Conditional Lower Bounds for Dynamic Geometric Measure Problems

TL;DR

This paper establishes new polynomial conditional lower bounds for various dynamic geometric measure problems in computational geometry, based on the hardness of well-known problems like 3SUM, APSP, and OMV, highlighting fundamental computational limitations.

Contribution

It provides the first set of conditional lower bounds for dynamic geometric measure problems, connecting them to classic hardness assumptions and extending understanding of their computational complexity.

Findings

Lower bounds for counting extremal points in R^3

Lower bounds for variants of Klee's Measure Problem

Conditional lower bound for dynamic approximate square set cover

Abstract

We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector Multiplication problem [Henzinger et al., STOC 2015]. In particular we get lower bounds in the incremental and fully-dynamic settings for counting maximal or extremal points in R^3, different variants of Klee's Measure Problem, problems related to finding the largest empty disk in a set of points, and querying the size of the i'th convex layer in a planar set of points. We also answer a question of Chan et al. [SODA 2022] by giving a conditional lower bound for dynamic approximate square set cover. While many conditional lower bounds for dynamic data structures have been proven since the seminal work of Patrascu [STOC 2010], few of them relate to computational geometry problems. This is the first paper focusing on this topic. Most problems we consider can be solved in O(n log n) time in the static case and their dynamic versions have only been approached from the perspective of improving known upper bounds. One exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010] gave an unconditional ${\Omega}(\sqrt{n})$ lower bound on the worst-case update time. By a similar approach, we show that such a lower bound also holds for an important special case of Klee's measure problem in R^3 known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.