Error Resilient Space Partitioning

TL;DR

This paper introduces a new space partitioning method in Euclidean space that enables error-resilient rounding by using colored tiles with controlled distances, and characterizes the tradeoffs between the number of colors and tile separation.

Contribution

It provides a novel partitioning framework with theoretical bounds on color and distance tradeoffs, advancing understanding of error-resilient space tilings in discrete geometry.

Findings

Using d+1 colors is necessary and sufficient for positive separation in .

Sharp asymptotic bounds on separation distance for dimensions 3, 4, 8, 24.

Multiple mathematical techniques used to derive bounds.

Abstract

A major research area in discrete geometry is to consider the best way to partition the $d$-dimensional Euclidean space $\mathbb{R}^d$ under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space $\mathbb{R}^d$ to a discrete subset of representative values. Specifically, we study partitions of $\mathbb{R}^d$ into bounded-size tiles colored by one of $k$ colors, such that tiles of the same color have a distance of at least $t$ from each other. Such tilings allow for \emph{error-resilient} rounding, as two points of the same color and distance less than $t$ from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors $k$ and the distance $t$, for various dimensions $d$. On the qualitative side, we show that in $\mathbb{R}^d$, using $k=d+1$ colors is both sufficient and necessary to achieve $t>0$. On the quantitative side, we achieve numerous upper and lower bounds on $t$ as a function of $k$. In particular, for $d=3,4,8,24$, we obtain sharp asymptotic bounds on $t$, as $k \to \infty$. We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat's connector-free lemma, and \v{C}ech cohomology.