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Conditional Lower Bounds for Dynamic Geometric Measure Problems

Authors Justin Dallant , John Iacono



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Justin Dallant
  • Université libre de Bruxelles, Belgium
John Iacono
  • Université libre de Bruxelles, Belgium

Acknowledgements

We thank Jean Cardinal for helpful discussions about the topic of this paper.

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Justin Dallant and John Iacono. Conditional Lower Bounds for Dynamic Geometric Measure Problems. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 39:1-39:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.39

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 ℝ³, 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(nlog 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 ℝ², 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 ℝ³ known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Computational geometry
  • Fine-grained complexity
  • Dynamic data structures

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