Data-Specific Analysis of String Sorting

Seidel, Raimund

doi:10.1137/1.9781611973075.102

Cited by 5 publications

(7 citation statements)

References 5 publications

(8 reference statements)

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“…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%

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%

Towards a Realistic Analysis of the QuickSelect Algorithm

et al. 2015

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“…Seidel [24] proves the following result, which is now described in our framework. Consider the subset U w of U which gathers all the words which begin by the prefix w, and the set P (U) of common prefixes of U, defined as the prefixes w for which the cardinality |U w | is at least equal to 2.…”

Section: Another Way To Relate the Number Of Key Comparisons And Symbmentioning

confidence: 65%

“…(e) We then discuss the faithfulness of the algorithms. This notion was recently introduced by Seidel [24], and we obtain here a natural characterization of this notion, from which we easily prove that the algorithms QuickSort and InsSort are faithful, whereas the algorithm BubSort is not faithful. Seidel used the faithfulness property to obtain an interesting relation between the two measures of interest -mean number of key comparisons and mean number of symbol comparisons -in the case of a faithful algorithm.…”

Section: (D)mentioning

confidence: 86%

“…In Section 7.1 we recall the approach due to Seidel [24], together with his notion of faithfulness (Section 7.2). We show that this approach leads to analysis of trie parameters (Section 7.3).…”

Section: Alternative Proofs and Lower Boundsmentioning

confidence: 99%

“…If it is not k-good, it is said to be k-bad. We will use the following lemma due to Seidel [24]. Lemma 7.6 (Seidel).…”

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confidence: 99%

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Towards a Realistic Analysis of Some Popular Sorting Algorithms

Clément¹,

Thi²,

Vallée³

2014

Combinator. Probab. Comp.

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International audienceWe describe a general framework for realistic analysis of sorting algorithms, and we applyit to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort,BubbleSort). Usually the analysis deals with the mean number of key comparisons, buthere we view keys as words produced by the same source, which are compared via theirsymbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total numberof symbol comparisons performed by the algorithm, and, in this context, the average-caseanalysis aims to provide estimates for the mean number of symbol comparisons used by thealgorithm. For sorting algorithms, and with respect to key comparisons, the average-casecomplexity of QuickSort is asymptotic to 2n log n, InsertionSort to n^2/4 and BubbleSort ton^2/2. With respect to symbol comparisons, we prove that their average-case complexitybecomes Θ(n log_2 n), Θ(n^2), Θ(n^2 log n). In these three cases, we describe the dominantconstants which exhibit the probabilistic behaviour of the source (namely entropy andcoincidence) with respect to the algorithm

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Upper Tail Analysis of Bucket Sort and Random Tries

Bercea

Even

2021

Lecture Notes in Computer Science

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Data-Specific Analysis of String Sorting

Cited by 5 publications

References 5 publications

Towards a Realistic Analysis of the QuickSelect Algorithm

Towards a Realistic Analysis of the QuickSelect Algorithm

Towards a Realistic Analysis of Some Popular Sorting Algorithms

Upper Tail Analysis of Bucket Sort and Random Tries

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