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24
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 70 (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.
New data structures for orthogonal range searching
 In Proc. 41st IEEE Symposium on Foundations of Computer Science
, 2000
"... ..."
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Implementation of the ROSE Algebra: Efficient Algorithms for RealmBased Spatial Data Types
 PROC. OF THE 4TH INTL. SYMPOSIUM ON LARGE SPATIAL DATABASES
, 1995
"... The ROSE algebra, defined earlier, is a system of spatial data types for use in spatial database systems. It offers data types to represent points, lines, and regions in the plane together with a comprehensive set of operations; semantics of types and operations have been formally defined. Values ..."
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Cited by 37 (14 self)
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The ROSE algebra, defined earlier, is a system of spatial data types for use in spatial database systems. It offers data types to represent points, lines, and regions in the plane together with a comprehensive set of operations; semantics of types and operations have been formally defined. Values of these data types have a quite general structure, e.g. an object of type regions may consist of several polygons with holes. All ROSE objects are realmbased which means all points and vertices of objects lie on an integer grid and no two distinct line segments of any two objects intersect in their interior. In this paper we describe the implementation of the ROSE algebra, providing data structures for the types and new realmbased geometric algorithms for the operations. The main techniques used are (parallel) traversal of objects, planesweep, and graph algorithms. All algorithms are analyzed with respect to their worst case time and space requirements. Due to the realm properties, these algorithms are relatively simple, efficient, and numerically completely robust. All data structures and algorithms have indeed been implemented in the ROSE system; the Modula2 source code is freely available from the authors for study or use.
Indexing and Dictionary Matching with One Error (Extended Abstract)
, 1999
"... The indexing problem is the one where a text is preprocessed and subsequent queries of the form: "Find all occurrences of pattern P in the text" are answered in time proportional to the length of the query and the number of occurrences. In the dictionary matching problem a set of patterns is preproc ..."
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Cited by 24 (2 self)
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The indexing problem is the one where a text is preprocessed and subsequent queries of the form: "Find all occurrences of pattern P in the text" are answered in time proportional to the length of the query and the number of occurrences. In the dictionary matching problem a set of patterns is preprocessed and subsequent queries of the form: "Find all occurrences of dictionary patterns in text T" are answered in time proportional to the length of the text and the number of occurrences. There exist efficient worstcase solutions for the indexing problem and the dictionary matching problem, but none that find approximate occurrences of the patterns, i.e. where the pattern is within a bound edit (or hamming...
Optimizing the Collision Detection Pipeline
 In The First International Game Technology Conference GTEC
, 2001
"... A general framework for collision detection is presented. Then, we look at each stage and compare different approaches by extensive benchmarks. The results suggest a way to optimize the performance of the overall framework. ..."
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Cited by 18 (4 self)
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A general framework for collision detection is presented. Then, we look at each stage and compare different approaches by extensive benchmarks. The results suggest a way to optimize the performance of the overall framework.
Range searching over tree cross products
 In Proc. 8th European Symposium on Algorithms (ESA
, 2000
"... Abstract. We introduce the tree crossproduct problem, which abstracts a data structure common to applications in graph visualization, string matching, and software analysis. We design solutions with a variety of tradeoffs, yielding improvements and new results for these applications. 1 ..."
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Cited by 18 (0 self)
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Abstract. We introduce the tree crossproduct problem, which abstracts a data structure common to applications in graph visualization, string matching, and software analysis. We design solutions with a variety of tradeoffs, yielding improvements and new results for these applications. 1
MultiMethod Dispatching: A Geometric Approach with Applications to String Matching Problems
, 1999
"... Current object oriented programming languages (OOPLs) rely on monomethod dispatching. Recent research has identified multimethods as a new, powerful feature to be added to OOPLs, and several experimental OOPLs now have multimethods. Their ultimate success and impact in practice depends, among ..."
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Cited by 15 (3 self)
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Current object oriented programming languages (OOPLs) rely on monomethod dispatching. Recent research has identified multimethods as a new, powerful feature to be added to OOPLs, and several experimental OOPLs now have multimethods. Their ultimate success and impact in practice depends, among other things, on whether multimethod dispatching can be supported efficiently. We show that the multimethod dispatching problem can be transformed to a geometric problem on multidimensional integer grids, for which we then develop a data structure that uses nearlinear space and has O(log log n) query time. This gives a solution whose performance almost matches that of the best known algorithm for standard monomethod dispatching. Our geometric data structure has other applications as well, namely in two string matching problems: matching multiple rectangular patterns against a rectangular query text, and approximate dictionary matching with edit distance at most one. Our results f...
Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing
"... We present a succinct representation of a set of n points on an n × n grid using n lg n + o(nlg n) bits 3 to support orthogonal range counting in O(lg n / lg lg n) time, and range reporting in O(k lg n/lg lg n) time, where k is the size of the output. This achieves an improvement on query time by ..."
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Cited by 15 (0 self)
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We present a succinct representation of a set of n points on an n × n grid using n lg n + o(nlg n) bits 3 to support orthogonal range counting in O(lg n / lg lg n) time, and range reporting in O(k lg n/lg lg n) time, where k is the size of the output. This achieves an improvement on query time by a factor of lg lg n upon the previous result of Mäkinen and Navarro [15], while using essentially the informationtheoretic minimum space. Our data structure not only can be used as a key component in solutions to the general orthogonal range search problem to save storage cost, but also has applications in text indexing. In particular, we apply it to improve two previous spaceefficient text indexes that support substring search [7] and positionrestricted substring search [15]. We also use it to extend previous results on succinct representations of sequences of small integers, and to design succinct data structures supporting certain types of orthogonal range query in the plane.
Orthogonal Range Searching on the RAM, Revisited
, 2011
"... We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and ..."
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Cited by 15 (4 self)
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We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves the previous results by Alstrup, Brodal, and Rauhe (FOCS’00), with O(n lg ε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. Our second data structure uses O(n) space and answers queries in O(lg ε n) time. The best previous O(n)space data structure, due to Nekrich (WADS’07), answers queries in O(lg n / lg lg n) time. 2. We give a data structure for 3d orthogonal range reporting with O(n lg 1+ε n) space and O(lg lg n+ k) query time for points in rank space, for any constant ε> 0. This improves the previous results by Afshani (ESA’08), Karpinski and Nekrich (COCOON’09), and Chan (SODA’11), with O(n lg 3 n) space and O(lg lg n + k) query time, or with O(n lg 1+ε n) space and O(lg 2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.