Conditional Lower Bounds for Dynamic Geometric Measure Problems

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 Pătraşcu [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 Ω( √ 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. discussions about the topic of this paper.