More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time

TL;DR

This paper introduces a novel dynamic data structure for geometric set cover problems that maintains an approximate solution efficiently under insertions and deletions, specifically for axis-aligned squares in 2D, disks in 2D, and halfspaces in 3D.

Contribution

It presents the first dynamic data structure with sublinear update time for 2D axis-aligned squares and extends techniques to disks and halfspaces, also providing improved static algorithms.

Findings

Achieves $O(n^{2/3+ ext{small }\delta})$ randomized update time for 2D axis-aligned squares.

Provides sublinear amortized update time for disks in 2D and halfspaces in 3D.

Offers an optimal $O(n ext{log} n)$ randomized algorithm for static set cover in 2D disks and 3D halfspaces.

Abstract

We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+\delta})$ for an arbitrarily small constant $\delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(n\log n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(n\log n(\log\log n)^{O(1)})$ result [SoCG 2020].