No-dimensional Tverberg Partitions Revisited
Sariel Har-Peled, Eliot W. Robson
TL;DR
We study algorithmic no-dimensional Tverberg partitions: partitions of a point set in $\mathbb{R}^d$ into blocks of size independent of dimension whose convex hulls intersect a ball of radius $\delta \mathrm{diam}(P)$. Our approach combines mean-sampling and halving techniques to produce partitions with provable guarantees and dimension-free parameters, yielding a randomized linear-time algorithm with an alteration step and a deterministic $O(dn\log n)$ algorithm. We derive two key applications: a no-dimensional centerball, providing a dimension-free centerpoint-like ball computable in $O(nd/\delta^2)$ time, and a no-dimensional weak $\varepsilon$-net with improved constants. The methods also support derandomization, enhancing practicality for high-dimensional data analysis and clustering contexts where dimension independence is crucial.
Abstract
$ \newcommand{\epsA}{\Mhδ} \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \renewcommand¶{P} \newcommand{\diam}Δ \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\PP}{P} \newcommand{\ptq}{q} \newcommand{\pts}{s}$Given a set $P \subset \Re^d$ of $n$ points, with diameter $\diam$, and a parameter $\epsA \in (0,1)$, it is known that there is a partition of $P$ into sets $P_1, \ldots, P_t$, each of size $O(1/\epsA^2)$, such that their convex-hulls all intersect a common ball of radius $\epsA \diam$. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm. We also provide a deterministic algorithm with running time $O( dn \log n)$. Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a ``fuzzy'' centerpoint, and prove a no-dimensional weak $\eps$-net theorem with an improved constant.