[Paper Review] An inventory of three-dimensional Hilbert space-filling curves

HJ Herman Haverkort|arXiv (Cornell University)|Sep 11, 2011

Algebraic Geometry and Number Theory9 references18 citations

TL;DR

This paper presents a comprehensive enumeration and analysis of 10,694,807 distinct three-dimensional Hilbert space-filling curves, identifying new curves with superior locality properties. By generalizing Hilbert's recursive construction using multiple recursive rules and optimizing for $L_\infty$, $L_2$, and $L_1$ locality measures, the study achieves significant improvements—reducing the $L_\infty$-locality bound to 9.45, the best known to date, and proving its optimality among poly-Hilbert curves.

ABSTRACT

Hilbert's two-dimensional space-filling curve is appreciated for its good locality properties for many applications. However, it is not clear what is the best way to generalize this curve to filling higher-dimensional spaces. We argue that the properties that make Hilbert's curve unique in two dimensions, are shared by 10694807 structurally different space-filling curves in three dimensions. These include several curves that have, in some sense, better locality properties than any generalized Hilbert curve that has been considered in the literature before.

Motivation & Objective

• To identify and enumerate all structurally distinct three-dimensional space-filling curves that satisfy the minimal properties of Hilbert curves, including vertex-continuity and recursive subdivision.
• To evaluate and compare the locality and geometric quality measures—such as $L_\infty$, $L_2$, $L_1$, surface-to-volume ratio, and bounding-box quality—of these curves.
• To discover new space-filling curves with better locality properties than previously known generalized Hilbert curves in three dimensions.
• To establish tighter upper and lower bounds on the best achievable locality measures for 3D space-filling curves, particularly for $L_\infty$, $L_2$, and $L_1$.
• To demonstrate that poly-Hilbert curves using multiple recursive rules and face-gating can significantly outperform traditional mono-Hilbert curves in terms of locality and bounding-box efficiency.

Proposed method

• The paper defines a generalized framework for constructing 3D space-filling curves using recursive subdivision of a cube into eight octants, with rules specifying the traversal order and recursive application.
• It introduces a formal classification of curves based on properties like vertex-continuity, face-continuity, order-preservation, symmetry, and downward-compatibility.
• The authors use a computational enumeration strategy to systematically explore all valid curve configurations under the defined constraints, leveraging symmetry reduction (modulo rotation, reflection, and reversal).
• For each curve, they compute key quality measures: $L_\infty$, $L_2$, $L_1$ locality, surface-to-volume ratio, and bounding-box quality metrics (WBV, WBS, WS).
• The computation of locality measures is accelerated using an optimized algorithm developed by Sasburg, enabling precise evaluation across millions of candidate curves.
• The study employs a poly-Hilbert curve model that allows multiple recursive rules and gated traversal (e.g., on faces), enabling curves with bends no larger than $\sqrt{2}$, which are shown to maximize locality.

Experimental results

Research questions

• RQ1 What is the complete set of structurally distinct three-dimensional space-filling curves that satisfy the minimal properties of Hilbert curves, such as vertex-continuity and recursive subdivision?
• RQ2 Which 3D space-filling curves achieve the best locality properties in terms of $L_\infty$, $L_2$, and $L_1$ measures compared to previously known constructions?
• RQ3 Can the use of multiple recursive rules and face-gating in the curve construction lead to significant improvements in locality and bounding-box quality?
• RQ4 Is the best achievable $L_\infty$-locality for 3D generalized Hilbert curves bounded, and if so, what is its optimal value?
• RQ5 How do the new curves compare to existing ones in terms of surface-to-volume ratio and bounding-box efficiency, particularly for applications requiring spatial locality?

Key findings

• There exist 10,694,807 structurally distinct three-dimensional Hilbert space-filling curves that satisfy the minimal requirements of vertex-continuity and recursive subdivision, modulo rotation, reflection, and reversal.
• The curve named 'Neptunus' achieves the optimal $L_\infty$-locality of 9.45 among all poly-Hilbert curves, and this value is proven to be the best possible.
• The $L_2$-locality is improved to 18.3, reducing the previous best upper bound of 33.2 by 45%, and the $L_1$-locality to 75.6, reducing the prior bound of 98.4 by 23.6%.
• The surface-to-volume ratio (WS) is reduced to 3.53 for the Neptunus curve, representing the best known value among all 3D space-filling curves.
• The bounding-box quality measures WBV and WBS are improved to 2.89 and 3.53, respectively, with the latter being the tightest known upper bound.
• The study confirms that curves using multiple recursive rules and face-gating—especially those with maximal bending (bends ≤ $\sqrt{2}$)—yield the best locality and geometric quality, significantly outperforming traditional mono-Hilbert curves.