Towards a Realistic Analysis of Some Popular Sorting Algorithms

Clément, Julien; Thi, Thu Hien Nguyen; Vallée, Brigitte

doi:10.1017/s0963548314000649

Cited by 8 publications

(20 citation statements)

References 23 publications

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“…We exhibit a striking fact: the constants change but the complexity remains linear. The situation is completely different for sorting algorithms: in [4], we study for instance the QuickSort algorithm, and we show that the mean number of comparisons goes from Θpn log nq to Θpn log 2 nq depending if we consider key or symbol comparisons on a "nice" source. Moreover, there are intermittent sources (of type described in Section 2.4) on which the complexity of QuickSort becomes Θpn β q for some β ą 1.…”

Section: Resultsmentioning

confidence: 99%

“…We have conducted such a study in [4] for the mean number of comparisons performed by sorting algorithms: we start with the decision tree used in the classical key comparison model, and we "mix" it with a trie, along the approach described by Seidel [24]. However, for the selection problem, the lower bound is already more intricate in the classical key comparison model, at least for a general rank m, even though there are some results given in [16], for instance.…”

Section: Resultsmentioning

confidence: 99%

“…Proof This proof is an easy extension of the proof which is done in [4] for the QuickMin algorithm. Using the reflected source p S described in Section 2.6, it is sufficient to deal with one of the two cases for U.…”

Section: Study Of Integrals Over Trianglesmentioning

confidence: 93%

“…Then, there are two long papers which are journal versions of the previous ones: The paper [4] is the journal version of [3], whereas the present paper can be viewed as the journal version of [26].…”

Section: Introductionmentioning

confidence: 99%

“…The paper [4] focuses on the three sorting algorithms QuickSort, InsSort, BubSort, whereas the present paper adapts the general method to the algorithms of the QuickSelect class, for which it provides a complete analysis. This analysis is particular and somehow indirect, as was already announced in [3]: it does not seem possible to directly analyse QuickSelect, and there is a "detour" via the QuickVal algorithm.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Study Of Integrals Over Trianglesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations