P. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient Searching with Linear Constraints. In Proc. of PODS, pages 169--178, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with large volumes of GADT data, queries cannot be efficiently processed by naively scanning relations; we need efficient access methods. Fast access to GADT data can be achieved by translating GADT queries into queries by linear constraints (QBLC) Goldstein et al. GRSY97] and Agarwal et al. AAE98] have recently shown that QBLC can be processed efficiently using standard indexing structures such as the R tree. be a GADT instance with J is logically equivalent to the pair , and any condition imposed on is equivalent to a constraint on . If the condition is given by ....

Pankaj K. Agarwal, Lars Arge, and Jeff Erickson. Efficient searching with linear constraints. In PODS, pages 169--178, 1998.

....in Section 2 can be answered in ### ######### ### time. Here # is the number of dimensions of the space where the objects move, # is the number of data blocks, and # is the size in blocks of a query answer. To achieve this bound, an external memory version of partition trees may be used [1]. It is argued that, although having good asymptotic performance bounds, partition trees are not practical due to the large constant factors involved. Basch et al. 4] propose so called kinetic main memory data structures for mobile objects. The idea is to schedule future events that update a ....

P. K. Agarwal et al. Efficient Searching with Linear Constraints. In Proc. of the PODS Conf., pp. 169--178 (1998).

....Department of Computer Science, University of Aarhus, Denmark and I.N.R.I.A. Sophia Antipolis, France. Email: jsv cs.duke.edu. 1 Introduction There has recently been much effort toward developing worst case I O efficient external memory data structures for range searching in two dimensions [1, 2, 4, 8, 12, 13, 20, 26, 28, 29]. In their pioneering work, Kanellakis et al. 13] showed that the problem of indexing in new data models (such as constraint, temporal, and object models) can be reduced to special cases of twodimensional indexing. Refer to Figure 1) In particular they identified the 3 sided range searching ....

....the TPIE system [27] There are several interesting range searching problems in external memory that remain open, such as higher dimensional range searching and non orthogonal queries. Some recent work has been done on I Oefficient three dimensional range searching and halfspace range searching [1, 28, 29]. ....

P. K. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient searching with linear constraints. In Proc. ACM Symp. Principles of Database Systems, pages 169--178, 1998.

....on how to filter those local constraints that are unaffected by the update. Others[5] only consider a particular class of local constraints and updates. Thus, none of the techniques is applicable to our case. The n conditions we intend to check are all linear conditions, making related techniques[2, 16, 12] developed in computation geometry also applicable. However, in our case, the linear conditions change frequently, making the cost of reconstructing the data structures [2, 12] outweigh the benefits. 7 Conclusion In this paper, we argue for efficiently bounding numerical error to support ....

....is applicable to our case. The n conditions we intend to check are all linear conditions, making related techniques[2, 16, 12] developed in computation geometry also applicable. However, in our case, the linear conditions change frequently, making the cost of reconstructing the data structures [2, 12] outweigh the benefits. 7 Conclusion In this paper, we argue for efficiently bounding numerical error to support replicated network services. Two algorithms, Split Weight AE and Compound Weight AE, are proposed to bound absolute error. They can be combined to achieve good performance and low ....

Pankaj K. Agarwal, Lars Arge, Jeff Erickson Paolo G. Fanciosa, and Jeffrey Scott Vitter. Efficient Searching with Linear Constraints. In Proceedings of the 17th ACM Symposium on Principles of Database Systems, 1998.

....query to predict traffic congestion or to guarantee the fixed travel time. Without it, we can predict at least how long and until which exit the congestion stays so that we can flexibly open or close exits in between. Several researchers have investigated the problem of indexing line segments [12, 10, 1] in the field of line segments databases. Although they did not consider their work in the context of moving objects, their results are closely related to moving objects databases. Jagadish [12] considered the problem of indexing line segments that (i) go through a specified point or (ii) ....

Pankaj Agarwal, Lars Arge, Jeff Erickson, Paolo Franciosa, and Jeffrey Vitter. Efficient Searching with Linear Constraints. In Proceedings of ACM Symp. on Principles of Database Systems, Seattle, Washington, 1998.

....in the third case we recurse on the triangle. The number of triangles that the query can cross is bounded however, since each line crosses at most O(jSj 1 4 ) triangles at the root. The query time is O(N 1 2 ffl K) with the constant factor depending on the choice of ffl. Agarwal et al. [1] give an external memory version of static partition trees that answers queries in O(n 1 2 ffl k) I Os. To adapt this structure to our environment, we have to make it dynamic. Using a standard technique by Overmars ( 28] for decomposable problems we can show that we can insert or delete ....

P.K. Agarwal, L. Arge, J. Erickson, P. Franciosa and J.S. Vitter. Efficient Searching with Linear Constraints In Proc. 17th ACM PODS Symposium on Principles of Database Systems,pp. 169-178 1998.

....in Section 2 can be answered in O(n (2d 1) 2d k) time. Here d is the number of dimensions of the space where the objects move, n is the number of data blocks, and k is the size in blocks of a query answer. To achieve this bound, an external memory version of partition trees may be used [1]. It is argued that, although having good asymptotic performance bounds, partition trees are not practical due to the large constant factors involved. Basch et al. 4] propose so called kinetic main memory data structures for mobile objects. The idea is to schedule future events that update a ....

P. K. Agarwal et al. Efficient Searching with Linear Constraints. In Proc. of the PODS Conf., pp. 169--178 (1998).

....in Section 2 can be answered in O(n (2d Gamma1) 2d k) time. Here d is the number of dimensions of the space where the objects move, n is the number of data blocks, and k is the size in blocks of a query answer. To achieve this bound, an external memory version of partition trees may be used [Agar98]. It is argued that, although having good asymptotic performance bounds, partition trees are not practical due to the large constant factors involved. 6 The problem of indexing moving point objects is related to the problem of indexing now relative temporal data. The GR tree [BJSS98a] an R tree ....

P. K. Agarwal et al. Efficient Searching with Linear Constraints. In Proc. of the PODS Conf., pp. 169--178 (1998).

....both selections. However, if the query angular coefficient does not belong to the predefined set, the logarithmic time complexity can be retained but the space complexity is no longer linear. The use of the dual representation for indexing constraint databases has also been recently considered in [1], to determine all points located above a given line. Our contribution. In this paper, we propose two approximation techniques based on the dual representation of objects, which are essentially different from any approximation technique usually adopted for object representation. Both ....

....P . The faces of P that are a subset of some supporting hyperplane with = and a d 0 ( and a d 0) form the upper hull (lower hull) of P ; 2. 1 The dual transformation The dual transformation is a basic operation applied to d dimensional objects in different geometrical algorithms [1, 8, 10, 13]. Here we assume that none of the considered half planes is vertical. 4 Under this hypothesis, a hyperplane a 1 x 1 : a d x d c = 0 intersects the d th coordinate in a unique point represented by the equation: x d = b 1 x 1 : b d Gamma1 x d Gamma1 b d where b i = Gammaa i =a ....

P. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient Searching with Linear Constraints. In Proc. of PODS, pages 169--178, 1998.

....in the third case we recurse on the triangle. The number of triangles that the query can cross is bounded however, since each line crosses at most O(jSj 1=4 ) triangles at the root. The query time is O(N 1=2 ffl T ) with the constant factor depending on the choice of ffl. Agarwal et al. [1] give an external memory version of static partition trees that answers queries in O(n 1=2 ffl k) I Os. To adapt this stucture to our environment, we have to make it dynamic. We show the following lemma: Lemma 1 We can insert or delete points in a partition tree in O( p (N) log(N) I Os, ....

P.K. Agarwal, L. Arge, J. Erickson, P. Franciosa and J.S. Vitter. Efficient Searching with Linear Constraints In Proc. 17th ACM PODS Symposium on Principles of Database Systems, Seattle, 1998.

....Although a number of practical methods have been proposed for accessing and searching moving objects [19, 25, 26, 43, 48] they all require Omega Gamma n) I Os in the worst case, even if the query range is empty. Kollios et al. 23] proposed an efficient indexing scheme, based on partition trees [2, 3], for storing a set of points moving on the real line. It uses O(n) disk blocks, answers a 1 dimensional query of type Q1 or Q2 using O(n 1=2 k) I Os 1 , and allows a point to be inserted or deleted in O(log 2 2 n) I Os. They also present a scheme that uses O(Nn) disk blocks and answers ....

....line segment. 2. 2 External partition trees Partition trees are one of the most commonly used internal memory data structures for geometric range searching [3, 28] Our first indexing scheme is based on partition trees, which were originally described in Matousek [28] and later extended in [2] to the external memory setting. We briefly summarize them here with an emphasis on insertion deletion operations, as we slightly improve their performance compared to [2] In order to describe an algorithm for constructing a partition tree, we need to define two concepts, 1=r) cuttings and ....

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P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter, Efficient searching with linear constraints, Proc. 17th Annu. ACM Sympos. Principles Database Syst., 169--178, 1998.

....Note that the query bound consists of an O(log B N) search term corresponding to the familiar O(log N) internal memory search term, and an O(T=B) reporting term accounting for the O(T=B) I Os needed to report T elements. Recently, the above bounds have been obtained for a number of problems (e. g [30, 26, 149, 5, 47, 87]) but higher lower bounds have also been established for some problems [141, 26, 93, 101, 106, 135, 102] We discuss these results in later sections. B trees come in several variants, like B and B trees (see e.g. 35, 63, 95, 30, 104, 3] and their references) A basic B tree is a ....

....one side of a query hyperplane. Halfspace range searching is the simplest form of non isothetic (non orthogonal) range searching. The problem was first considered in external memory by Franciosa and Talamo [83, 82] Based on an internal memory structure due to Chazelle et al. 55] Agarwal et al. [5] described an optimal O(log B N T=B) query and linear space structure for the 2 dimensional case. Using ideas from an internal memory result of Chan [50] they described a structure for the 3 dimensional case, answering queries in O(log B N T=B) expected I Os but requiring O( N=B) log(N=B) space. ....

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P. K. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient searching with linear constraints. Journal of Computer and System Sciences, 61(2):194--216, 2000.

.... handling moving objects (see [20, 16] and the references therein) Almost all of them require Omega (N=B) I Os in the worst case to answer a Q1 or Q2 query even if the query output size is O(1) Kollios et al. 13] proposed the first provably efficient data structure, based on partition trees [2, 3, 15], for queries of type Q1. The structure uses O(N=B) disk blocks and answers queries in O( N=B) 1=2 ffl K=B) I Os, for any ffl 0. Agarwal et al. 1] extended the result to Q2 queries. Kollios et al. 13] also present a scheme that answers a Q1 query using optimal O(log B N K=B) I Os but ....

....axis. Dey [10] showed that the maximum number of vertices on the k level in an arrangement of N lines in the plane is O(Nk 1=3 ) Recently, T oth proved a lower bound of Omega (N2 p log k ) on the complexity of a k level [17] Using a result by Edelsbrunner and Welzl [11] Agarwal et al. [2] discussed how a given level of an arrangement of lines can be computed I Oefficiently. Lemma 1 (Agarwal et al. 2] A given level with T vertices in an arrangement of N lines can be computed in O(N log 2 N T log 2 N log B N) I Os. B trees. A B tree, one of the most fundamental external data ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter, Efficient searching with linear constraints, Journal of Computer and System Sciences, 61 (2000), 194-- 216.

.... and searching moving objects (see [36, 30, 21, 26, 24] and the references therein) almost all of them require Omega Gamma N=B) I Os in the worst case even if the query output size is O(1) Kollios et al. 20] proposed the first provably efficient indexing scheme, based on partition trees [3, 4], for queries of type Q1. The structure uses O(N=B) disk blocks and answers queries in O( N=B) 1=2 ffl K=B) I Os, for an arbitrarily small constant ffl 0. Agarwal et al. 2] extended the result to Q2 queries. Kollios et al. 20] also present a scheme that answers a Q1 query using optimal ....

.... Dey [13] showed that the maximum number of vertices on the k level in an arrangement of N lines in the plane is O(Nk 1=3 ) Recently, T oth proved a lower bound of Omega Gamma N2 p log k ) on the complexity of a k level [31] Using a result by Edelsbrunner and Welzl [14] Agarwal et al. [3] discussed how a given level of an arrangement of lines with T vertices can be computed I O efficiently. 4 Lemma 2.1 ( 3] A given level with T vertices in an arrangement of N lines can be computed in O(N log 2 N T log 2 N log B N) I Os. B trees. A B tree, one of the most fundamental indexing ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter. Efficient searching with linear constraints. In Proc. Annu. ACM Sympos. Principles Database Syst., 1998. 169--178.

.... Although a number of practical methods have been proposed for accessing and searching moving objects [38, 33, 20] they all require Omega Gamma n) I Os in the worst case even if the query output size is O(1) Kollios et al. 18] proposed an efficient indexing scheme, based on partition trees [1, 2], for storing a set of points moving on the real line. It uses O(n) disk blocks, answers a 1 dimensional query of type Q2 using O(n 1=2 k) I Os, for an arbitrarily small constant 0, and inserts or deletes a point using O(log 2 2 n) I Os. They also present another scheme that uses O(Nn) ....

....line segment. 2. 2 External partition trees Partition trees are one of the most commonly used internal memory data structures for geometric range searching [2, 21] Our first indexing scheme is based on partition trees, which were originally described in Matousek [21] and later extended in [1] to the external memory setting. We briefly summarize them here with an emphasis on insertion deletion operations, as we slightly improve their performance compared to [1] Let S be a set of N points in R 2 . A simplicial partition of S is a set of pairs Pi = f(S 1 ; 4 1 ) S 2 ; 4 2 ) ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter, Efficient searching with linear constraints, Proc. 17th Annu. ACM Sympos. Principles Database Syst., 169--178, 1998.

.... a number of practical methods have been proposed for accessing and searching moving objects [35, 31, 30, 18, 19] they all require Omega Gamma n) I Os in the worst case even if the query output is O(1) Kollios et al. 16] proposed an efficient indexing scheme, based on partition trees [1, 2], for storing a set of points moving on the x axis with a fixed speed. It uses O(n) disk blocks, answers a 1 dimensional query of type Q2 using O(n 1=2 k) I Os, for an arbitrarily small constant 0, and inserts or deletes a point using O(log 2 2 n) I Os. They also present another scheme ....

.... External partition trees Partition trees are one of the most commonly used internal memory data structures for geometric range searching [2, 20] Our first indexing scheme is based on partition trees, which were originally described Indexing Moving Points 4 in Matousek [20] and later extended in [1] to the external memory setting. We briefly summarize them here with an emphasis on insertion deletion operations, as we slightly improve their performance. Let S be a set of N points in R 2 . A simplicial partition of S is a set of pairs Pi = f(S 1 ; 4 1 ) S 2 ; 4 2 ) S r ; 4 r ....

[Article contains additional citation context not shown here]

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter, Efficient searching with linear constraints, Proc. 17th Annu. ACM Sympos. Principles Database Syst., 1998, pp. 169--178.

....d dimensional simplex queries) efficiently in the worst case. For the d = 3 case, we also show how to obtain tradeoffs between space and query time. An extended abstract of this paper appeared in Proceedings of the 15th ACM SIGACT SIGMOD SIGART Symposium on Principles of Database Systems [1]. y Center for Geometric Computing, Department of Computer Science, Duke University, Box 90129, Durham, NC 27708 0129; pankaj cs:duke:edu; http: www:cs:duke:edu pankaj. Supported in part by National Science Foundation research grants EIA 9870724, and CCR 9732787, by Army Research Office ....

.... t) O(n log B n) O(n 1 Gamma1=d t) O(n) Table 1. Our main results. In Section 4, we describe a data structure that uses O(n log 2 n) disk blocks and answers a three dimensional halfspace range query using O(log B n t) expected I Os. In the conference version of this paper [1], we described another data structure with optimal worst case query time O(log B n t) but using O(N(log 2 n) log B n) space. As part of our result we also develop a data structure that uses O(n log 2 n) space to store N points in the plane and that can be used to find the k nearest neighbors of ....

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter, Efficient searching with linear constraints, Proc. 17th Annu. ACM Sympos. Principles Database Syst., 1998, pp. 169--178.

....a data structure that can answer a halfspace range reporting query in time O(log n t) using O(n bd=2c log c n) space, for some constant c. He also developed a data structure that can answer a query in time O(n 1 Gamma1=bd=2c log c n t) using O(n log log n) space [253] See also [6, 109]. Using linearization, a semialgebraic range searching query, where one wants to report all points of S lying inside a semialgebraic set of constant description complexity, can be answered efficiently using some of the halfspace range searching data structures [18, 343] Point location in ....

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter, Efficient searching with linear constraints, Proc. Annu. ACM Sympos. Principles Database Syst., 1998. to appear.

.... are optimal for (1; 1; 1) sided queries (i.e. k = 0) and (2; 1; 1) sided queries (i.e. k = 1) The result also provides optimal O(log N Z) time query performance in the RAM model using linear space for answering (1; 1; 1) sided queries, improving upon the result in [39] Agarwal et al. [4] give optimal bounds for static halfspace range searching in two dimensions and some variants in higher dimensions. The number of I Os needed to build the 3 D and halfspace data structures is rather large (more than order N ) Still, the structures shed useful light on the complexity of range ....

P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter. Efficient searching with linear constraints. In Proc. 17th ACM Symposium on Principles of Database Systems, 169--178, 1998.

....bound for sorting in internal memory. Work has also been done on matrix algebra and related problems arising in scientific computation [3, 51, 52] More recently, researchers have designed external memory algorithms for a number of problems in different areas, such as in computational geometry [32, 5, 53, 31, 2, 11, 34, 44, 47, 12, 50, 17, 1], string processing [28, 29, 9] and graph theoretic computation [6, 24, 38, 35] Some encouraging experimental results regarding the practical merits of the developed algorithms have also been obtained [23, 51, 11, 33] Recent surveys can be found in [7, 8] 1.3 Our Results In this paper, we ....

P. K. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient searching with linear constraints. Manuscript, 1997.

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P. Agarwal, L. Arge, J. Erickson, P. Franciosa, and J. Vitter. Efficient Searching with Linear Constraints. In Proc. of PODS, pages 169--178, 1998.

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P. K. Agarwal, L. Arge, J. Erickson, P. G. Franciosa, and J. S. Vitter. Efficient searching with linear constraints. In Proceedings of the 17th ACM Symposium on Principles of Database Systems, pages 169--178, 1998.

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P.K. Agarwal, L. Arge, J. Erickson, P. Franciosa and J.S. Vitter. Efficient Searching with Linear Constraints In Proc. 17th ACM PODS Symposium on Principles of Database Systems,pp. 169-178 1998.

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Agarwal, P., Arge, L., Erickson, J., Franciosa, P., Vitter, J. Efficient Searching with Linear Constraints. Journal of Computer and System Sciences, 61(2): 1942.

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P. Agarwal, L. Arge, J. Erickson, P.G. Franciosa, and J.S. Vitter. Efficient searching with linear constraints. In PODS '98, 1998.

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