Engineering Fully Dynamic Convex Hulls

2026-03-31

AI Summary

This work addresses the challenge of efficiently maintaining the convex hull of a dynamically changing point set under frequent insertions and deletions while supporting non-decomposable geometric queries such as point location. By integrating the logarithmic method with a deletion-only convex hull structure, the paper presents the first stable and non-trivial fully dynamic convex hull algorithm. The proposed approach achieves an amortized update time of $O(\log n \log \log n)$ and a query time of $O(\log^2 n)$. Experimental evaluation demonstrates that it significantly outperforms existing techniques on both update-intensive workloads and real-world datasets, offering strong robustness alongside practical performance.

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. $Θ(n^\varepsilon)$ 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.

Problem

• dynamic convex hull
• fully dynamic
• geometric queries
• robustness
• point-location

Innovation

• fully dynamic convex hull
• logarithmic method
• amortised update time
• point-location queries
• robust implementation