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15
A functional approach to data structures and its use in multidimensional searching
 SIAM J. Comput
, 1988
"... Abstract. We establish new upperbounds on the complexity ofmultidimensional 3earching. Our results include, in particular, linearsize data structures for range and rectangle counting in two dimensions with logarithmic query time. More generally, we give improved data structures for rectangle proble ..."
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Cited by 132 (3 self)
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Abstract. We establish new upperbounds on the complexity ofmultidimensional 3earching. Our results include, in particular, linearsize data structures for range and rectangle counting in two dimensions with logarithmic query time. More generally, we give improved data structures for rectangle problems in any dimension, in a static as well as a dynamic setting. Several ofthe algorithms we give are simple to implement and might be the solutions of choice in practice. Central to this paper is the nonstandard approach followed to achieve these results. At its rootwe find a redefinition ofdata structures interms offunctional specifications.
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 79 (36 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.
Efficient Aggregation over Objects with Extent (Extended Abstract)
 TechReport UCR CS 01 01, CS Dept
, 2002
"... We examine the problem of efficiently computing sum/count/avg aggregates over... ..."
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Cited by 33 (8 self)
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We examine the problem of efficiently computing sum/count/avg aggregates over...
SpaceEfficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting
 PROCEEDINGS OF THE 15TH ISAAC, VOLUME 3341 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We present linearspace sublogarithmic algorithms for handling the 3dimensional dominance reporting and the 2dimensional dominance counting problems. Under the RAM model as described in [M. L. Fredman and D. E. Willard. “Surpassing the information theoretic bound with fusion trees”, Journal of C ..."
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Cited by 29 (1 self)
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We present linearspace sublogarithmic algorithms for handling the 3dimensional dominance reporting and the 2dimensional dominance counting problems. Under the RAM model as described in [M. L. Fredman and D. E. Willard. “Surpassing the information theoretic bound with fusion trees”, Journal of Computer and System Sciences, 47:424– 436, 1993], our algorithms achieve O(log n / log log n + f) query time for the 3dimensional dominance reporting problem, where f is the output size, and O(log n / log log n) query time for the 2dimensional dominance counting problem. We extend these results to any constant dimension d ≥ 3, achieving O(n(log n / log log n) d−3) space and O((log n / log log n) d−2 + f) query time for the reporting case and O(n(log n / log log n) d−2) space and O((log n / log log n) d−1) query time for the counting case.
The expected size of some graphs in computational geometry
 COMPUTERS & MATHEMATICS WITH APPLICATIONS
, 1988
"... We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rec ..."
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Cited by 24 (4 self)
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We consider n independent points with a common but arbitrary density fin R a. Two points (X~, Xj) are joined by an edge when a certain set S(X~, Xj) does not contain any other data points. The expected number E(N) of edges in the graph depends upon n, f and the definition of S. Examples include rectangles, spheres and loons; these lead to the graph of all dominance pairs, the Gabriel graph and the relative neighborhood graph, respectively. Other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph. In all cases, we obtain asymptotic lower bounds that do not depend uponf(and are hence useful in all applications involving these graphs, since we usually do not know f). For sparse graphs, exact asymptotic constants are obtained for E(N) that are valid for all densities.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
CRBTree: An Efficient Indexing Scheme for Range Aggregate Queries
 IN PROC. INTERNATIONAL CONFERENCE ON DATABASE THEORY
, 2003
"... We propose a new indexing scheme, called the CRBtree, for efficiently answering rangeaggregate queries. The rangeaggregate problem is defined as follows: Given a set of weighted points in R , compute the aggregate of weights of points that lie inside a ddimensional query rectangle. In this ..."
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Cited by 12 (1 self)
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We propose a new indexing scheme, called the CRBtree, for efficiently answering rangeaggregate queries. The rangeaggregate problem is defined as follows: Given a set of weighted points in R , compute the aggregate of weights of points that lie inside a ddimensional query rectangle. In this paper we focus on COUNT, SUM, AVG aggregates. First, we develop an indexing scheme for answering twodimensional rangeCOUNT queries that uses O(N=B) disk blocks and answers a query in O(log B N) I/Os, where N is the number of input points and B is the disk block size. This is the first optimal index structure for the 2D rangeCOUNT problem. The index can be extended to obtain a nearlinearsize indexing structure for answering rangeSUM queries using O(log B N) I/Os. We also obtain similar bounds for rectangleintersection aggregate queries, in which the input is a set of weighted rectangles and a query asks to compute the aggregate of the weights of those input rectangles that overlap with the query rectangle. This result immediately improves a recent result on temporalaggregate queries. Our indexing scheme can be dynamized and extended to higher dimensions. Finally, we demonstrate the practical efficiency of our index by comparing its performance against kdBtree. For a dataset of around 100 million points, the CRBtree query time is 810 times faster than the kdBtree query time. Furthermore, unlike other indexing schemes, the query performance of CRBtree is oblivious to the distribution of the input points and placement, shape and size of the query rectangle.
Efficient MaximaFinding Algorithms for Random Planar Samples
 Discrete Mathematics and Theoretical Computer Science (Electronic
, 2003
"... this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithmsone with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient t ..."
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Cited by 5 (2 self)
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this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithmsone with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient than existing ones
Partial Orders and Euclidean Geometry
 In Algorithms and Order, I. Rival
, 1987
"... The study of simple families of geometric objects on the plane has always been of great interest in mathematics. Incidence relations among families of points, lines and circles on the Euclidean plane were studied intensely long ago. In fact, most of what is now considered as basic Euclidean Geometry ..."
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Cited by 4 (1 self)
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The study of simple families of geometric objects on the plane has always been of great interest in mathematics. Incidence relations among families of points, lines and circles on the Euclidean plane were studied intensely long ago. In fact, most of what is now considered as basic Euclidean Geometry was developed a few centuries before the advent of the Christian Era.