In the constraint database model, spatial and spatio-temporal data are stored by boolean combinations of polynomial equalities and inequalities over the real numbers. The relational calculus augmented with polynomial constraints is the standard first-order query language for constraint databases. Although the expressive power of this query language has been studied extensively, the difficulty of the efficient evaluation of queries, usually involving some form of quantifier elimination, has received considerably less attention. The inefficiency of existing quantifier-elimination software and the intrinsic difficulty of quantifier elimination have proven to be a bottle-neck for for real-world implementations of constraint database systems. In this paper, we focus on a particular query, called the \emph{alibi query}, that asks whether two moving objects whose positions are known at certain moments in time, could have possibly met, given certain speed constraints. This query can be seen as a constraint database query and its evaluation relies on the elimination of a block of three existential quantifiers. Implementations of general purpose elimination algorithms are in the specific case, for practical purposes, too slow in answering the alibi query and fail completely in the parametric case. The main contribution of this paper is an analytical solution to the parametric alibi query, which can be used to answer this query in the specific case in constant time. We also give an analytic solution to the alibi query at a fixed moment in time. The solutions we propose are based on geometric argumentation and they illustrate the fact that some practical problems require creative solutions, where at least in theory, existing systems could provide a solution.
Deep Dive into A case study of the difficulty of quantifier elimination in constraint databases: the alibi query in moving object databases.
In the constraint database model, spatial and spatio-temporal data are stored by boolean combinations of polynomial equalities and inequalities over the real numbers. The relational calculus augmented with polynomial constraints is the standard first-order query language for constraint databases. Although the expressive power of this query language has been studied extensively, the difficulty of the efficient evaluation of queries, usually involving some form of quantifier elimination, has received considerably less attention. The inefficiency of existing quantifier-elimination software and the intrinsic difficulty of quantifier elimination have proven to be a bottle-neck for for real-world implementations of constraint database systems. In this paper, we focus on a particular query, called the \emph{alibi query}, that asks whether two moving objects whose positions are known at certain moments in time, could have possibly met, given certain speed constraints. This query can be seen as
The framework of constraint databases was introduced in 1990 by Kanellakis, Kuper and Revesz [Kanellakis et al. 1995] as an extension of the relational database model that allows the use of infinite, but first-order definable relations rather than just finite relations. It provides an elegant and powerful model for applications that • deal with infinite sets of points in some real affine space R n , such as spatial and spatio-temporal databases [Paredaens et al. 1994]. In the setting of the constraint model, infinite relations in spatial or spatio-temporal databases are finitely represented as boolean combinations of polynomial equalities and inequalities, which are interpreted over the real numbers.
Various aspects of the constraint database model are well-studied by now. For an overview of research results we refer to [Paredaens et al. 2000] and the textbook [Revesz 2002]. The relational calculus augmented with polynomial constraints, or equivalently, first-order logic over the reals augmented with relation predicates to address the database relations R 1 , . . . , R m , FO(+, ×, <, 0, 1, R 1 , . . . , R m ) for short, is the standard first-order query language for constraint databases. The expressive power of first-order logic over the reals, as a constraint database query language, has been studied extensively [Paredaens et al. 2000]. It is well-known that first-order constraint queries can be effectively evaluated [Tarski 1951;Paredaens et al. 2000]. However, the difficulty of the efficient evaluation of first-order queries, usually involving some form of quantifier elimination, has been largely neglected [Heintz and Kuijpers 2004]. The existing constraint database systems or prototypes, such as Dedale and Disco [Paredaens et al. 2000, Chapters 17 and 18] are based on general purpose quantifier-elimination algorithms and are, in most cases, restricted to work with linear data, i.e., they use first-order logic over the reals without multiplication [Paredaens et al. 2000, Part IV]. Of the general purpose elimination algorithms [Basu et al. 1996;Collins 1975;Grigor’ev and Vorobjov 1988;Heintz et al. 1990;Renegar 1992], some are now available in software packages such as QEPCAD [Hong 1990], Redlog [Sturm 2000] and Mathematica [Wolfram 2007]. But the intrinsic inefficiency of quantifier elimination and the inefficiency of its current implementations represent a bottle-neck for real-world implementations of constraint database systems [Heintz and Kuijpers 2004].
In this paper, we focus on a case study of quantifier elimination in constraint databases. Our example is the alibi query in moving object databases, which was introduced and studied in the area of geographic information systems (GIS) [Pfoser and Jensen 1999;Egenhofer 2003;Hornsby and Egenhofer 2002;Miller 2005]. This query can be expressed in the constraint database formalism and, at least in theory could be answered, both in the specific and the parametric case, by existing implementations of quantifier elimination over the reals. The evaluation of the alibi query adds up to the elimination of a block of three existential quantifiers. It turns out that packages such as QEPCAD [Hong 1990], Redlog [Sturm 2000] and Mathematica [Wolfram 2007], can only solve the alibi query in specific cases, with a running time that is not acceptable for moving object database users. In the parametric case, these quantifier-elimination implementations fail miserably. The main contribution of this paper is a theoretic and practical solution to the alibi query in the parametric case (and thus the specific cases).
The research on spatial databases, which started in the 1980s from work in geographic information systems, was extended in the second half of the 1990s to deal with spatio-temporal data. In this field, one particular line of research, started by Wolfson, concentrates on moving object databases (MODs) [Güting and Schneider 2005;Wolfson 2002], a field in which several data models and query languages have been proposed to deal with moving objects whose position is recorded at discrete moments in time. Some of these models are geared towards handling uncertainty that may come from various sources (measurements of locations, interpolation, …) and several query formalisms have been proposed [Su et al. 2001;Geerts 2004;Kuijpers and Othman 2007]. For an overview of models and techniques for MODs, we refer to the book by Güting and Schneider [Güting and Schneider 2005].
In this paper, we focus on the trajectories that are produced by moving objects and which are stored in a database as a collection of tuples (t i , x i , y i ), i = 0, …, N , i.e., as a finite sample of time-stamped locations in the plane. These samples may have been obtained by GPS-measurements or from other location aware devices.
One particular model for the management of the uncertainty of the moving object’s position in between sample points is provided by the bead model. In this model, it
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