Squarish <i>k</i>-<i>d</i> Trees

Devroye, Luc; Jabbour, Jean; Zamora-Cura, Carlos

doi:10.1137/s0097539799358926

Cited by 29 publications

(34 citation statements)

References 14 publications

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“…, K − 1}. The squarish K-d trees of Devroye et al [DJZC00] try to achieve a more balanced partition of the space by discriminating along the coordinate for which the bounding box of the node is most elongated.…”

Section: K-d Trees and Partial Match Queriesmentioning

confidence: 99%

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On the Cost of Fixed Partial Match Queries in K-d Trees

2015

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Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as K-d trees or quadtrees. Given a query q = (q0, . . . , qK−1) where s of the coordinates are specified and K − s are left unspecified (qi = * ), a partial match search returns the subset of data points x = (x0, . . . , xK−1) in the data structure that match the given query, that is, the data points such that xi = qi whenever qi = * . There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query q. This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost Pn,q for a given fixed query q. From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for Pn,qin many multidimensional data structures, which differ only in the exponent α and the constant ν, both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in q as well. Although it is tempting to conjecture that this functional shape is "universal", we have shown experimentally that it seems not to be true for a variant of K-d trees called squarish K-d trees.

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Section: K-d Trees and Partial Match Queriesmentioning

confidence: 99%

“…Theorem 3 (Devroye et al [DJZC00]). The expected cost P n,s,K of a random PM query where s out of the K coordinates of the query are specified and the other K − s are not, in a random squarish K-d tree of size n is P n,s,K = Θ(n α(s/K) ), with α(x) = 1 − x.…”

Section: Introductionmentioning

confidence: 99%

On the Cost of Fixed Partial Match Queries in K-d Trees

2015

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“…Other possible structures of varying complexity include the random quadtree (which behaves as the random 2-d tree and yields similar expected times), the squarish 2-d tree of Devroye, Jabbour and Zamora-Cura [18] (for which the expected complexity is unknown but probably O(log 2 n)), and the random 2-d tree (or region quadtree).…”

Section: Resultsmentioning

confidence: 99%

Expected time analysis for Delaunay point location

Devroye

Lemaire

Moreau

2004

Computational Geometry

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We consider point location in Delaunay triangulations with the aid of simple data structures. In particular, we analyze methods in which a simple data structure is used to first locate a point close to the query point. For points uniformly distributed on the unit square, we show that the expected point location complexities are ( √ n ) for the Green-Sibson rectilinear search, (n 1/3 ) for Jump and Walk, (n 1/4 ) for BinSearch and Walk (which uses a 1-dimensional search tree), (n 0.056... ) for search based on a random 2-d tree, and (log n) for search aided by a 2-d median tree.

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“…k-d trees provide a hierarchic spatial relationship between data points, which can be used to find nearest neighbours, locate points within a zone, and for other searching operations, such as construction and point insertion of Delaunay triangulation [23,24].…”

Section: Construction Of Kd-treementioning

confidence: 99%

3D Delaunay triangulation of non-uniform point distributions

2014

Finite Elements in Analysis and Design

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a b s t r a c tIn view of the simplicity and the linearity of regular grid insertion, a multi-grid insertion scheme is proposed for the three-dimensional Delaunay triangulation of non-uniform point distributions by recursive application of the regular grid insertion to an arbitrary subset of the original point set. The fundamentals and difficulties of three-dimensional Delaunay triangulation of highly non-uniformly distributed points by the insertion method are reviewed. Current strategies and methods of point insertions for non-uniformly distributed spatial points are discussed. An enhanced kd-tree insertion algorithm with a specified number of points in a cell and its natural sequence derived from a sandwich insertion scheme is also presented.The regular grid insertion, the enhanced kd-tree insertion and the multi-grid insertion have been rigorously studied with benchmark non-uniform distributions of 0.4-20 million points. It is found that the kd-tree insertion is more efficient in locating the base tetrahedron, but it is also more sensitive to the triangulation of non-uniform point distributions with a large amount of conflicting elongated tetrahedra. Including the grid construction time, multi-grid insertion is the most stable and efficient for all the uniform and non-uniform point distributions tested.

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Squarish k-d Trees

Cited by 29 publications

References 14 publications

On the Cost of Fixed Partial Match Queries in K-d Trees

On the Cost of Fixed Partial Match Queries in K-d Trees

Expected time analysis for Delaunay point location

3D Delaunay triangulation of non-uniform point distributions

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