Dynamic Geometric Set Cover, Revisited

TL;DR

This paper introduces new dynamic geometric set cover data structures in 1D and 2D, significantly improving update times and extending capabilities to weighted versions, with applications in computational geometry.

Contribution

It presents the first polylogarithmic update time structure for approximate dynamic interval set cover and improves update times for unit-square, square, and halfplane set covers, including weighted cases.

Findings

Polylogarithmic update time for (1+ε)-approximate dynamic interval set cover.

Substantially improved update times for unit-square and square set covers.

First sublinear results for weighted dynamic geometric set cover.

Abstract

Geometric set cover is a classical problem in computational geometry, which has been extensively studied in the past. In the dynamic version of the problem, points and ranges may be inserted and deleted, and our goal is to efficiently maintain a set cover solution (satisfying certain quality requirement). In this paper, we give a plethora of new dynamic geometric set cover data structures in 1D and 2D, which significantly improve and extend the previous results: 1. The first data structure for $(1+\varepsilon)$-approximate dynamic interval set cover with polylogarithmic amortized update time. Specifically, we achieve an update time of $O(\log^3 n/\varepsilon)$, improving the $O(n^\delta/\varepsilon)$ bound of Agarwal et al. [SoCG'20], where $\delta>0$ denotes an arbitrarily small constant. 2. A data structure for $O(1)$-approximate dynamic unit-square set cover with $2^{O(\sqrt{\log n})}$ amortized update time, substantially improving the $O(n^{1/2+\delta})$ update time of Agarwal et al. [SoCG'20]. 3. A data structure for $O(1)$-approximate dynamic square set cover with $O(n^{1/2+\delta})$ randomized amortized update time, improving the $O(n^{2/3+\delta})$ update time of Chan and He [SoCG'21]. 4. A data structure for $O(1)$-approximate dynamic 2D halfplane set cover with $O(n^{17/23+\delta})$ randomized amortized update time. The previous solution for halfplane set cover by Chan and He [SoCG'21] is slower and can only report the size of the approximate solution. 5. The first sublinear results for the \textit{weighted} version of dynamic geometric set cover. Specifically, we give a data structure for $(3+o(1))$-approximate dynamic weighted interval set cover with $2^{O(\sqrt{\log n \log\log n})}$ amortized update time and a data structure for $O(1)$-approximate dynamic weighted unit-square set cover with $O(n^\delta)$ amortized update time.