Orthogonal Range Reporting in Three and Higher Dimensions

Afshani, Peyman; Arge, Lars; Larsen, Kasper Dalgaard

doi:10.1109/focs.2009.58

Cited by 31 publications

(64 citation statements)

References 29 publications

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“…The space bound is optimal if the query cost is O(log c B N + T /B), where c > 0 is any arbitrary constant [1]. For threedimensions, there is also another tradeoff with optimal O(log B N +T /B) query cost, using O(N (log N/ log log B N ) 3 ) space [1].…”

Section: Higher Dimensionsmentioning

confidence: 99%

“…This is optimal if points can be stored only once [13]. If one is willing to spend super-linear space, the best known data structure uses O(N/B · (log N/ log log B N ) d−1 ) space and answers queries in O(log B N (log N/ log log B N ) d−2 + T /B) I/Os [1]. The space bound is optimal if the query cost is O(log c B N + T /B), where c > 0 is any arbitrary constant [1].…”

Section: Higher Dimensionsmentioning

confidence: 99%

“…If one is willing to spend super-linear space, the best known data structure uses O(N/B · (log N/ log log B N ) d−1 ) space and answers queries in O(log B N (log N/ log log B N ) d−2 + T /B) I/Os [1]. The space bound is optimal if the query cost is O(log c B N + T /B), where c > 0 is any arbitrary constant [1]. For threedimensions, there is also another tradeoff with optimal O(log B N +T /B) query cost, using O(N (log N/ log log B N ) 3 ) space [1].…”

Section: Higher Dimensionsmentioning

confidence: 99%

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I/O-efficient spatial data structures for range queries

Arge

Larsen

2012

SIGSPATIAL Special

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Range reporting is a one of the most fundamental topics in spatial databases and computational geometry. In this class of problems, the input consists of a set of geometric objects, such as points, line segments, rectangles etc. The goal is to preprocess the input set into a data structure, such that given a query range, one can efficiently report all input objects intersecting the range. The ranges most commonly considered are axis-parallel rectangles, halfspaces, points, simplices and balls.

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Section: Higher Dimensionsmentioning

confidence: 99%

Section: Higher Dimensionsmentioning

confidence: 99%

Section: Higher Dimensionsmentioning

confidence: 99%

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I/O-efficient spatial data structures for range queries

Arge

Larsen

2012

SIGSPATIAL Special

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“…Hence, we can use the data structure of Afshani et al [2], which uses O(n log n/ log log n) storage and has O(log n + #answers) query time. For data structure D 4 we use the point-enclosure data structure developed by Chazelle [6], which uses O(n) storage and can be used to report all rectangles in O containing a query point in O(log n + #answers) time.…”

Section: The Planar Casementioning

confidence: 99%

Finding Pairwise Intersections Inside a Query Range

2017

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We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size O(n · polylog n) and with query time O((k + 1) · polylog n) time, where k is the number of reported pairs, for two classes of objects in R 2 : axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R 3 , we present a data structure of size O(n √ n · polylog n) and query time O(( √ n + k) · polylog n). When the objects and query are fat, we obtain O((k + 1) · polylog n) query time using O(n · polylog n) storage.

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“…Recently, Afshani et al [3,4] presented an optimal structure for the general problem in three dimensions, namely a structure that uses O(n(log n/ log log n)…”

Section: Orthogonal Range Reportingmentioning

confidence: 99%

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

Afshani

Arge

Larsen

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in ddimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n + t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log 2 n/ log log n + t) has not been improved for decades.We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log 2 n + t) query time in four dimensions. More precisely, we develop a structure that uses O(n(log n/ log log n) d ) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O(log n(log n/ log log n) d−4+1/(d−2) +t) time. Ignoring log log n factors, this speeds up the best previous query time by a log 1−1/(d−2) n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n(log n/ log h) d−2 + t) time * This research was done while the author was a postdoctoral researcher at Dalhousie University, and was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through the Postdoctoral Fellowships Program (PDF). † Is supported in part by MADALGO and in part by a Google Fellowship in Search and Information Retrieval ‡ Center for Massive Data Algorithmics, a center of the Danish National Research Foundation.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ log d−2+ε n where ε > 0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.

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Orthogonal Range Reporting in Three and Higher Dimensions

Abstract: In orthogonal range reporting we are to preprocess

Cited by 31 publications

References 29 publications

I/O-efficient spatial data structures for range queries

I/O-efficient spatial data structures for range queries

Finding Pairwise Intersections Inside a Query Range

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

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