Dynamic Geometric Set Cover, Revisited (2111.01196v1)

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.