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Optimal Sampling Strategies in Quicksort and Quickselect
 PROC. OF THE 25TH INTERNATIONAL COLLOQUIUM (ICALP98), VOLUME 1443 OF LNCS
, 1998
"... It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median ..."
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Cited by 28 (4 self)
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It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median of the sample makes a more accurate estimate of the median of the array to be sorted, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show that the optimal sample size to minimize the average total cost of quicksort (which includes both comparisons and exchanges) is s = a \Delta p n + o( p n ). We also give a closed expression for the constant factor a, which depends on the medianfinding algorithm and the costs of elementary comparisons and exchanges. The result above holds in most situations, unless the cost of an exchange exceeds by far the cost of a comparison. In that particular case, it is better to select not the median of...
An improved master theorem for divideandconquer recurrences
 In Automata, languages and programming
, 1997
"... Abstract. This paper presents new theorems to analyze divideandconquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll fun ..."
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Cited by 12 (2 self)
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Abstract. This paper presents new theorems to analyze divideandconquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll functions and weight distributions, stochastic recurrences included.
Asymptotic normality of recursive algorithms via martingale difference arrays
 Discrete Mathematics and Theoretical Computer Science
, 2000
"... We propose martingale central limit theorems as an appropriate tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs LN, then L ..."
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Cited by 7 (0 self)
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We propose martingale central limit theorems as an appropriate tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs LN, then LN D = Ln + ¯LN−n + RN for N ≥ n0 ≥ 2, where n follows a certain distribution PN on the integers {0,...,N} and Lk D = ¯Lk for k ≥ 0. Ln, LN−n and RN are independent, conditional on n, and RN are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N − n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to ZN: = LN−IE LN √. Under certain Var LN compatibility assumptions on the sequence (PN)N≥0 we show that a pair of sufficient conditions (of Lyapunov type) for ZN D → N (0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (IE LN)N≥0. In the case that the PN are binomial distributions with the same parameter p, and for deterministic RN, we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (RN)N≥0 (and for the scale RN = N α a characterization of those α) leading to asymptotic normality of ZN.