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136
On Indexing Mobile Objects
, 1999
"... We show how to index mobile objects in one and two dimensions using efficient dynamic external memory data structures. The problem is motivated by real life applications in traffic monitoring, intelligent navigation and mobile communications domains. For the 1dimensional case, we give (i) a dynamic ..."
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Cited by 223 (16 self)
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We show how to index mobile objects in one and two dimensions using efficient dynamic external memory data structures. The problem is motivated by real life applications in traffic monitoring, intelligent navigation and mobile communications domains. For the 1dimensional case, we give (i) a dynamic, external memory algorithm with guaranteed worst case performance and linear space and (ii) a practical approximation algorithm also in the dynamic, external memory setting, which has linear space and expected logarithmic query time. We also give an algorithm with guaranteed logarithmic query time for a restricted version of the problem. We present extensions of our techniques to two dimensions. In addition we give a lower bound on the number of I/O's needed to answer the ddimensional problem. Initial experimental results and comparisons to traditional indexing approaches are also included. 1 Introduction Traditional database management systems assume that data stored in the database rem...
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 191 (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
Fractional cascading: I. A data structuring technique
 Algorithmica
, 1986
"... Abstract. In computational geometry many search problems and range queries can be solved by performing an iterative search for the same key in separate ordered lists. In this paper we show that, if these ordered lists can be put in a onetoone correspondence with the nodes of a graph of degree d so ..."
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Cited by 174 (6 self)
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Abstract. In computational geometry many search problems and range queries can be solved by performing an iterative search for the same key in separate ordered lists. In this paper we show that, if these ordered lists can be put in a onetoone correspondence with the nodes of a graph of degree d so that the iterative search always proceeds along edges of that graph, then we can do much better than the obvious sequence of binary searches. Without expanding the storage by more than a constant factor, we can build a datastructure, called a fractional cascading structure, in which all original searches after the first can be carried out at only log d extra cost per search. Several results related to the dynamization of this structure are also presented. A companion paper gives numerous applications of this technique to geometric problems.
On TwoDimensional Indexability and Optimal Range Search Indexing (Extended Abstract)
, 1999
"... Lars Arge Vasilis Samoladas y Jeffrey Scott Vitter z Abstract In this paper we settle several longstanding open problems in theory of indexability and external orthogonal range searching. In the first part of the paper, we apply the theory of indexability to the problem of twodimensional rang ..."
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Cited by 90 (26 self)
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Lars Arge Vasilis Samoladas y Jeffrey Scott Vitter z Abstract In this paper we settle several longstanding open problems in theory of indexability and external orthogonal range searching. In the first part of the paper, we apply the theory of indexability to the problem of twodimensional range searching. We show that the special case of 3sided querying can be solved with constant redundancy and access overhead. From this, we derive indexing schemes for general 4sided range queries that exhibit an optimal tradeoff between redundancy and access overhead.
Optimal Dynamic Interval Management in External Memory (Extended Abstract)
 IN PROC. IEEE SYMP. ON FOUNDATIONS OF COMP. SCI
, 1996
"... We present a space and I/Ooptimal externalmemory data structure for answering stabbing queries on a set of dynamically maintained intervals. Our data structure settles an open problem in databases and I/O algorithms by providing the first optimal externalmemory solution to the dynamic interval m ..."
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Cited by 81 (20 self)
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We present a space and I/Ooptimal externalmemory data structure for answering stabbing queries on a set of dynamically maintained intervals. Our data structure settles an open problem in databases and I/O algorithms by providing the first optimal externalmemory solution to the dynamic interval management problem, which is a special case of 2dimensional range searching and a central problem for objectoriented and temporal databases and for constraint logic programming. Our data structure simultaneously uses optimal linear space (that is, O(N/B) blocks of disk space) and achieves the optimal O(log B N + T/B) I/O query bound and O(log B N ) I/O update bound, where B is the I/O block size and T the number of elements in the answer to a query. Our structure is also the first optimal external data structure for a 2dimensional range searching problem that has worstcase as opposed to amortized update bounds. Part of the data structure uses a novel balancing technique for efficient worstcase manipulation of balanced trees, which is of independent interest.
External Memory Data Structures
, 2001
"... In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynami ..."
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Cited by 76 (32 self)
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In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynamic data structures. We also briefly discuss some of the most popular external data structures used in practice.
ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 73 (13 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 73 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
An Optimal Algorithm for the Distinct Elements Problem
"... We give the first optimal algorithm for estimating the number of distinct elements in a data stream, closing a long line of theoretical research on this problem begun by Flajolet and Martin in their seminal paper in FOCS 1983. This problem has applications to query optimization, Internet routing, ne ..."
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Cited by 70 (6 self)
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We give the first optimal algorithm for estimating the number of distinct elements in a data stream, closing a long line of theoretical research on this problem begun by Flajolet and Martin in their seminal paper in FOCS 1983. This problem has applications to query optimization, Internet routing, network topology, and data mining. For a stream of indices in {1,..., n}, our algorithm computes a (1 ± ε)approximation using an optimal O(ε −2 +log(n)) bits of space with 2/3 success probability, where 0 < ε < 1 is given. This probability can be amplified by independent repetition. Furthermore, our algorithm processes each stream update in O(1) worstcase time, and can report an estimate at any point midstream in O(1) worstcase time, thus settling both the space and time complexities simultaneously.
Lower bounds for orthogonal range searching: I. the reporting case
 Journal of the ACM
, 1990
"... Abstract. We establish lower bounds on the complexity of orthogonal range reporting in the static case. Given a collection of n points in dspace and a box [a,, b,] x. x [ad, bd], report every point whose ith coordinate lies in [a,, biJ, for each i = 1,..., d. The collection of points is fixed once ..."
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Cited by 69 (4 self)
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Abstract. We establish lower bounds on the complexity of orthogonal range reporting in the static case. Given a collection of n points in dspace and a box [a,, b,] x. x [ad, bd], report every point whose ith coordinate lies in [a,, biJ, for each i = 1,..., d. The collection of points is fixed once and for all and can be preprocessed. The box, on the other hand, constitutes a query that must be answered online. It is shown that on a pointer machine a query time of O(k + polylog(n)), where k is the number of points to be reported, can only be achieved at the expense of fl(n(logn/loglogn)d‘) storage. Interestingly, these bounds are optimal in the pointer machine model, but they can be improved (ever so slightly) on a random access machine. In a companion paper, we address the related problem of adding up weights assigned to the points in the query box.