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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 ..."
Abstract

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.
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 24 (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.
Concatenable Structures for Decomposable Problems
, 1989
"... Given a data structure and an ordering on the objects it contains, we study methods for obtaining a modified data structure that allows for splits and concatenations with respect to that ordering. A general technique will be given, which works for all data structures for decomposable searching probl ..."
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Given a data structure and an ordering on the objects it contains, we study methods for obtaining a modified data structure that allows for splits and concatenations with respect to that ordering. A general technique will be given, which works for all data structures for decomposable searching problems and order decomposable set problems. Furthermore, the results imply a new method for adding range restrictions to data structures. Applications include e.g. a version of an interval tree that allows for splitting and concatenating on the length of the intervals, a version of the ddimensional kd tree that allows for splitting and concatenating on all coordinates, and a data structure on points in the plane that allows for reporting the convex hull of the points in a given query rectangle.