Online Unit Covering in Euclidean Space

We revisit the online Unit Covering problem in higher dimensions: Given a set of $n$ points in $\mathbb{R}^d$, 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 $\mathbb{R}^d$ using Euclidean distance. The current best competitive ratio of an online algorithm, $O(2^d d \log{d})$, is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio $O(1.321^d)$, thereby sharply improving on the earlier record by a large exponential factor. In particular, the competitive ratios are $5$ for the plane and $12$ for $3$-space (the previous ratios were $7$ and $21$, respectively). For $d=3$, 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 remarkable (and rather unusual). (II) We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in $\mathbb{R}^d$ under the $L_{2}$ norm is at least $d+1$ for every $d \geq 1$. This greatly improves upon the previous best lower bound, $\Omega(\log{d} / \log{\log{\log{d}}})$, due to Charikar et al. (2004). (III) We obtain lower bounds of $4$ and $5$ for the competitive ratio of any deterministic algorithm for online Unit Covering in $\mathbb{R}^2$ and respectively $\mathbb{R}^3$; the previous best lower bounds were both $3$. (IV) When the input points are taken from the square or hexagonal lattices in $\mathbb{R}^2$, we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of $3$.