Engineering Fully Dynamic Convex Hulls

TL;DR

This paper introduces a new fully dynamic convex hull algorithm that efficiently handles insertions and deletions, supports complex queries, and outperforms existing methods in practical and update-heavy scenarios.

Contribution

The authors develop a robust, efficient fully dynamic convex hull algorithm combining advanced data structures, with proven theoretical bounds and superior practical performance.

Findings

Achieves amortised update time of O(log n log log n) and query time of O(log^2 n)

Outperforms existing techniques on polynomially large data sets

Remains stable and robust across various real-world data sets

Abstract

We present a new fully dynamic algorithm for maintaining convex hulls under insertions and deletions while supporting geometric queries. Our approach combines the logarithmic method with a deletion-only convex hull data structure, achieving amortised update times of $O(\log n \log \log n)$ and query times of $O(\log^2 n)$. We provide a robust and non-trivial implementation that supports point-location queries, a challenging and non-decomposable class of convex hull queries. We evaluate our implementation against the state of the art, including a new naive baseline that rebuilds the convex hull whenever an update affects it. On hulls that include polynomially many data points (e.g. $\Theta(n^\varepsilon)$ for some $\varepsilon$), such as the ones that often occur in practice, our method outperforms all other techniques. Update-heavy workloads strongly favour our approach, which is in line with our theoretical guarantees. Yet, our method remains competitive all the way down to when the update to query ratio is $1$ to $10$. Experiments on real-world data sets furthermore reveal that existing fully dynamic techniques suffer from significant robustness issues. In contrast, our implementation remains stable across all tested inputs.