## Fractals for Secondary Key Retrieval

### Cached

### Download Links

Citations: | 153 - 20 self |

### BibTeX

@MISC{Faloutsos_fractalsfor,

author = {Christos Faloutsos and Shari Roseman},

title = {Fractals for Secondary Key Retrieval},

year = {}

}

### OpenURL

### Abstract

In this paper we propose the use of fractals and especially the Hilbert curve, in order to design good distance-preserving mappings. Such mappings improve the performance of secondary-key- and spatial- access methods, where multi-dimensional points have to be stored on an 1-dimensional medium (e.g., disk). Good clustering reduces the number of disk accesses on retrieval, improving the response time. Our experiments on range queries and nearest neighbor queries showed that the proposed Hilbert curve achieves better clustering than older methods ("bit-shuffling", or Peano curve), for every situation we tried.

### Citations

1979 |
The Fractal Geometry of Nature
- Mandelbrot
(Show Context)
Citation Context ...preserve the distance, that is, points which are close in space and represent similar data should be stored close together in the linear order. The space-filling curves are a special case of fractals =-=[10]-=-. Distance preserving mappings are useful in two general situations: a) when we have to manage multi-dimensional points (geometric data) and b) when we have to store a k-dimensional array on the disk.... |

398 | The grid file: an adaptable, symmetric multikey file structure
- Nievergelt, Hinterberger, et al.
- 1984
(Show Context)
Citation Context ...alary Figure 1.1. Records of employees correspond to multidimensional points. preserving mapping achieves. Moreover, older secondary key methods can benefit from the present work: E.g., the grid file =-=[12]-=- needs to store a k-d directory on the disk; a good way to store it would definitely reduce the number of disk accesses on partial match and range queries. Additional applications for distance-preserv... |

179 |
Spatial Query Processing in an Object-Oriented Database System
- Orenstein
- 1986
(Show Context)
Citation Context ... store many thousands of rectangles [15] representing electronic gates and higher level elements. Rectangles can be divided in pieces; each piece is assigned a "z-value", according to the Peano curve =-=[13]-=-. 3) Computer vision and robotics. 4) Retrieval in large knowledge bases [9], [11], [18]. 5) Clustering of data in data base machines [3], [4]. 6) In numerical analysis, large k-d arrays that have to ... |

154 | Direct spatial search on pictorial databases using packed R-trees
- Roussopoulos, Leifker
(Show Context)
Citation Context ...queries. Additional applications for distance-preserving mappings include the following: 1) Cartography. Maps could be stored and searched electronically, answering efficiently geometric queries [5], =-=[17]-=-. In the TIGER project at the U.S. Bureau of Census, the map of the United States will eventually be stored in a database [19]; the "bit-shuffling" method is used for a distance-preserving mapping. 2)... |

151 |
A class of data structures for associative searching
- Orenstein, Merett
- 1984
(Show Context)
Citation Context ... with k attributes corresponds to a point in a k-d space (see Figure 1.1). A class of secondary key methods is based on the idea of distance preserving mappings, using "bit-shuffling" or "z-ordering" =-=[14]-=-, which is essentially the Peano curve. The binary representations of the attribute values are combined ("shuffled") to create a single value, the "zvalue", which can be used as the primary key in con... |

69 |
Partial match retrieval algorithms
- Rivest
- 1976
(Show Context)
Citation Context ...rtial match retrieval. The hypothesis is that the proposed Hilbert curve yields a better distance preserving mapping, because it avoids long jumps between points. Among the different types of queries =-=[16]-=-, we focus on range queries and nearest neighbor queries; based on them, we derive measures to quantify the "goodness" of a distance preserving mapping. The structure of the paper is as follows: Secti... |

63 |
Space-filling curves: Their generation and their application to bandwidth reduction
- Bially
- 1969
(Show Context)
Citation Context ...nth point in the ordering.) The H1 algorithm has complexity O(n2) and may only be used for two-dimensional curves. An O(n) algorithm to compute the h-values of the Hilbert curve was derived by Bially =-=[2]-=-. He describes a way to create state transition machines to determine h-values for any dimensionality Hilbert curves. Fig. A.2 shows a state transition machine he created for the three-dimensional Hil... |

42 | Magic: A VLSI layout system - Ousterhout, Hamachi, et al. - 1984 |

41 |
Algorithms and Data Structures
- Wirth
- 1986
(Show Context)
Citation Context ...he same rotation and reflection pattern at each vertex of the basic curve. Fig. 2.3 also shows the Hilbert curves of order 2 and 3. An algorithm to draw this curve is given in Griffiths [8] and Wirth =-=[20]-=-. 0 1 2 3 H1 H2 H3 Fig. 2.3 Hilbert curves of order 1, 2, and 3 The path of a space-filling curve imposes a linear ordering, which may be calculated by starting at one end of the curve and following t... |

35 |
Logic for data description
- Kowalski
- 1978
(Show Context)
Citation Context ...gher level elements. Rectangles can be divided in pieces; each piece is assigned a "z-value", according to the Peano curve [13]. 3) Computer vision and robotics. 4) Retrieval in large knowledge bases =-=[9]-=-, [11], [18]. 5) Clustering of data in data base machines [3], [4]. 6) In numerical analysis, large k-d arrays that have to be stored on disk [6]. 7) In computational geometry. Heuristics in geometric... |

31 |
Codes for Partial Match and Range Queries
- Faloutsos, ‘‘Gray
- 1988
(Show Context)
Citation Context ...salesman problem, the cities are ordered in a linear ordering, and visited in this order. [1]. The three space-filling curves we compare are the Peano curve, the reflected binary graycode (RBG) curve =-=[7]-=- and the Hilbert curve. The first two have been used before as distance preserving mappings for partial match retrieval. The hypothesis is that the proposed Hilbert curve yields a better distance pres... |

25 |
Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core
- Duff
- 1984
(Show Context)
Citation Context ...on and robotics. 4) Retrieval in large knowledge bases [9], [11], [18]. 5) Clustering of data in data base machines [3], [4]. 6) In numerical analysis, large k-d arrays that have to be stored on disk =-=[6]-=-. 7) In computational geometry. Heuristics in geometric complexity problems use distancepreserving mappings: E.g., to solve the traveling salesman problem, the cities are ordered in a linear ordering,... |

15 |
Database Structure and Manipulation Capabilities of a Picture Database Management System (PICDMS
- Chock, Cardenas, et al.
(Show Context)
Citation Context ...ange queries. Additional applications for distance-preserving mappings include the following: 1) Cartography. Maps could be stored and searched electronically, answering efficiently geometric queries =-=[5]-=-, [17]. In the TIGER project at the U.S. Bureau of Census, the map of the United States will eventually be stored in a database [19]; the "bit-shuffling" method is used for a distance-preserving mappi... |

14 |
N-trees: Large ordered Indexes for Multi-dimensional Space
- White
- 1981
(Show Context)
Citation Context ...and searched electronically, answering efficiently geometric queries [5], [17]. In the TIGER project at the U.S. Bureau of Census, the map of the United States will eventually be stored in a database =-=[19]-=-; the "bit-shuffling" method is used for a distance-preserving mapping. 2) Computer-Aided Design (CAD). For example, VLSI design systems need to store many thousands of rectangles [15] representing el... |

13 |
An Algorithm for Displaying a Class of Space-filling Curves
- Griffiths
- 1986
(Show Context)
Citation Context ...by following the same rotation and reflection pattern at each vertex of the basic curve. Fig. 2.3 also shows the Hilbert curves of order 2 and 3. An algorithm to draw this curve is given in Griffiths =-=[8]-=- and Wirth [20]. 0 1 2 3 H1 H2 H3 Fig. 2.3 Hilbert curves of order 1, 2, and 3 The path of a space-filling curve imposes a linear ordering, which may be calculated by starting at one end of the curve ... |

9 |
Heuristics Based on Spacefilling Curves for
- Bartholdi, Platzman
- 1986
(Show Context)
Citation Context ...ry. Heuristics in geometric complexity problems use distancepreserving mappings: E.g., to solve the traveling salesman problem, the cities are ordered in a linear ordering, and visited in this order. =-=[1]-=-. The three space-filling curves we compare are the Peano curve, the reflected binary graycode (RBG) curve [7] and the Hilbert curve. The first two have been used before as distance preserving mapping... |

9 |
A Superjoin Algorithm for Deductive Databases
- Thom, Ramamohanarao, et al.
- 1986
(Show Context)
Citation Context ...elements. Rectangles can be divided in pieces; each piece is assigned a "z-value", according to the Peano curve [13]. 3) Computer vision and robotics. 4) Retrieval in large knowledge bases [9], [11], =-=[18]-=-. 5) Clustering of data in data base machines [3], [4]. 6) In numerical analysis, large k-d arrays that have to be stored on disk [6]. 7) In computational geometry. Heuristics in geometric complexity ... |

6 |
An Experimental Relational Data Base System Based on
- Minker
- 1978
(Show Context)
Citation Context ...level elements. Rectangles can be divided in pieces; each piece is assigned a "z-value", according to the Peano curve [13]. 3) Computer vision and robotics. 4) Retrieval in large knowledge bases [9], =-=[11]-=-, [18]. 5) Clustering of data in data base machines [3], [4]. 6) In numerical analysis, large k-d arrays that have to be stored on disk [6]. 7) In computational geometry. Heuristics in geometric compl... |

4 | Database Machine Morphology - Boral, Redfield - 1985 |

2 |
A Reliable Parallel Backend Using Multiattribute Clustering and
- Cheiney, Faudemay, et al.
- 1986
(Show Context)
Citation Context ...ce is assigned a "z-value", according to the Peano curve [13]. 3) Computer vision and robotics. 4) Retrieval in large knowledge bases [9], [11], [18]. 5) Clustering of data in data base machines [3], =-=[4]-=-. 6) In numerical analysis, large k-d arrays that have to be stored on disk [6]. 7) In computational geometry. Heuristics in geometric complexity problems use distancepreserving mappings: E.g., to sol... |