Multidimensional Access Methods VOLKER GAEDE IC-Parc, Imperial College, London AND OLIVER GU¨ NTHER Humboldt-Universita¨t, Berlin 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 than ten years of spatial database research have resulted in a great variety of multidimensional access methods to support such operations. We give an overview of that work. After a brief survey of spatial data management in general, we first present the class of point access methods, which are used to search sets of points in two or more dimensions. The second part of the paper is devoted to spatial access methods to handle extended objects, such as rectangles or polyhedra. We conclude with a discussion of theoretical and experimental results concerning the relative performance of various approaches. Categories and Subject Descriptors: H.2.2 [Database Management]: Physical Design—access methods; H.2.4 [Database Management]: Systems; H.2.8 [Database Management]: Database Applications—spatial databases and GIS; H.3.3 [Information Storage and Retrieval]: Information Search and Retrieval— search process, selection process General Terms: Design, Experimentation, Performance Additional Key Words and Phrases: Data structures, multidimensional access methods 1. INTRODUCTION nated in the geosciences and mechani- cal CAD, the range of possible applica- With an increasing number of computer tions has expanded to areas such as applications that rely heavily on multi- robotics, visual perception, autonomous dimensional data, the database commu- navigation, environmental protection, nity has recently devoted considerable and medical imaging [Gu¨ nther and attention to spatial data management. Buchmann 1990]. Although the main motivation origi- The range of interpretation given to This work was partially supported by the German Research Society (DFG/SFB 373) and by the ESPRIT Working Group CONTESSA (8666). Authors’ address: Institut fu¨ r Wirtschaftsinformatik, Humboldt-Universita¨t zu Berlin, Spandauer Str. 1, 10178 Berlin, Germany; email: ^{gaede,guenther}@wiwi.hu-berlin.de&. 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The objects are usually repre- 2.2 Definitions and Queries sented in some vector-based format, and 3. BASIC DATA STRUCTURES 3.1 One-Dimensional Access Methods their relative positions may be explicit 3.2 Main Memory Structures or implicit (i.e., derivable from the in- 4. POINT ACCESS METHODS ternal representation of their absolute 4.1 Multidimensional Hashing positions). Image databases often place 4.2 Hierarchical Access Methods 4.3 Space-Filling Curves for Point Data less emphasis on data analysis. They 5. SPATIAL ACCESS METHODS provide storage and retrieval for unana- 5.1 Transformation lyzed pictorial data, which are typically 5.2 Overlapping Regions represented in some raster format. 5.3 Clipping Techniques developed for the storage 5.4 Multiple Layers 6. COMPARATIVE STUDIES and manipulation of image data can be 7. CONCLUSIONS applied to other media as well, such as infrared sensor signals or sound. In this survey we assume that the goal is to manipulate analyzed multidi- the term spatial data management is mensional data and that unanalyzed just as broad as the range of applica- images are handled only as the source tions. In VLSI CAD and cartography, from which spatial data can be derived. this term refers to applications that rely The challenge for the developers of a mostly on two-dimensional or layered spatial database system lies not so two-dimensional data. VLSI data are much in providing yet another collection usually represented by rectilinear poly- of special-purpose data structures. lines or polygons whose edges are iso- Rather, one has to find abstractions and oriented, that is, parallel to the coordi- architectures to implement generic sys- nate axes. Typical operations include tems, that is, to build systems with ge- intersection and geometric routing neric spatial data-management capabil- [Shekhar and Liu 1995]. Cartographic ities that can be tailored to the data are also two-dimensional with requirements of a particular application points, lines, and regions as basic prim- domain. Important issues in this con- itives. In contrast to VLSI CAD, how- text include the handling of spatial rep- ever, the shapes are often characterized resentations and data models, multidi- by extreme irregularities. Common op- mensional access methods, and pictorial erations include spatial searches and or spatial query languages and their map overlay, as well as distance-related optimization. operations. In mechanical CAD, on the This article is a survey of multidimen- other hand, data objects are usually sional access methods to support search three-dimensional solids. They may be operations in spatial databases. Figure represented in a variety of data formats, 1, which was inspired by a similar including cell decomposition schemes, graph by Lu and Ooi [1993], gives a first constructive solid geometry (CSG), and overview of the diversity of existing boundary representations [Kemper and multidimensional access methods. The Wallrath 1987]. Yet other applications goal is not to describe all of these struc- emphasize the processing of unanalyzed tures, but to discuss the most prominent images, such as X-rays and satellite im- ones, to present possible taxonomies, agery, from which features are ex- and to establish references to other lit- tracted. In those areas, the terms spa- erature. ACM Computing Surveys, Vol. 30, No. 2, June 1998 172 • V. Gaede and O. Gu¨ nther Figure 1. History of multidimensional access methods. Several shorter surveys have been had an impact on the design of multidi- published previously in various Ph.D. mensional access methods. Sections 4 theses such as Ooi [1990], Kolovson and 5 form the core of this survey, pre- [1990], Oosterom [1990], and Schiwietz senting a variety of point access meth- [1993]. Widmayer [1991] gives an over- ods (PAMs) and spatial access methods view of work published before 1991. (SAMs), respectively. Some remarks Like the thesis by Schiwietz, however, about theoretical and experimental his survey is available only in German. analyses are contained in Section 6, and Samet’s books [1989, 1990] present the Section 7 concludes the article. state of the art until 1989. However, they primarily cover quadtrees and re- 2. ORGANIZATION OF SPATIAL DATA lated data structures. Lomet [1991] dis- cusses the field from a systems-oriented 2.1 What Is Special About Spatial? point of view. The remainder of the article is orga- To obtain a better understanding of the nized as follows. Section 2 discusses requirements in spatial database sys- some basic properties of spatial data tems, we first discuss some basic prop- and their implications for the design erties of spatial data. First, spatial data and implementation of spatial data- have a complex structure. A spatial bases. Section 3 gives an overview of data object may be composed of a single some traditional data structures that point or several thousands of polygons, ACM Computing Surveys, Vol. 30, No. 2, June 1998 Multidimensional Access Methods • 173 arbitrarily distributed across space. It one needs special multidimensional ac- is usually not possible to store collec- cess methods. The main problem in the tions of such objects in a single rela- design of such methods, however, is tional table with a fixed tuple size. Sec- that there exists no total ordering ond, spatial data are often dynamic. among spatial objects that preserves Insertions and deletions are interleaved spatial proximity. In other words, there with updates, and data structures used is no mapping from two- or higher-di- in this context have to support this dy- mensional space into one-dimensional namic behavior without deteriorating space such that any two objects that are over time. Third, spatial databases tend spatially close in the higher-dimen- to be large. Geographic maps, for exam- sional space are also close to each other ple, typically occupy several gigabytes in the one-dimensional sorted sequence. of storage. The integration of secondary This makes the design of efficient ac- and tertiary memory is therefore essen- cess methods in the spatial domain tial for efficient processing [Chen et al. much more difficult than in traditional 1995]. Fourth, there is no standard al- databases, where a broad range of effi- gebra defined on spatial data, although cient and well-understood access meth- several proposals have been made in the ods is available. Examples for such one- past [Egenhofer 1989; Gu¨ ting 1989; dimensional access methods (also called Scholl and Voisard 1989; Gu¨ ting and single key structures, although that Schneider 1993]. This means in particu- term is somewhat misleading) include lar that there is no standardized set of the B-tree [Bayer and McCreight 1972] base operators. The set of operators de- and extendible hashing [Fagin et al. pends heavily on the given application 1979]; see Section 3.1 for a brief discus- domain, although some operators (such sion.