ contiguous vectors, greatly improving cache performance. Through empirical evaluation on real-world and synthetic data sets, we uncover surprising trends. Let $h$ denote the size of the convex hull. We show that a na\"ive $O(h)$ insertion-only algorithm based on Graham scan consistently outperforms both theoretical and practical state-of-the-art methods under realistic workloads, even on data sets with rather large convex hulls. While tree-based methods with $O(\log h)$ update times offer solid theoretical guarantees, they are never optimal in practice. In contrast, our vector-based logarithmic method, despite its theoretically inferior bounds, is highly competitive across all tested scenarios. It is optimal whenever the convex hull becomes large.