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 the Word-RAM model, conditioned on the hardness of either the 3SUM problem or the Online Matrix-Vector Mutliplication 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. 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. The problems we consider can all be solved in O(n log n) time in the static case and their dynamic versions have mostly 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 this also holds for an important special case of Klee's measure problem in R^3 known as the Hypervolume Indicator problem.