[Paper Review] Recursive tilings and space-filling curves with little fragmentation

Herman Haverkort|arXiv (Cornell University)|Feb 9, 2010

Algorithms and Data Compression10 references4 citations

TL;DR

This paper introduces recursive tilings and space-filling curves with optimal Arrwwid numbers, minimizing the number of tiles or curve fragments needed to cover any query region with total volume proportional to the region’s volume. It proves that for d ≥ 3, regular cube tilings are suboptimal and presents constructions with lower Arrwwid numbers, achieving optimal worst-case performance for spatial data access patterns in disk, memory, and distributed systems.

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.

Motivation & Objective

• To address the challenge of minimizing disk, memory, or network I/O during spatial queries by reducing fragmentation in data layout.
• To define and optimize the Arrwwid number as a measure of worst-case fragmentation in recursive tilings and space-filling curves.
• To demonstrate that regular cube-based tilings and curves are suboptimal in 3D and higher dimensions, and to construct alternatives with better Arrwwid numbers.
• To explore how the order of tile storage affects query efficiency, particularly in reducing seek operations during data retrieval.
• To analyze the trade-off between worst-case performance (Arrwwid number) and average-case access cost, especially in practical storage and rendering scenarios.

Proposed method

• Defines the Arrwwid number as the smallest number a such that any ball Q can be covered by a tiles with total volume O(volume(Q)).
• Proposes recursive tilings where each tile is subdivided recursively, enabling adaptive resolution based on query size.
• Introduces the concept of scanning order (e.g., coil order, Dekking’s order) to minimize disk head movements between consecutive tiles in a query cover.
• Constructs alternative tilings—such as the recursified shifted cube tiling and the Kochel curve—for d ≥ 3 that achieve lower Arrwwid numbers than regular cube tilings.
• Uses geometric and combinatorial analysis to bound the cover ratio and prove optimality of the proposed constructions.
• Applies base-5 coordinate indexing to enable efficient tile location in high-dimensional recursive tilings, supporting practical deployment.

Experimental results

Research questions

• RQ1 What is the minimal Arrwwid number achievable for recursive tilings and space-filling curves in d ≥ 3 dimensions?
• RQ2 Can regular cube-based tilings and curves be improved upon in terms of Arrwwid number, especially in higher dimensions?
• RQ3 How does the scanning order of tiles affect the number of disk head movements during spatial query processing?
• RQ4 What is the trade-off between worst-case Arrwwid number and average-case performance in practical I/O scenarios?
• RQ5 Can efficient algorithms be designed to locate a point in a recursive tiling using coordinate digit analysis, particularly in high dimensions?

Key findings

• For d ≥ 3, regular cube tilings and space-filling curves cannot achieve optimal Arrwwid numbers, as they inherently suffer from higher fragmentation.
• The paper constructs recursive tilings and space-filling curves with Arrwwid number three in 2D and optimal Arrwwid numbers in higher dimensions, improving upon the standard four of Hilbert and Peano curves.
• The recursified shifted cube tiling achieves a lower Arrwwid number than regular cube tilings in d ≥ 3, demonstrating that non-regular recursive structures are necessary for optimality.
• Dekking’s scanning order ensures that neighboring tiles in a recursive tiling are close in storage order, reducing seek overhead when covering a query region.
• The cover ratio for query regions can be bounded by 36/π ≈ 11.46 for certain configurations, significantly better than naive bounds when scanning order is considered.
• While Arrwwid number three offers theoretical optimality, preliminary experiments show only up to 10% performance difference between curves with Arrwwid number three and four, suggesting practical gains may be limited in 2D.