## On Multi-Dimensional Hilbert Indexings (1998)

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Venue: | Theory of Computing Systems |

Citations: | 13 - 1 self |

### BibTeX

@INPROCEEDINGS{Alber98onmulti-dimensional,

author = {Jochen Alber and Rolf Niedermeier},

title = {On Multi-Dimensional Hilbert Indexings},

booktitle = {Theory of Computing Systems},

year = {1998},

pages = {329--338},

publisher = {Springer}

}

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### Abstract

Indexing schemes for grids based on space-filling curves (e.g., Hilbert indexings) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multi-dimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling indexing scheme. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability. We define and analyze in a precise mathematical way r-dimensional Hilbert indexings for arbitrary r 2. Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multi-dimensional grids. As we show, Hilbert indexings can be completely described and analyzed by "generating elements of order 1", thus, in comparison with previous work, reducing their structural comp...

### Citations

272 |
Space-Filling Curves
- Sagan
- 1994
(Show Context)
Citation Context ... Analogously a curve C has order k if its range is a cubic grid of order k. Fig. 1 shows the smallest 2D continuous curve indexing a grid of size 4. This curve can be found in Hilbert's original work =-=[9, 16]-=- as a constructing unit for a whole family of curves. Fig. 2 shows the general construction principle for these so-called Hilbert curves: For any ks1 four Hilbert indexings of size 4 k are combined in... |

186 |
Linear clustering of objects with multiple attributes. A
- Jagadish
- 1990
(Show Context)
Citation Context ...e kind of locality, that is, close-by points in the space are mapped to close-by numbers or vice versa. For this purpose, indexing schemes based on space-filling curves have shown to be of high value =-=[1, 4, 5, 7, 6, 8, 10, 11, 12, 13, 14, 17]-=-. 1 In this paper we study Hilbert indexings [9], perhaps the most popular space-filling indexing schemes. Properties of 2D and 3D Hilbert indexings have been extensively studied recently [4, 5, 6, 8,... |

185 | Hilbert r-tree: an improved r-tree using fractals
- Kamel, Faloutsos
- 1994
(Show Context)
Citation Context ...e kind of locality, that is, close-by points in the space are mapped to close-by numbers or vice versa. For this purpose, indexing schemes based on space-filling curves have shown to be of high value =-=[1, 4, 5, 7, 6, 8, 10, 11, 12, 13, 14, 17]-=-. 1 In this paper we study Hilbert indexings [9], perhaps the most popular space-filling indexing schemes. Properties of 2D and 3D Hilbert indexings have been extensively studied recently [4, 5, 6, 8,... |

140 | FractaJs for secondary key retrieval
- Faloutsos, Roseman
- 1989
(Show Context)
Citation Context ...e kind of locality, that is, close-by points in the space are mapped to close-by numbers or vice versa. For this purpose, indexing schemes based on space-filling curves have shown to be of high value =-=[1, 4, 5, 7, 6, 8, 10, 11, 12, 13, 14, 17]-=-. 1 In this paper we study Hilbert indexings [9], perhaps the most popular space-filling indexing schemes. Properties of 2D and 3D Hilbert indexings have been extensively studied recently [4, 5, 6, 8,... |

96 | Über die stetige Abbildung einer Linie auf Flächenstück - Hilbert |

50 | Locality-Preserving Hashing in Multidimensional Spaces
- Indyk, Motwani, et al.
- 1997
(Show Context)
Citation Context |

40 | Spacefilling Curves and the Planar Travelling Salesman Problem - Platzman, Bartholdi - 1989 |

37 | On the Metric Properties of Discrete Space-Filling Curves
- Gotsman, Lindenbaum
- 1996
(Show Context)
Citation Context |

32 | Space-filling curves and their use in the design of geometric data structures - Asano, Ranjan, et al. - 1997 |

31 | Towards optimal locality in meshindexings
- Niedermeier, Reinhardt, et al.
- 1997
(Show Context)
Citation Context |

26 |
Analysis of the Hilbert Curve for Representing Two-Dimensional Space
- Jagadish
- 1997
(Show Context)
Citation Context ...e kind of locality, that is, close-by points in the space are mapped to close-by numbers or vice versa. For this purpose, indexing schemes based on space-filling curves have shown to be of high value =-=[2, 5, 6, 8, 7, 9, 11, 12, 13, 14, 15, 16, 19]-=-. In this paper, we study Hilbert indexings [10], perhaps the most popular space-filling indexing schemes. Properties of 2D and 3D Hilbert indexings have been extensively studied recently [5, 6, 7, 9,... |

17 | Space filling curves and their use in the design of geometric data structures
- Asano, Ranjan, et al.
- 1997
(Show Context)
Citation Context |

12 |
Convergence with Hilbert’s space filling curve
- Butz
- 1969
(Show Context)
Citation Context ...sm that allows combinatorial studies of multi-dimensional Hilbert indexings. For reasons of (geometrical) clearness, we base our formalism on permutations instead of e.g. matrices or other formalisms =-=[2, 3, 4, 15]-=-. So we obtain the following insight: Space-filling curves with Hilbert property can be completely described by simple generating elements and permutations operating on them. Structural questions for ... |

12 | Implementing the hierarchical PRAM on the 2D mesh: Analyses and experiments - Chochia, Cole, et al. - 1995 |

11 |
Space filling curves and mathematical programming
- Butz
- 1968
(Show Context)
Citation Context ...sm that allows combinatorial studies of multi-dimensional Hilbert indexings. For reasons of (geometrical) clearness, we base our formalism on permutations instead of e.g. matrices or other formalisms =-=[2, 3, 4, 15]-=-. So we obtain the following insight: Space-filling curves with Hilbert property can be completely described by simple generating elements and permutations operating on them. Structural questions for ... |

6 |
Recursive 3D mesh indexing with improved locality
- Chochia, Cole
- 1997
(Show Context)
Citation Context |

5 | On the metric properties of discrete space- curves - Gotsman, Lindenbaum - 1996 |

4 |
A three-dimensional Hilbert curve
- Sagan
- 1993
(Show Context)
Citation Context ... 13, 14, 17]. 1 In this paper we study Hilbert indexings [9], perhaps the most popular space-filling indexing schemes. Properties of 2D and 3D Hilbert indexings have been extensively studied recently =-=[4, 5, 6, 8, 11, 12, 13, 15]-=-. However, most of the work so far has focused on empirical studies. Up to now, little attention has been paid to the theoretical study of structural properties of multi-dimensional Hilbert curves, th... |

4 |
Locality properties of discrete space-filling curves: Results with relevance for computer science (in German). Studienarbeit, Universitat Tubingen, WilhelmSchickard -Institut fur Informatik
- Alber
- 1997
(Show Context)
Citation Context ...ves one even can get closer results. Gotsman and Lindenbaum present a result (cf. [9, Theorem 4]), which---by a more detailed investigation---can be improved to the following theorem. It is proved in =-=[1]-=-. Theorem 6. Given any 2D CHP C = fC k j ks1g. Then 6(1 \Gamma O(2 \Gammak ))sL 1 (C k )s6 1 2 6(1 \Gamma O(2 \Gammak ))sL1 (C k )s6 2 5 for all C k 2 C with ord(C k ) = k. 4 This result is completed ... |

3 |
Efficient massively parallel quicksort
- Sanders, Hansch
- 1997
(Show Context)
Citation Context |

2 | Stereoscopic families of permutations, and their applications
- Feige, Krauthgamer
- 1997
(Show Context)
Citation Context |

2 | Locality properties of discrete space- curves: Results with relevance for computer science - Alber - 1997 |

2 | Convergence with Hilbert's space- curve - Butz - 1969 |

1 | Space curves and mathematical programming - Butz - 1968 |