Probabilistic Analysis of MULTIPLE QUICK SELECT

Mahmoud, Hosam M.; Smythe, R. T.

doi:10.1007/pl00009241

Cited by 14 publications

(7 citation statements)

References 10 publications

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“…The GD(θ) distributions (particularly for θ = 1 and θ = 2) also appear as the limits of certain random variables in Hoare's FIND algorithm on random permutations and its variants (see e.g. [12,16]). They also appear in the study of perpetuities (see [8]).…”

Section: Probabilistic Properties Of the Gd Distributionsmentioning

confidence: 99%

Random minimal directed spanning trees and Dickman-type distributions

Penrose

Wade

2004

Adv. Appl. Probab.

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Publisher's copyright statement:Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractIn Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a southwesterly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

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Section: Probabilistic Properties Of the Gd Distributionsmentioning

confidence: 99%

Random minimal directed spanning trees and Dickman-type distributions

Penrose

Wade

2004

Adv. Appl. Probab.

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“…Similarly, if k < m, then the algorithm recursively operates on the set of keys larger than the pivot and returns the (k − m)-th smallest key from the subset. Although previous studies (e.g., Knuth [11], Mahmoud et al [15], Prodinger [18], Grübel and U. Rösler [7], Lent and Mahmoud [14], Mahmoud and Smythe [16], Devroye [1], Hwang and Tsai [9]) examined Quickselect with regard to key comparisons, this study is the first to analyze the bit complexity of the algorithm.…”

Section: Introductionmentioning

confidence: 98%

Analysis of the Expected Number of Bit Comparisons Required by Quickselect

Fill

Nakama

2009

Algorithmica

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When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.

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“…There have been many studies of the random variables K n,m , including [2], [17], [10], [14], [9], [3], [12], [4], and [6], and several corresponding studies, including [19], [15], and [16], of the number(s) of key comparisons for an extension of QuickSelect called MultipleQuickSelect that searches simultaneously for multiple order statistics. Grübel and Rösler [10] analysed a modified version of QuickSelect that splits the collection of keys into two sets, those smaller than the pivot and those greater than or equal to the pivot, rather than into three sets (one of which has the pivot as its only element) as considered in this paper.…”

Section: Introductionmentioning

confidence: 99%

QuickSelect Tree Process Convergence, With an Application to Distributional Convergence for the Number of Symbol Comparisons Used by Worst-Case Find

Fill¹,

Matterer²

2014

Combinator. Probab. Comp.

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We dedicate this paper to the memory of our colleague and friend Philippe Flajolet.We define a sequence of tree-indexed processes closely related to the operation of the QuickSelect search algorithm (also known as Find) for all the various values of n (the number of input keys) and m (the rank of the desired order statistic among the keys). As a 'master theorem' we establish convergence of these processes in a certain Banach space, from which known distributional convergence results as n → ∞ about (1) the number of key comparisons required are easily recovered (a) when m/n → α ∈ [0, 1], and (b) in the worst case over the choice of m.From the master theorem it is also easy, for distributional convergence of (2) the number of symbol comparisons required, both to recover the known result in the case (a) of fixed quantile α and to establish our main new result in the case (b) of worst-case Find.Our techniques allow us to unify the treatment of cases (1) and (2) and indeed to consider many other cost functions as well. Further, all our results provide a stronger mode of convergence (namely, convergence in L p or almost surely) than convergence in distribution. Extensions to MultipleQuickSelect are discussed briefly.

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Probabilistic Analysis of MULTIPLE QUICK SELECT

Cited by 14 publications

References 10 publications

Random minimal directed spanning trees and Dickman-type distributions

Random minimal directed spanning trees and Dickman-type distributions

Analysis of the Expected Number of Bit Comparisons Required by Quickselect

QuickSelect Tree Process Convergence, With an Application to Distributional Convergence for the Number of Symbol Comparisons Used by Worst-Case Find

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