Dynamic 3D Convex Hulls Revisited and Applications

TL;DR

This paper revisits and refines dynamic 3D convex hull data structures, achieving faster deletion times and improved overall efficiency for maintaining convex hulls and related geometric queries.

Contribution

It introduces a modified data structure that reduces deletion time to $O(\log^3 n \log\log n)$ while maintaining fast insertion, improving efficiency over previous methods.

Findings

Reduced deletion time to $O(\log^3 n \\log\\log n)$

Maintained $O(\log^2 n)$ amortized insertion time

Improved overall performance when deletions dominate

Abstract

Chan [JACM, 2010] gave a data structure for maintaining the convex hull of a dynamic set of 3D points under insertions and deletions, supporting extreme-point queries. Subsequent refinements by Kaplan, Mulzer, Roditty, Seiferth, and Sharir [DCG, 2020] and Chan [DCG, 2020] preserved the framework while improving bounds. The current best result achieves $O(\log^2 n)$ amortized insertion time, $O(\log^4 n)$ amortized deletion time, and $O(\log^2 n)$ worst-case query time. These techniques also yield dynamic 2D Euclidean nearest neighbor searching via duality, where the problem becomes maintaining the lower envelope of 3D planes for vertical ray-shooting queries. Using randomized vertical shallow cuttings, Kaplan et al. [DCG, 2020] and Liu [SICOMP, 2022] extended the framework to dynamic lower envelopes of general 3D surfaces, obtaining the same asymptotic bounds and enabling nearest neighbor searching under general distance functions. We revisit Chan's framework and present a modified structure that reduces the deletion time to $O(\log^3 n \log\log n)$, while retaining $O(\log^2 n)$ amortized insertion time, at the cost of increasing the query time to $O(\log^3 n / \log\log n)$. When the overall complexity is dominated by deletions, this yields an improvement of roughly a logarithmic factor over previous results.