Page usage in a quadtree index

Hoshi, Mamoru; Flajolet, Philippe

doi:10.1007/bf02074876

Cited by 18 publications

(25 citation statements)

References 13 publications

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“…We push the analysis further and derive a closed form for the generating functions of D n and en using the hypergeometric equation known to play a crucial role in similar analyses [10], [20]. In this way the distribution of search costs becomes expressible as a complicated convolution of Stirling numbers and asymptotic normality results.…”

Section: •mentioning

confidence: 99%

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Search costs in quadtrees and singularity perturbation asymptotics

Flajolet

Lafforgue

1994

Discrete Comput Geom

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Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees.The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions

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Section: •mentioning

confidence: 99%

“…They are discussed in classical treatises on algorithms [18], [31] and examined in great detail in Samet's reference books [29], [20]. Their analysis has made tangible progress over recent years [7], [10], [12], [20], [23], [27].…”

Section: Introductionmentioning

confidence: 99%

Search costs in quadtrees and singularity perturbation asymptotics

Flajolet

Lafforgue

1994

Discrete Comput Geom

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“…[] The techniques of this paper have been applied recently by Hoshi and Flajolet [20] to determine the storage occupation of paged quadtrees used as indexes for two-dimensional data. As a special case, the proportion of leaves in a randomly grown standard quadtree is asymptotically (4re z -39).…”

mentioning

confidence: 99%

Analytic variations on quadtrees

Flajolet

Gonnet²,

Puech

et al. 1993

Algorithmica

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Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtrees--fully specified and partial-match queries. The data model assumes random points with independently drawn coordinate values.The analysis leads to a class of "full-history" divide-and-conquer recurrences. These recurrences are solved using generating functions, either exactly for dimension d = 2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions rely on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques.These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.

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“…Papers [95], [105], [108], [119], and [124] deal with quadtrees; a quadtree is a tree to store a geometric object of dimension 2 (there are also generalizations to higher dimensions). Unlike a binary search tree, where an element goes either left or right depending on whether it is smaller or larger than the element in the root, there are now four subtrees, conveniently denoted by northwest, northeast, southwest, and southeast.…”

Section: Quadtreesmentioning

confidence: 99%

“…Unlike a binary search tree, where an element goes either left or right depending on whether it is smaller or larger than the element in the root, there are now four subtrees, conveniently denoted by northwest, northeast, southwest, and southeast. Paper [105] studies the idea of paging; with a design parameter b, a node can hold up to b data (they are stored sequentially, as they come in). Then the expected number of pages is asymptotic to γ b · n, with…”

Section: Quadtreesmentioning

confidence: 99%

Philippe Flajolet's Research in Analysis of Algorithms and Combinatorics

Prodinger¹,

Szpankowski²

1998

Algorithmica

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Philippe Flajolet's research in theoretical computer science spans over more than 20 years. He made lasting contributions to the analysis of algorithms and analytic combinatorics. Among many of his results we mention here some in such diversified topics as enumeration, number theory, formal languages, continued fractions, automatic analysis of algorithms, Mellin transform, digital sums, recurrences, trees, random generation of combinatorial objects, random graphs and mappings, polynomial factorization, communications, codes, graphics, etc. This gives only a small snapshot of his work, and we encourage the reader to visit Flajolet's homepage http://www-rocq.inria.fr/algo/flajolet/index.html for a fuller account. The bibliography was taken from his homepage and may not be totally complete, although we added a few items. Our paper was written without giving any prior notice to Philippe Flajolet. It reflects the view of the authors and any misunderstandings and shortcomings should be put on their account. Register Function and Gray Code. We start with a section about the register function and Gray codes since Flajolet's research in the analysis of algorithms started with papers [13] and[20] about the register function of binary trees (see also [58] for an extension to more general trees). The register function reg(t) of a binary tree is the minimal number of auxiliary registers to evaluate the tree t. If t has subtrees t 1 and t 2 , thenwith the initial value reg( ) = 0. Assume now that all binary trees with n internal nodes (= operators) are equally likely (there are (1/(n + 1)) 2n n of them), then the problem is to compute the average register function R n . It turns out to be R n = (n + 1) k≥1

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Page usage in a quadtree index

Cited by 18 publications

References 13 publications

Search costs in quadtrees and singularity perturbation asymptotics

Search costs in quadtrees and singularity perturbation asymptotics

Analytic variations on quadtrees

Philippe Flajolet's Research in Analysis of Algorithms and Combinatorics

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