Box-trees and R-trees with near-optimal query time

Agarwal, Pankaj K.; Berg, Mark de; Gudmundsson, Joachim; Hammar, Mikael; Haverkort, Herman

doi:10.1145/378583.378645

Cited by 24 publications

(24 citation statements)

References 21 publications

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“…Our goal is to achieve the same performance as for a constant stabbing number, that is, O( √ N/B + T /B) memory transfers for rectangle queries, and O((N/B) ε + T /B) memory transfers for point queries. Unfortunately, the lower bounds of Agarwal et al [3] show that any R-tree must do Ω( √ N/B +T /B) memory transfers in the worst case for point queries if the stabbing number can be arbitrarily high. Therefore we slightly relax the definition of our bounding-box hierarchy:…”

Section: Our Resultsmentioning

confidence: 99%

“…In the internal-memory model, Agarwal et al [3] developed a box-tree structure, called the kd-interval tree, that answers a query Q in O( √ N + T ) time, where T is the number of reported rectangles [22]. They also proved that this is optimal for rectangle queries.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, if σ is the stabbing number of S, that is, the maximum number of rectangles in S containing any query point p, the kd-interval tree answers point queries in O(log 2 N) time for all values σ = O(log N); as σ increases to Θ(N), the point query time gradually degrades to O( √ N + T ). See [3,22] for surveys of earlier internal-memory box-tree results.…”

Section: Introductionmentioning

confidence: 99%

“…While many R-tree variants have been proposed-see for example [11,23,28] or refer to the surveys in [20,25]-it is only recently that a worst-case-efficient R-tree has been developed. Agarwal et al [3] described how to modify their kdinterval tree structure to obtain an external-memory structure that answers queries using O( √ N/B + T ) memory transfers. Very recently, Arge et al [8] developed the so-called priority R-tree (or PR-tree) that improves the bound to O( √ N/B + T /B), which is optimal for rectangle queries and for point queries on data with high stabbing number [3,24].…”

Section: Introductionmentioning

confidence: 99%

“…Agarwal et al [3] described how to modify their kdinterval tree structure to obtain an external-memory structure that answers queries using O( √ N/B + T ) memory transfers. Very recently, Arge et al [8] developed the so-called priority R-tree (or PR-tree) that improves the bound to O( √ N/B + T /B), which is optimal for rectangle queries and for point queries on data with high stabbing number [3,24]. For point queries on data with low stabbing number, no better bounds than the internal-memory results are known.…”

Section: Introductionmentioning

confidence: 99%

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Cache-Oblivious R-Trees

2007

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We develop a cache-oblivious data structure for storing a set S of N axisaligned rectangles in the plane, such that all rectangles in S intersecting a query rectangle or point can be found efficiently. Our structure is an axis-aligned bounding-box hierarchy and as such it is the first cache-oblivious R-tree with provable performance guarantees. If no point in the plane is contained in more than a constant number of rectangles in S, we can construct, for any constant ε, a structure that answers a rectangle query using O( √ N/B + T /B) memory transfers and a point query using O((N/B) ε ) memory transfers, where T is the number of reported rectangles and BL. 51 is the block size of memory transfers between any two levels of a multilevel memory hierarchy. We also develop a variant of our structure that achieves the same performance on input sets with arbitrary overlap among the rectangles. The rectangle query bound matches the bound of the best known linear-space cache-aware structure.

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Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cache-Oblivious R-Trees

2007

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Computational algorithms for tracking dynamic fluid–structure interfaces in embedded boundary methods

Wang¹,

Gretarsson²,

Main³

et al. 2012

Numerical Methods in Fluids

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SUMMARY A robust, accurate, and computationally efficient interface tracking algorithm is a key component of an embedded computational framework for the solution of fluid–structure interaction problems with complex and deformable geometries. To a large extent, the design of such an algorithm has focused on the case of a closed embedded interface and a Cartesian computational fluid dynamics grid. Here, two robust and efficient interface tracking computational algorithms capable of operating on structured as well as unstructured three‐dimensional computational fluid dynamics grids are presented. The first one is based on a projection approach, whereas the second one is based on a collision approach. The first algorithm is faster. However, it is restricted to closed interfaces and resolved enclosed volumes. The second algorithm is therefore slower. However, it can handle open shell surfaces and underresolved enclosed volumes. Both computational algorithms exploit the bounding box hierarchy technique and its parallel distributed implementation to efficiently store and retrieve the elements of the discretized embedded interface. They are illustrated, and their respective performances are assessed and contrasted, with the solution of three‐dimensional, nonlinear, dynamic fluid–structure interaction problems pertaining to aeroelastic and underwater implosion applications. Copyright © 2012 John Wiley & Sons, Ltd.

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Fast computation of the wall distance in unsteady Eulerian fluid‐structure computations

Grimberg

Farhat

2018

Numerical Methods in Fluids

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Summary Highly nonlinear, turbulent, dynamic, fluid‐structure interaction problems characterized by large structural displacements and deformations, as well as self‐contact and topological changes, are encountered in many applications. For such problems, the Eulerian computational framework, which is often equipped with an embedded (or immersed) boundary method for computational fluid dynamics, is often the most appropriate framework. In many circumstances, it requires the computation of the time‐dependent distance from each active mesh vertex of the embedding mesh to the nearest embedded discrete surface. Such circumstances include, for example, modeling turbulence using the Spalart‐Allmaras or detached eddy simulation turbulence models and performing adaptive mesh refinement in order to track the boundary layer. Evaluating at each time step the distance to the wall is computationally prohibitive, particularly in the context of explicit‐explicit fluid‐structure time‐integration schemes. Hence, this paper presents two complementary approaches for reducing this computational cost. The first one recognizes that many quantities depending on the wall distance are relatively insensitive to its inaccurate evaluation in the far field. Therefore, it simplifies a state‐of‐the‐art algorithm for computing the wall distance accordingly. The second approach relies on an effective wall distance error estimator to update the evaluation of the wall distance function only when otherwise, a quantity of interest that depends on it would become tainted by an unacceptable level of error. The potential of combining both approaches for dramatically accelerating the computation of the wall distance is demonstrated with the Eulerian simulation of the inflation of a disk‐gap‐band parachute system in a supersonic airstream.

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Box-trees and R-trees with near-optimal query time

Cited by 24 publications

References 21 publications

Cache-Oblivious R-Trees

Cache-Oblivious R-Trees

Computational algorithms for tracking dynamic fluid–structure interfaces in embedded boundary methods

Fast computation of the wall distance in unsteady Eulerian fluid‐structure computations

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