Search operations in databases require special support at the physical level. This is true for conventional databases as well as spatial databases, where typical search operations include the point query (find all objects that contain a given search point) and the region query (find all objects that overlap a given search region). More

An overview is presented of the use of spatial data structures in spatial databases. The focus is on hierarchical data structures, including a number of variants of quadtrees, which sort the data with respect to the space occupied by it. Suchtechniques are known as spatial indexing methods. Hierarchical data structures are based on the principle of recursive decomposition. They are attractive because they are compact and depending on the nature of the data they save space as well as time and also facilitate operations such as search. Examples are given of the use of these data structures in the representation of different data types such as regions, points, rectangles, lines, and volumes.

A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles. Rectangles are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object. They arise in applications in cartography as well as very large-scale integration (VLSI) design rule checking. The different data structures are discussed in terms of how they support the execution of queries involving proximity relations. The focus is on intersection and subset queries. Several types of representations are described. Some are designed for use with the plane-sweep paradigm, which works well for static collections of rectangles. Others are oriented toward dynamic collections. In this case, one representation reduces each rectangle to a point in a higher multidimensional space and treats the problem as one involving point data. The other representation is area based-that is, it depends on the physical extent of each rectangle.

Techniques are presented for navigating between adjacent triangles of greater or equal size in a hierarchical triangle mesh where the triangles are obtained by a recursive quadtree-like subdivision of the underlying space into four equilateral triangles. These techniques are useful in a number of applications including finite element analysis, ray tracing, and the modeling of spherical data. The operations are implemented in a manner analogous to that used in a quadtree representation of data on the two-dimensional plane where the underlying space is tessellated into a square mesh. A new technique is described for labeling the triangles which is useful in implementing the quadtree triangle mesh as a linear quadtree (i.e., a pointer-less quadtree); the navigation can then take place in this linear quadtree. When the neighbors are of equal size, the algorithms take constant time. The algorithms are very efficient, as they make use of just a few bit manipulation operations and can be impl...

There is mounting evidence [Man77, Sch91] that real datasets are statistically self-similar, and thus, `fractal'. This is an important insight since it permits a compact statistical description of spatial datasets; subsequently, as we show, it also forms the basis for the theoretical analysis of spatial access methods, without using the typical, but unrealistic, uniformity assumption. In this paper, we focus on the estimation of the number of quadtree blocks that a real, spatial dataset will require. Using the the well-known Hausdorff fractal dimension, we derive some closed formulas which allow us to predict the number of quadtree blocks, given some few parameters. Using our formulas, it is possible to predict the space overhead and the response time of linear quadtrees/z-ordering [OM88], which are widely used in practice. In order to verify our analytical model, we performed an extensive experimental investigation using several real datasets coming from different domains. In these ex...

This paper shows how scale-based clustering can be done using the Radial Basis Function (RBF) Network, with the RBF width as the scale parameter and a dummy target as the desired output. The technique suggests the "right" scale at which the given data set should be clustered, thereby providing a solution to the problem of determining the number of RBF units and the widths required to get a good network solution. The network compares favorably with other standard techniques on benchmark clustering examples. Properties that are required of non-gaussian basis functions, if they are to serve in alternative clustering networks, are identified. The work on the whole points out an important role played by the width parameter in RBFN, when observed over several scales, and provides a fundamental link to the scale space theory developed in computational vision. The work described here is supported in part by the National Science Foundation under grant ECS-9307632 and in part by ONR Contract N...

Abstract. Emerging database applications require the use of new indexing structures beyond B-trees and R-trees. Examples are the k-D tree, the trie, the quadtree, and their variants. They are often proposed as supporting structures in data mining, GIS, and CAD/CAM applications. A common feature of all these indexes is that they recursively divide the space into partitions. A new extensible index structure, termed SP-GiST is presented that supports this class of data structures, mainly the class of space partitioning unbalanced trees. Simple method implementations are provided that demonstrate how SP-GiST can behave as a k-D tree, a trie, a quadtree, or any of their variants. Issues related to clustering tree nodes into pages as well as concurrency control for SP-GiST are addressed. A dynamic minimum-height clustering technique is applied to minimize disk accesses and to make using such trees in database systems possible and efficient. A prototype implementation of SP-GiST is presented as well as performance studies of the various SP-GiST’s tuning parameters. Keywords: space-partitioning trees, spatial databases, extensible index, generalized search trees, clustering

In a spatial database, an object may extend arbitrarily in space. As a result, many spatial data structures (e.g., the quadtree, the cell tree, the R +-tree) represent an object by partitioning it into multiple, yet simple, pieces, each of which is stored separately inside the data structure. Many operations on these data structures are likely to produce duplicate results because of the multiplicity of object pieces. A novel approach for duplicate processing based on proximity of spatial objects is presented. This is di erent from conventional duplicate elimination in database systems because, with spatial databases, di erent pieces of the same object can span multiple buckets of the underlying data structure. Example algorithms are presented to perform duplicate processing using proximity for a quadtree representation of line segments and arbitrary rectangles. The complexity of the algorithms is seen to depend on a geometric classi cation of di erent instances of the spatial objects. By using proximity and the spatial properties of the objects, the number of disk-I/O requests as well as the run-time storage during duplicate processing can be reduced. 1

A variety of techniques for performing a spatial join are reviewed. Instead of just summarizing the literature and presenting each technique in its entirety, distinct components of the different techniques are described and each is decomposed into an overall framework for performing a spatial join. A typical spatial join technique consists of the following components: partitioning the data, performing internal-memory spatial joins on subsets of the data, and checking if the full polygons intersect. Each technique is decomposed into these components and each component addressed in a separate section so as to compare and contrast similar aspects of each technique. The goal of this survey is to describe the algorithms within each component in detail, comparing and contrasting competing methods, thereby enabling further analysis and experimentation with each component and allowing the best algorithms for a particular situation to be built piecemeal, or, even better, enabling an optimizer to choose which algorithms to use. Categories and Subject Descriptors: H.2.4 [Database Management]: Systems—Query processing; H.2.8 [Database Management]: Database Applications—Spatial databases and GIS

Abstract. The use of bounding structures in the form of orthogonal polygons (also known as rectilinear polygons) with a varying number of vertices in contrast with a minimum bounding rectangle (an orthogonal polygon with just 4 vertices in two dimensions) as an object approximation method is presented. Orthogonal polygons can be used to improve the performance of the re ne step in the lterre ne query processing strategy employed in spatial databases. The orthogonal polygons are represented using the vertex representation implemented as a vertex list. The advantage of the vertex representation implemented as a vertex list is that it can be used to represent orthogonal polygons in arbitrary dimensions using just their vertices. This is in contrast to conventional methods such as the chain code which only work in two dimensions and cannot be extended to deal with higher dimensional data. Algorithms are given for varying the number of vertices used to represent the objects. It is shown that the use of non-trivial orthogonal polygons (i.e., with more than four vertices) is of bene t when a spatial index is used in the lter step for processing spatial queries such as point-in-object and windowing. If no spatial index is used, then all objects must be examined. In this case, many of the objects are small thereby not bene ting from the variation in the number of vertices that they have as the simple bounding box is adequate. 1