Error Resilient Space Partitioning

TL;DR

This work introduces error-resilient space partitions: partitioning $\,\mathbb{R}^d$ into bounded, connected tiles colored with $k$ colors so that equi-colored tiles are at distance at least $t$, enabling robust rounding under small measurement errors when the color is known. It establishes that $k=d+1$ colors are necessary and sufficient for positive resilience in general, and provides both upper and lower bounds on $t$ across dimensions, including sharp asymptotics for $d\in\{2,3,8,24\}$ as $k\to\infty$ via sphere-packing densities. The paper also delivers concrete constructions (brick-wall tilings, honeycomb-inspired tilings, close-packing tilings, and a dimension-reducing tiling) to realize substantial $t$, and presents topological proofs (Bapat’s connector-free lemma and Čech cohomology) to justify the minimal color requirement. Together, these results illuminate the tradeoffs between color count and error resilience, linking discrete geometry, topology, and sphere packing to rounding resilience in high dimensions and suggesting directions for practical tiling schemes in applications requiring robust quantization. The findings have potential implications for error-resilient rounding schemes and robust discretization in computational geometry and signal processing.

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 Čech cohomology.

Figures (4)

• Figure 1: A 3-colored hexagonal tiling of the plane, and a maximal non-intersecting inflation of the tiles colored 1
• Figure 2: The 2-dimensional brick wall tiling for $k=3,4,5$ colors. The ratio between the width and the height of each tile is $2:(k-2)$.
• Figure 3: The 3-dimensional brick wall tiling
• Figure 4: The honeycomb of rectangles tiling for $d=2$ and $k=16$ colors. The boundaries of the basic 'large rectangles' are depicted in bold. The placement of the tiles in each single color resembles the honeycomb lattice.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4: Sperner's lemma
• Proposition 3.5
• Definition 3.6
• Proposition 3.7
• Remark 3.8