article

SFCGen: A framework for efficient generation of multi-dimensional space-filling curves by recursion

Authors:

John Mellor-CrummeyAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 31, Issue 1

Pages 120 - 148

https://doi.org/10.1145/1055531.1055537

Published: 01 March 2005 Publication History

Abstract

Because they are continuous and self-similar, space-filling curves have been widely used in mathematics to transform multi-dimensional problems into one-dimensional forms. For scientific applications, reordering computation by certain space-filling curves can significantly improve data reuse because of the locality properties of these curves. However, when space-filling curves are used in programs for reordering data, traversal or indexing of the curves must be efficient. To address this problem, we present the table-driven framework SFCGen to efficiently generate multi-dimensional space-filling curves on the fly. The framework is general and easy enough to be used in any application that can be partitioned recursively in multiple dimensions. We describe a movement specification table, a universal turtle algorithm to enumerate points along a space-filling curve, a table-based indexing algorithm to transform coordinates of a point into its position along the curve and an algorithm to pregenerate the table automatically. As examples, we show how high-dimensional Hilbert, Morton, and Peano curves and a two-dimensional Sierpiński curve can be generated with our algorithms. We present performance results for Hilbert, Morton, and Peano curves and compare the efficiency of our curve generation algorithm with the most recent work on generating Hilbert curves. Our experimental results on three modern microprocessor-based platforms show that SFCGen performs up to 63% faster than the most recent recursive algorithm on 2D curve generation and up to a factor of 132 faster than two previous byte-oriented non-recursive implementations. On curve indexing, SFCGen performs as much as a factor of three faster than the byte-oriented implementation. Our results on 4D space-filling curves also show that SFCGen scales very well with curve level for higher dimensional spaces.

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Index Terms

SFCGen: A framework for efficient generation of multi-dimensional space-filling curves by recursion
1. Computing methodologies
  1. Computer graphics
    1. Image manipulation
    2. Shape modeling
      1. Parametric curve and surface models
2. Mathematics of computing
  1. Mathematical software

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 31, Issue 1

March 2005

185 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/1055531

Issue’s Table of Contents

Copyright © 2005 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 March 2005

Published in TOMS Volume 31, Issue 1

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Franco PMullot ROwczarek V(2025)Pareto multi-objective optimization for high locality-preserving space-filling curve identificationSwarm and Evolutionary Computation10.1016/j.swevo.2024.10179792(101797)Online publication date: Feb-2025
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Kountchev RMironov RKountcheva R(2023)Analysis of the Recursive Locally-Adaptive Filtration of 3D Tensor ImagesSymmetry10.3390/sym1508149315:8(1493)Online publication date: 27-Jul-2023
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Liu JSu ZWang CXu Z(2022)Effect of an Adaptive-Density Filling Structure on the Mechanical Properties of FDM Parts with a Variable Cross-SectionMaterials10.3390/ma1524874615:24(8746)Online publication date: 7-Dec-2022
https://doi.org/10.3390/ma15248746
Guan XVan Oosterom PCheng B(2018)A Parallel N-Dimensional Space-Filling Curve Library and Its Application in Massive Point Cloud ManagementISPRS International Journal of Geo-Information10.3390/ijgi70803277:8(327)Online publication date: 15-Aug-2018
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