Analysis of quickselect : an algorithm for order statistics

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

doi:10.1051/ita/1995290402551

Cited by 56 publications

(55 citation statements)

References 11 publications

(14 reference statements)

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“…The asymptotic grand average 2(H p + 1)n from Prodinger (1995) and Lent and Mahmoud (1996) suggests the use of this factor for centering. We use n as a scale factor since it is the order of the standard deviation for the case p = 1 (see Mahmoud et al, 1995), and we anticipate it to be the order of the variance for higher p as well; we shall see that indeed n is the exact order of the standard deviation for all fixed p. We normalize the random variable C…”

Section: Discussionmentioning

confidence: 99%

“…where we used the boundary condition v 1 = 1, from Mahmoud et al (1995). The sums involved in g p vary in complexity.…”

Section: The Variancementioning

confidence: 99%

“…The limit distribution of QUICK SORT has been characterized as the fixed-point of a contraction mapping (Rösler, 1991); an existential but nonconstructive proof based on martingales can also be found in Régnier (1989). For QUICK SELECT (the special case p = 1 of MQS), the limit distribution has been found to be infinitely divisible (Mahmoud et al, 1995). By an argument based on Rösler (1991) and a coupling between Y ( p) n and its proposed limit Y ( p) that generalizes that of the case p = 1, we shall prove that Y ( p) n D −→ Y ( p) .…”

Section: The Variancementioning

confidence: 99%

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Probabilistic Analysis of MULTIPLE QUICK SELECT

Mahmoud

Smythe

1998

Algorithmica

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We investigate the distribution of the number of comparisons made by MULTIPLE QUICK SELECT (a variant of QUICK SORT for finding order statistics). By convergence in the Wasserstein metric space, we show that a limit distribution exists for a suitably normalized version of the number of comparisons. We characterize the limiting distribution by an inductive convolution and find its variance. We show that the limiting distribution is smooth and prove that it has a continuous density with unbounded support.

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Section: Discussionmentioning

confidence: 99%

“…where we used the boundary condition v 1 = 1, from Mahmoud et al (1995). The sums involved in g p vary in complexity.…”

Section: The Variancementioning

confidence: 99%

Section: The Variancementioning

confidence: 99%

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Probabilistic Analysis of MULTIPLE QUICK SELECT

Mahmoud

Smythe

1998

Algorithmica

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“…We seek a grand average of all averages, a grand variance of all variances, and a grand (average) distribution of all distributions with a fixed r as a global measure over all possible order statistics. This smoothing technique was introduced in Mahmoud et al (1995), and was used successfully in Lent and Mahmoud (1996) and Prodinger (1995).…”

Section: The Data Modelmentioning

confidence: 99%

Analysis of swaps in radix selection

Elmasry

Mahmoud

2011

Adv. Appl. Probab.

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Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p = 1 2 ) and the unbiased case (p = 1 2 ). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.Keywords: Random structure; algorithm; order statistic; recurrence; perpetuity; phase change; digital data; digital sorting; selection; Mellin transform; poissonization; depoissonization 2010 Mathematics Subject Classification: Primary 60C05 Secondary 60F05; 68P10 Radix methodsRadix Sort is a sorting technique based on analyzing the digital composition of keys. Digits (possibly bits at the lowest machine level) are extracted from keys and used to classify the keys. Radix Sort dates back to the nineteenth century and can be found in the work of Hollerith (1894) on tabulating machines. It provides a good alternative for comparison-based sorting algorithms like Quick Sort and Merge Sort, where keys are compared according to an ordering relation; see Knuth (1998, pp. 116-119), Mahmoud (2000, pp. 148-151), and Mehlhorn (1984, pp. 42-68) for a broad discussion of these sorting algorithms.Two different variants of Radix Sort are known, the least-significant-digit (LSD) Radix Sort and the most-significant-digit (MSD) Radix Sort. The LSD Radix Sort starts with the leastsignificant digit and orders the keys accordingly. It repeatedly orders the keys moving towards the most-significant digit and using one digit at a time; see, for example, Cormen et al. (2001, p. 178) for details. To guarantee the correctness of the LSD variant, every ordering round

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“…Further we refer to the limit law of Quickselect, see Mahmoud, Modarres and Smythe [12] and Hwang and Tsai [8], which is also Dickman's infinitely divisible distribution.…”

Section: Introductionmentioning

confidence: 99%