Online Unit Covering in Euclidean Space
Adrian Dumitrescu, Anirban Ghosh, Csaba D. T\'oth

TL;DR
This paper improves online algorithms for covering points with unit balls in high-dimensional Euclidean spaces, providing better competitive ratios, establishing new lower bounds, and achieving optimal ratios for lattice-based inputs.
Contribution
It introduces a deterministic online algorithm with exponential improvement in competitive ratio, establishes new lower bounds, and offers optimal algorithms for lattice point sets.
Findings
Deterministic algorithm with competitive ratio $O(1.321^d)$
Lower bounds of $d+1$ for general $d$
Optimal ratio of 3 for lattice-based points in 2D
Abstract
We revisit the online Unit Covering problem in higher dimensions: Given a set of points in , that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in using Euclidean distance. The current best competitive ratio of an online algorithm, , is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio , thereby sharply improving on the earlier record by a large exponential factor. In particular, the competitive ratios are for the plane and for -space (the previous ratios were and , respectively). For , the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is…
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