Recursive tilings and space-filling curves with little fragmentation

TL;DR

This paper introduces the Arrwwid number to evaluate recursive tilings and space-filling curves, presenting constructions with optimal numbers for higher dimensions to improve spatial data access efficiency.

Contribution

It defines the Arrwwid number and constructs recursive tilings and space-filling curves with optimal values, especially for dimensions three and above.

Findings

Regular cube tilings are suboptimal for d >= 3

New constructions achieve better Arrwwid numbers in higher dimensions

Optimal Arrwwid numbers improve data access patterns

Abstract

This paper defines the Arrwwid number of a recursive tiling (or space-filling curve) as the smallest number w such that any ball Q can be covered by w tiles (or curve sections) with total volume O(vol(Q)). Recursive tilings and space-filling curves with low Arrwwid numbers can be applied to optimise disk, memory or server access patterns when processing sets of points in d-dimensional space. This paper presents recursive tilings and space-filling curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube tilings and space-filling curves cannot have optimal Arrwwid number, and we see how to construct alternatives with better Arrwwid numbers.