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Foundations of Computational Mathematics

, Volume 19, Issue 4, pp 843–868 | Cite as

On the Number of Face-Connected Components of Morton-Type Space-Filling Curves

  • Carsten BursteddeEmail author
  • Johannes Holke
  • Tobin Isaac
Article

Abstract

The Morton- or z-curve is one example for a space-filling curve: Given a level of refinement \(L \in \mathbb {N}_0\), it maps the interval \([0, 2^{dL}) \cap \mathbb {Z}\) one-to-one to a set of d-dimensional cubes of edge length \(2^{-L}\) that form a subdivision of the unit cube. Similar curves have been proposed for triangular and tetrahedral unit domains. In contrast to the Hilbert curve that is continuous, the Morton-type curves produce jumps between disconnected subdomains. We prove that any contiguous subinterval of the curve divides the domain into a bounded number of face-connected subdomains. For the hypercube case in arbitrary dimension, the subdomains are star-shaped and the bound is indeed two. For the simplicial case in dimension 2, the bound is \(2(L - 1)\), and in dimension 3 it is \(2L + 1\), where L is the depth of refinement. We supplement the paper with theoretical and computational studies on the distribution of the number of jumps. For the hypercube curve, we can characterize the distribution by the fraction of segments of a given length that have no jump, and find that the fraction has a lower bound of \(1/(2^d -1)\) and an asymptotic upper bound of 1 / 2. For the simplicial curve, over 90% of all segments have three components or less.

Keywords

Space-filling curve Adaptive mesh refinement Morton code 

Mathematics Subject Classification

65M50 65D18 

Notes

Acknowledgements

Burstedde would like to thank Andreas Dedner for the invitation to the ICMS workshop on Galerkin methods with applications in weather and climate forecasting, which provided motivation to get going proving this conjecture. The authors would like to thank Michael Bader and Herman Haverkort for suggesting additional relevant literature. Burstedde and Holke acknowledge travel support by the Hausdorff Center for Mathematics (HCM) at Bonn University funded by the German Research Foundation (DFG). Isaac gratefully acknowledges the support of the Intel Parallel Computing Center at the University of Chicago. Holke gratefully acknowledges the scholarship support by the Bonn International Graduate School for Mathematics (BIGS) as part of HCM. We would also like to thank two anonymous referees for their thoughtful and constructive comments.

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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Carsten Burstedde
    • 1
    Email author
  • Johannes Holke
    • 2
  • Tobin Isaac
    • 3
  1. 1.Institut für Numerische Simulation (INS)BonnGermany
  2. 2.German Aerospace Center (DLR)KölnGermany
  3. 3.College of ComputingGeorgia Institute of TechnologyAtlantaUSA

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