Analysis of the Expected Number of Bit Comparisons Required by Quickselect

Fill, James Allen; Nakama, Takéhiko

doi:10.1007/s00453-009-9294-3

Cited by 16 publications

(10 citation statements)

References 18 publications

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“…Thus, in these cases the expected number of bit comparisons is asymptotically larger than that of key comparisons required to complete the same task only by a constant factor, since the expectation for key comparisons is asymptotically 2n. Fill and Nakama [8] also found that the expected number of bit comparisons required by QuickRand is also asymptotically linear in n (with slope approximately 8.207 31), as for key comparisons (with slope 3).…”

Section: Introductionmentioning

confidence: 91%

“…Many studies have examined this algorithm to quantify its execution costs-a nonexhaustive list of references is [5], [8]- [10], [12]- [16], [18], [23]; all of them except for Fill and Nakama [8] and Vallée et al [23] have conducted the quantification with regard to the number of key comparisons required by the algorithm to achieve its task. As a result, most of the theoretical results on the complexity of QuickSelect are about expectations or distributions for the number of required key comparisons.…”

Section: Introductionmentioning

confidence: 99%

“…By closely following [7], Fill and Nakama [8] studied the expected number of bit comparisons required by QuickSelect. More precisely, they treated the case of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

“…Vallée et al [23] extended the average-case analyses of [7] and [8] to keys represented by sequences of general symbols generated by any of a wide variety of sources that include memoryless, Markov, and other dynamical sources. They broadly extended the results of [8] in another direction as well by treating QuickQuant(n, α) for general α ∈ [0, 1], not just QuickMin, QuickMax, and QuickRand. Here the algorithm QuickQuant(n, α) (for 'quick quantile') refers to QuickSelect(n, m n ) with m n /n → α.…”

Section: Introductionmentioning

confidence: 99%

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Distributional Convergence for the Number of Symbol Comparisons Used by Quickselect

Fill

Nakama

2013

Advances in Applied Probability

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Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness" condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by n. Additionally, under a condition that grows more restrictive as p increases, we have convergence of moments of orders p and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased ("fair") bits, the mean exhibits periodic fluctuations of order n.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

“…By closely following [7], Fill and Nakama [8] studied the expected number of bit comparisons required by QuickSelect. More precisely, they treated the case of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distributional Convergence for the Number of Symbol Comparisons Used by Quickselect

Fill

Nakama

2013

Advances in Applied Probability

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“…So, it is not straightforward to compare Quick Sort to a radix-based algorithm such as Radix Sort without considering the number of bit comparisons in both. We refer the reader to the recent work of Fill and Janson (2004), Fill and Nakama (2008), and Vallée et al (2009). This renewed interest in the analysis of algorithms from a bits point of view provided the motivation for us to consider radix selection, and to perform an analysis of the number of swaps, to further put in perspective a meaningful distinction between radix-based methods and comparison-based methods.…”

Section: Radix Methodsmentioning

confidence: 99%

Analysis of swaps in radix selection

Elmasry

Mahmoud

2011

Advances in Applied Probability

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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 ≠ ½) and the unbiased case (p = ½). 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.

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Towards a Realistic Analysis of the QuickSelect Algorithm

et al. 2015

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We revisit the analysis of the classical QuickSelect algorithm. Usually, the analysis deals with the mean number of key comparisons, but here we view keys as words produced by a source, and words are compared via their symbols in lexicographic order. Our probabilistic models belong to a broad category of information sources that encompasses memoryless (i.e., independentsymbols) and Markov sources, as well as many unbounded-correlation sources. The "realistic" cost of the algorithm is here the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons. For the QuickSort algorithm, known average-case complexity results are of Θpn log nq in the case of key comparisons, and Θpn log 2 nq for symbol comparisons. For QuickSelect algorithms, and with respect to key comparisons, the average-case complexity is Θpnq. In this present article, we prove that, with respect to symbol comparisons, QuickSelect's average-case complexity remains Θpnq. In each case, we provide explicit expressions for the dominant constants, closely related to the probabilistic behaviour of the source.We began investigating this research topic with Philippe Flajolet, and the short version of the present paper (the ICALP'2009 paper) was written with him. As usual, Philippe played a central role, notably on the following points: introduction of the QuickVal algorithm, tameness of sources, and use of the Rice's method. He also made many experiments exhibiting the asymptotic slope ρpαq and plotted nice graphs, which are reproduced in this paper. Even though the extended abstract does not provide any proof of the analysis of the algorithm QuickQuant, Philippe also devised with us a precise plan for this proof which has now completely been written. For all these reasons, we could have added (and certainly would have liked to add) Philippe as a co-author of this paper. On the other hand, Philippe was extremely exacting of how his papers were to be written and organised, and we cannot be sure that he would have liked or validated our editing choices. In the end, this is why we have decided not to include him as a co-author, but instead, to dedicate, with deference and affection, this paper to his memory. Thank you, Philippe!

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Analysis of the Expected Number of Bit Comparisons Required by Quickselect

Cited by 16 publications

References 18 publications

Distributional Convergence for the Number of Symbol Comparisons Used by Quickselect

Distributional Convergence for the Number of Symbol Comparisons Used by Quickselect

Analysis of swaps in radix selection

Towards a Realistic Analysis of the QuickSelect Algorithm

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