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Indexing moving points
, 2003
"... We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an in ..."
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Cited by 168 (13 self)
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We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an indexing structure that, for any given constant e> 0; uses OðN=BÞ disk blocks and answers a query in OððN=BÞ 1=2þe þ K=BÞ I/Os, where B is the block size. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in Oðlog 2 B NÞ I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a tradeoff between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between the query time stamp t and the current time. Finally, we develop an efficient indexing scheme to answer approximate
Backtracking Algorithms for Disjunctions of Temporal Constraints
 Artificial Intelligence
, 1998
"... We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. W ..."
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Cited by 106 (2 self)
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We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. We have implemented four progressively more efficient algorithms for the consistency checking problem for this class of temporal constraints. We have partially ordered those algorithms according to the number of visited search nodes and the number of performed consistency checks. Finally, we have carried out a series of experimental results on the location of the hard region. The results show that hard problems occur at a critical value of the ratio of disjunctions to variables. This value is between 6 and 7. Introduction Reasoning with temporal constraints has been a hot research topic for the last fifteen years. The importance of this problem has been demonstrated in many areas of artifici...
SATbased Procedures for Temporal Reasoning
, 1999
"... In this paper we study the consistency problem for a set of disjunctive temporal constraints [Stergiou and Koubarakis, 1998]. We propose two SATbased procedures, and show thaton sets of binary randomly generated disjunctive constraintsthey perform up to 2 orders of magnitude less consistency ..."
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Cited by 53 (6 self)
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In this paper we study the consistency problem for a set of disjunctive temporal constraints [Stergiou and Koubarakis, 1998]. We propose two SATbased procedures, and show thaton sets of binary randomly generated disjunctive constraintsthey perform up to 2 orders of magnitude less consistency checks than the best procedure presented in [Stergiou and Koubarakis, 1998]. On these tests, our experimental analysis conrms Stergiou and Koubarakis's result about the existence of an easyhardeasy pattern whose peak corresponds to a value in between 6 and 7 of the ratio of clauses to variables.
Tractable Disjunctions of Linear Constraints: Basic Results and Applications to Temporal Reasoning
 Theoretical Computer Science
, 1996
"... We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez an ..."
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Cited by 49 (2 self)
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We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and McAloon. We show that deciding consistency of a set of constraints in this class can be done in polynomial time. We also present a variable elimination algorithm which is similar to Fourier's algorithm for linear inequalities. Finally, we use these results to provide new temporal reasoning algorithms for the OrdHorn subclass of Allen's interval formalism. We also show that there is no low level of local consistency that can guarantee global consistency for the OrdHorn subclass. This property distinguishes the OrdHorn subclass from the pointizable subclass (for which strong 5consistency is sufficient to guarantee global consistency), and the continuous endpoint subclass (for whi...
Complexity Results for FirstOrder Theories of Temporal Constraints
 In Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR'94
, 1994
"... We study the complexity of quantifier elimination and decision in firstorder theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the firstorder theories of point and interval constraints over two time structures: the integer ..."
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Cited by 26 (8 self)
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We study the complexity of quantifier elimination and decision in firstorder theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the firstorder theories of point and interval constraints over two time structures: the integers and the rationals. We show that in all cases quantifierelimination can be done in PSPACE. We also show that the decision problem for arbitrarily quantified sentences is PSPACEcomplete while for 9 k sentences it is \Sigma p k complete. Our results must be of interest to researchers working on temporal constraints, computational complexity of logical theories, constraint databases and constraint logic programming. 1 INTRODUCTION The study of temporal constraints has recently received much attention from the AI community [All83, LM88, Lad88, VKvB89, vBC90, DMP91, KL91, Mei91, vB92, Kou92, GS93, SD93]. Much of this work draws upon concepts and techniques from the literature of general co...
Manipulating Interpolated Data is Easier than You Thought
, 2000
"... Data defined by interpolation is frequently found in new applications involving geographical entities, moving objects, or spatiotemporal data. These data lead to potentially infinite collections of items, (e.g., the elevation of any point in a map), whose definitions are based on the associati ..."
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Cited by 22 (2 self)
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Data defined by interpolation is frequently found in new applications involving geographical entities, moving objects, or spatiotemporal data. These data lead to potentially infinite collections of items, (e.g., the elevation of any point in a map), whose definitions are based on the association of a collection of samples with an interpolation function. The naive manipulation of the data through direct access to both the samples and the interpolation functions leads to cumbersome or inaccurate queries. It is desirable to hide the samples and the interpolation functions from the logical level, while their manipulation is performed automatically. We propose to model such data using infinite relations (e.g., the map with elevation yields an infinite ternary relation) which can be manipulated through standard relational query languages (e.g., SQL), with no mention of the interpolated definition. The clear separation between logical and physical levels ensures the accu...
On the consistency of cardinal directions constraints
 Artificial Intelligence
, 2005
"... We present a formal model for qualitative spatial reasoning with cardinal directions utilizing a coordinate system. Then, we study the problem of checking the consistency of a set of cardinal direction constraints. We introduce the first algorithm for this problem, prove its correctness and analyze ..."
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Cited by 9 (0 self)
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We present a formal model for qualitative spatial reasoning with cardinal directions utilizing a coordinate system. Then, we study the problem of checking the consistency of a set of cardinal direction constraints. We introduce the first algorithm for this problem, prove its correctness and analyze its computational complexity. Using the above algorithm, we prove that the consistency checking of a set of basic (i.e., nondisjunctive) cardinal direction constraints can be performed in O(n 5) time. We also show that the consistency checking of a set of unrestricted (i.e., disjunctive and nondisjunctive) cardinal direction constraints is N Pcomplete. Finally, we briefly discuss an extension to the basic model and outline an algorithm for the consistency checking problem of this extension. 1
Safe Datalog Queries with Linear Constraints
 In Proceedings of the 4th International Conference on Principles and Practice of Constraint Programming (CP98), number 1520 in LNCS
, 1998
"... . In this paper we consider Datalog queries with linear constraints. We identify several syntactical subcases of Datalog queries with linear constraints, called safe queries, and show that the least model of safe Datalog queries with linear constraints can be evaluated bottomup in closedform. These ..."
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Cited by 7 (2 self)
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. In this paper we consider Datalog queries with linear constraints. We identify several syntactical subcases of Datalog queries with linear constraints, called safe queries, and show that the least model of safe Datalog queries with linear constraints can be evaluated bottomup in closedform. These subcases include Datalog with only positive and upperbound or only negative and lower bound constraints or only halfaddition, upper and lower bound constraints. We also study other subcases where the recognition problem is decidable. 1 Introduction Constraint databases is an active area of current research. In particular, linear constraint databases have been used for modeling geometric data and in other applications [3, 14, 15, 22, 23]. There are several proposals to define query languages for linear constraint databases. Most query language proposals are based on firstorder logic. However, it has been found that firstorder languages even with real polynomial constraint databases are ...
A Minicourse on Temporal Databases
"... Temporal Databases 9 Basic Building Blocks 10 The Snapshot Model 11 Snapshots: Example 12 Histories 13 The Timestamp Model 14 Timestamp Example 15 Query Languages 16 Firstorder Temporal Connectives 17 Examples of Temporal Connectives 18 Propositional Temporal Logic 19 Firstorder Temporal Logic: sy ..."
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Temporal Databases 9 Basic Building Blocks 10 The Snapshot Model 11 Snapshots: Example 12 Histories 13 The Timestamp Model 14 Timestamp Example 15 Query Languages 16 Firstorder Temporal Connectives 17 Examples of Temporal Connectives 18 Propositional Temporal Logic 19 Firstorder Temporal Logic: syntax 20 FOTL: semantics 21 Examples 22 Temporal Relational Calculus: syntax 23 Temporal RC: Semantics 24 Examples 25 Examples (cont.) 26 Expressive Power 27 Expressive Power (cont.) 28 How do we prove it? 29 Scope of Temporal Variables 30 EhrenfeuchtFrass e Games 31 EhrenfeuchtFrass e Games (cont.) 32 EhrenfeuchtFrass e Games (cont.) 33 EhrenfeuchtFrass e Games (cont.) 34 EF Games and Temporal Logic 35 Compatibility of Variables in FOTL 36 Databases not distinguishable by FOTL 37 Communication Complexity 38 Consequences for Temporal Queries 39 Temporal Relational Algebra 40 TRA: example 41 Temporal Logic TL(FO) 42 Plan 43 Concrete Temporal Databases 44 Finite Encoding using Constraints 4...
A Minicourse on Temporal Databases
"... Temporal Databases 9 Basic Building Blocks 10 The Snapshot Model 11 Snapshots: Example 12 Histories 13 The Timestamp Model 14 Timestamp Example 15 Query Languages 16 Firstorder Temporal Connectives 17 Examples of Temporal Connectives 18 Propositional Temporal Logic 19 Firstorder Temporal Logic: sy ..."
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Temporal Databases 9 Basic Building Blocks 10 The Snapshot Model 11 Snapshots: Example 12 Histories 13 The Timestamp Model 14 Timestamp Example 15 Query Languages 16 Firstorder Temporal Connectives 17 Examples of Temporal Connectives 18 Propositional Temporal Logic 19 Firstorder Temporal Logic: syntax 20 FOTL: semantics 21 Examples 22 Temporal Relational Calculus: syntax 23 Temporal RC: Semantics 24 Examples 25 Examples (cont.) 26 Expressive Power 27 Expressive Power (cont.) 28 How do we prove it? 29 Scope of Temporal Variables 30 EhrenfeuchtFrassb Games 31 EhrenfeuchtFrassb Games (cont.) 32 EhrenfeuchtFrassb Games (cont.) 33 EhrenfeuchtFrassb Games (cont.) 34 EF Games and Temporal Logic 35 Compatibility of Variables in FOTL 36 Databases not distinguishable by FOTL 37 Communication Complexity 38 Consequences for Temporal Queries 39 Temporal Relational Algebra 40 TRA: example 41 Temporal Logic TL(FO) 42 Plan 43 Concrete Temporal Databases 44 Finite Encoding using Constraints 45 Interval Encoding 46 Interval Encoding (cont.) 47 Example 48 Why Intervals? 49 Interval Queries 50 Genericity 51 TSQL2 [Snodgrass, 1995] 52 Duplicate Semantics 53 TSQL2's Successors 54 Example 55 Coalescing 56 Example (cont,) 57 Failure of Coalescing 58 Folding and Unfolding 59 Other Proposals 60 Intervalsvs, True Intervals 61 Temporal Connectives in ,z 62 Translations 63 Closure for Intervals 64 SQL/TP [Toman, 1997] 65 SQL/TP: syntax 66 SQL/TP: encoding of time 67 SQL/TP: Query Evaluation 68 Data Definition Language 69 How do we compile Queries? 70 Closure for SQL/TP 71 Conditional Queries 72 Select Block: Join and Selection 73 Select Block: Duplicate Elimination 74 Timecompatible Queries 75 Normalization 76 Set Operations 77 Set Operations (cont,) 78 Set Operations (cont,) 79 Size of the ...