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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 140 (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.
LempelZiv parsing and sublinearsize index structures for string matching (Extended Abstract)
 Proc. 3rd South American Workshop on String Processing (WSP'96
, 1996
"... String matching over a long text can be significantly speeded up with an index structure formed by preprocessing the text. For very long texts, the size of such an index can be a problem. This paper presents the first sublinearsize index structure. The new structure is based on LempelZiv parsing ..."
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Cited by 55 (1 self)
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String matching over a long text can be significantly speeded up with an index structure formed by preprocessing the text. For very long texts, the size of such an index can be a problem. This paper presents the first sublinearsize index structure. The new structure is based on LempelZiv parsing of the text and has size linear in N, the size of the LempelZiv parse. For a text of length n, N = O(n = log n) and can be still smaller if the text is compressible. With the new index structure, all occurrences of a pattern string of length m can be found in time O(m 2
Efficient Data Structures for Range Searching on a Grid
, 1987
"... We consider the 2dimensional range searching problem in the case where all point lie on an integer grid. A new data structure is preented that solves range queries on a U U grid in O(k + loglog U) time using O(n log n) storage, where n is the number of points and k the number of reported answers ..."
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Cited by 39 (0 self)
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We consider the 2dimensional range searching problem in the case where all point lie on an integer grid. A new data structure is preented that solves range queries on a U U grid in O(k + loglog U) time using O(n log n) storage, where n is the number of points and k the number of reported answers. Although the query
Hierarchical representations of collections of small rectangles
 ACM Computing Surveys
, 1988
"... A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles. Rectangles are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object. They arise in applications in cartography as well as very larges ..."
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Cited by 28 (1 self)
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A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles. Rectangles are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object. They arise in applications in cartography as well as very largescale integration (VLSI) design rule checking. The different data structures are discussed in terms of how they support the execution of queries involving proximity relations. The focus is on intersection and subset queries. Several types of representations are described. Some are designed for use with the planesweep paradigm, which works well for static collections of rectangles. Others are oriented toward dynamic collections. In this case, one representation reduces each rectangle to a point in a higher multidimensional space and treats the problem as one involving point data. The other representation is area basedthat is, it depends on the physical extent of each rectangle.
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 23 (4 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.
Persistence in Computational Geometry
, 1995
"... We show how persistence can be used to solve a number of geometric problems where preprocessing is required to facilitate query answering. Efficient solutions for most of the problems discussed already exist in the literature; however, persistence provides an efficient and conceptually simpler al ..."
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Cited by 7 (0 self)
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We show how persistence can be used to solve a number of geometric problems where preprocessing is required to facilitate query answering. Efficient solutions for most of the problems discussed already exist in the literature; however, persistence provides an efficient and conceptually simpler alternative to existing solutions.
EXACT SIZE OF BINARY SPACE PARTITIONINGS AND IMPROVED RECTANGLE TILING ALGORITHMS
, 2002
"... We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n − 1 in general, and an upper bound of 2n − 1 if the rectangles tile the underlying space. This improves the upper bounds of ..."
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Cited by 5 (0 self)
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We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n − 1 in general, and an upper bound of 2n − 1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P.
The Rectangle Enclosure and PointDominance Problems Revisited
, 1994
"... We consider the problem of reporting the pairwise enclosures among a set of n axesparallel rectangles in IR 2 , which is equivalent to reporting dominance pairs in a set of n points in IR 4 . For more than ten years, it has been an open problem whether these problems can be solved faster tha ..."
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Cited by 3 (0 self)
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We consider the problem of reporting the pairwise enclosures among a set of n axesparallel rectangles in IR 2 , which is equivalent to reporting dominance pairs in a set of n points in IR 4 . For more than ten years, it has been an open problem whether these problems can be solved faster than in O(n log 2 n+k) time, where k denotes the number of reported pairs. First, we give a divideandconquer algorithm that matches the O(n) space and O(n log 2 n + k) time bounds of the algorithm of Lee and Preparata [LP82], but is simpler. Then we give another algorithm that uses O(n) space and runs in O(n log n log log n+k log log n) time. For the special case where the rectangles have at most ff different aspect ratios, we give an algorithm that runs in O(ffn log n + k) time and uses O(n) space. 1 Introduction The problem of computing intersections in a set of rectangles has received much attention. (See Chapter 8 of [PS88].) There are several variants of the problem depending...
under supervision of
, 2001
"... Nonparametric maximum likelihood estimation for bivariate censored data by ..."
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Nonparametric maximum likelihood estimation for bivariate censored data by
© 1996 SpringerVerlag New York Inc. Using Topological Sweep to Extract the Boundaries of Regions in Maps Represented by Region Quadtrees 1
"... Abstract. A variant of the planesweep paradigm known as topological sweep is adapted to solve geometric problems involving twodimensional regions when the underlying representation is a region quadtree. The utility of this technique is illustrated by showing how it can be used to extract the bound ..."
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Abstract. A variant of the planesweep paradigm known as topological sweep is adapted to solve geometric problems involving twodimensional regions when the underlying representation is a region quadtree. The utility of this technique is illustrated by showing how it can be used to extract the boundaries of a map in O(M) space and O(Mα(M)) time, where M is the number of quadtree blocks in the map, and α(·) is the (extremely slowly growing) inverse of Ackerman’s function. The algorithm works for maps that contain multiple regions as well as holes. The algorithm makes use of active objects (in the form of regions) and an active border. It keeps track of the current position in the active border so that at each step no search is necessary. The algorithm represents a considerable improvement over a previous approach whose worstcase execution time is proportional to the product of the number of blocks in the map and the resolution of the quadtree (i.e., the maximum level of decomposition). The algorithm works for many different quadtree representations including those where the quadtree is stored in external storage.