Online unit clustering in higher dimensions

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of $n$ points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in $\mathbb{R}^d$ using the $L_\infty$ norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension $d$. We also give a randomized online algorithm with competitive ratio $O(d^2)$ for Unit Clustering}of integer points (i.e., points in $\mathbb{Z}^d$, $d\in \mathbb{N}$, under $L_{\infty}$ norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least $2^d$. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can be replaced with $n$ replicas of $K$, and it is completely saturated if it is $n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced by $n-1$ replicas of $K$ without uncovering a portion of the space, and it is completely reduced if it is $n$-reduced for each $n\ge 1$. We prove that every body $K$ in $d$-dimensional Euclidean or hyperbolic space admits both an $n$-saturated packing and an $n$-reduced covering with replicas of $K$. Under some assumptions on $K\subset \mathbb{E}^d$ (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of $n$-saturated packings and $n$-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a $d+2$-reduced covering of $\mathbb{E}^d$ with congruent balls, and we produce some density bounds for the $n$-saturated packings and $n$-reduced coverings of the plane with congruent circles

Besicovitch type properties in metric spaces

We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, with emphasis on geometrically doubling spaces.Comment: 14 page

Volumes of balls in Riemannian manifolds and Uryson width

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $\epsilon_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.Comment: 26 page