Modeling Topological Spatial Relations

This paper investigates the processing of spatial queries with topological constraints, for which current database solutions are inappropriate. Topological relations, such as disjoint, meet, overlap, inside, and contains, have been well defined by the 9-intersection, a comprehensive model for binary topological relations. We focus on two types of queries: (1) “Which objects have a stated topological relation with a given spatial object?” and (2) “What is the topological relation between two given spatial objects?” Such queries are processed at two levels of detail. First, Minimum Bounding Rectangles are used as an approximation of the objects’ geometry and as a means of identifying candidates that might satisfy the query. Next, the nine intesections that determine the topological relations between candidate pairs are calculated. We present algorithms for minimizing these computations. Considerable performance can be gained by exploiting the semantics of spatial relations. We also compare the approach for a naive cost model, which assumes that all relations have the same frequency of occurrence, with a refined cost model, which considers the probability of occurrence of the topological relations. The strategies presented here have three key benefits: (1) they are based on a well-defined formalism; (2) they are customizable; and (3) they can take into account important statistical information about the data. 1 . Introduction Geographic Information Systems (GISs) contain high-level spatial operators that are uncommon in conventional database management systems (DBMSs) [1]. Spatial operators appear, for instance, as constraints in spatial queries to select spatial objects. They may include such simple selections as, “Retrieve all lakes in the state of Maine,” or more complex ones like, “Find the shortest path from Boston to Bangor based on travel time.” Traditionally, concerns about processing spatial queries have been addressed primarily at the level of spatial access methods in order to minimize the number of disk accesses by clustering spatial objects according to their spatial neighborhoods. This method supports requests like, “Display a map of Maine showing highways and cities with a population of more than 25,000” where the constraint is that objects intersect with a search window. For other types of queries in which the semantics of the constraints are more complex, the mere support of spatial access methods is insufficient to guarantee efficient processing of queries. Spatial access has to be complemented by methods that consider (1) the semantics of the spatial relations, i.e., how the relations are defined, (2) heuristics for evaluating spatial constraints, and (3) estimates of the distribution, i.e., the probability of occurrence of the relations. The semantics of the spatial relations allow for the inference of a relation from a set of given relations. For example, given that region A is disjoint from region B and that B contains region C, it can be inferred that A is disjoint from C [2]. Heuristics for evaluating spatial constraints are dependent on * This work was performed while on a leave of absence from Università di L’Aquila, Dipartimento di Ingegneria Elettrica, 67040 Poggio di Roio, L’Aquila, Italy. Eliseo Clementini is partially supported by the Italian National Council of Research (CNR) under grant No. 92.01574.PF69. † Jayant Sharma receives partial support from a University of Maine Graduate Research Assistantship and the NCGIA. § Max Egenhofer’s work is partially supported by NSF grant No. IRI-9309230, a grant from Intergraph Corporation, a University of Maine Summer Faculty Research Grant, and the NCGIA through NSF grant no. SBR-8810917. Computers and Graphics 18 (6): 815-822, 1994. the spatial data model and data structures used. An example of a heuristic would be the use of Minimum Bounding Rectangles (MBRs) as a first approximation of the objects’ geometry as a fast filter. Estimates of the distribution are important, because the application often determines what relations are feasible. For example, in a cadastral application the only possible topological relations between land parcels are disjoint or meet. This paper focuses on the processing and algebraic optimization of spatial queries with topological constraints. An example of such a query is, “Find all residential lots for sale adjacent to Branch Lake,” where adjacent is a topological relation. Such relations are usually not explicitly stored among spatial objects, but have to be inferred from the objects’ geometry. For example, the fact that two land parcels are adjacent would be inferred from the fact that the two regions have a part of their boundaries, but no interior, in common. While existing DBMSs do not support such complex relations, extensible DBMSs [3] have the provisions to incorporate them into query languages. To be successful as geographic databases, extensible DBMSs need models of how to process and optimize queries over spatial relations. The query processing strategies presented in this paper are based on the 9-intersection, a comprehensive model for binary topological relations among point-, line-, and area-objects [4, 5]. It identifies eight basic relations between two regions in ✩✲; 19 topological relations between a region and a simple line in ✩✲; and 33 relations between two simple lines in ✩✲. The strategies for processing queries with topological constraints are based on the observation that only a true subset of the nine intersections need to be determined in order to identify the topological relation between two spatial objects. For example, determining whether two regions are disjoint only requires determining that two intersections, the boundary-boundary and the interior-interior, are empty because in none of the other seven possible cases are these two intersections both empty. The optimization of topological queries is particularly challenging because terminology and semantics of the relations varies across application domains. As long as no formalizations of such spatial predicates exist that would match with humans’ interpretations, it is standard practice to define the set of topological relations for an application domain as disjunctions of a set of basic, mutually exclusive relations [6]. The sets of relations defined by the 9-intersection form base sets, from which database users and administrators can construct the non-primitive sets of relations relevant to their specific application domain. For example, an application domain may not need the distinction between covers and contains as defined in the 9-intersection model for region-region relations; therefore, for this user group, the set of relevant region-region relations would be {disjoint, containsOrCovers, insideOrCoveredBy, meet, equal, overlap}. It is important that any query optimization strategy selected will work independently of the particular combinations made. This general applicability is of great importance since initial tests with human subjects clearly demonstrate that humans group conceptually close relations and devise a prototypical representative of this group [7]. Two topological relations are conceptually close if there is a transition from one to another as a result of a gradual deformation applied to one object [8, 9]. Given this evidence it is unlikely that all users would need or grasp the nuances between the 19 distinct region-line or 33 line-line relations in their applications. The particular subset, or subgroupings, of these relations will necessarily vary with the application domain. Hence any query processing strategy based on the base sets of relations must be applicable to such domain specific groupings. In this paper we give examples of how the strategies presented can be directly applied to such cases, and therefore, these strategies provide a degree of flexibility and customizability not typically found in conventional spatial query processors. We assume an object-centered geographic database. Object-centered geographic databases [1] have a vector or topological data model and deal with spatial objects that have a distinct identity, e.g., in the form of simplicial complexes or cells [10, 11]. This spatial data model represents the geometry of geographic objects in terms of points, lines, and areas, and records explicitly boundary and coboundary relations among the geometric elements. Computers and Graphics 18 (6): 815-822, 1994. There are two types of queries, the processing of which requires the computation of the values of the nine intersections between the interiors, boundaries, and exteriors. They are: • “Find all objects that have the topological relation R to object A?” and • “What is the topological relation between objects A and B?” The latter type of query is less frequently asked though such a query is as important as the former in geographic applications. The results of these queries have been called “qualitative answers” [12]. The goal of this paper is to present the most promising strategies for processing topological queries. The novel result is an algorithm to determine the smallest subset of the nine intersections that have to be evaluated. We demonstrate that such spatial queries can be frequently determined with less effort than computing all nine intersections for a topological relation. We start with a “naive” model of query processing, assuming that the computation of all intersections is of equal complexity, that all relations are equally distributed, and that all relations are equally frequently queried. The naive model is then refined by assigning a probability distribution to the set of relations. For example, in a cadastral application the most frequent topological relation is “disjoint” followed by “meet.” The framework introduced in this paper allows a query processor to find the best strategy to assess a topological relation between two spatial entities. The 9-intersection model gives the primitives for describing a topological relation, and by using a combination of such primitives it is possible to model every set of topological relations. The same elementary tools can be used even when such complex geographic objects as regions with holes or 1-spheres are involved [13]. The remainder of this paper is structured as follows: after a brief summary of the pertinent work in spatial query optimization, we review the concepts of the 9-intersection as the model for which we will investigate query processing strategies. As a first step, we introduce the mappings from topological relations as defined by the 9-intersection onto relations between MBRs and show how such knowledge can be exploited as a fast filter to find candidates that would satisfy a particular topological relation. Subsequently, we design algorithms to select objects from the candidates for two types of spatial queries: (1) finding the set of objects that hold a particular topological relation, or set of topological relations, with respect to a given object; and (2) determining the topological relation between two given objects. For the latter we compare the approach for the naive cost model with a refined cost model. 2 . Previous Work in Spatial Query Optimization Most approaches to spatial query processing reported in the literature optimize spatial queries by transforming user queries into evaluation plans that take into account the extended physical storage mechanisms and access methods. This section reviews four such approaches and compares them with the goals of this paper. 2.1 Spatial and Non-Spatial Database (SAND) Aref and Samet [14, 15] describe strategies for constructing query evaluation plans and assessing their costs. SAND consists of separate spatial and non-spatial data stores, which have two-way links between records that describe the spatial and non-spatial attributes of some object. The strategies essentially extend traditional non-spatial approaches with a spatial selection, which retrieves records from the spatial data store that satisfy the given constraints. The evaluation plans involve reordering the selections, i.e., choosing between performing the spatial or non-spatial selection first; using indexes on both spatial and non-spatial data; and performing spatial operations while accessing the data rather than storing links in a temporary store and subsequently traversing these records to perform the desired operations. The major contribution of SAND is its extensibility because the techniques are applicable to various data types, not only to spatial data.