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The Number of Bit Comparisons Used by Quicksort: An Averagecase Analysis
, 2003
"... The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Q ..."
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Cited by 16 (7 self)
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The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed in terms of the number of key comparisons. We introduce the prospect of a fair comparison between algorithms of the two types by providing an averagecase analysis of the number of bit comparisons required by Quicksort. Counting bit comparisons rather than key comparisons introduces an extra logarithmic factor to the asymptotic average total. We also provide a new algorithm, "BitsQuick" that reduces this factor to constant order by eliminating needless bit comparisons.
THE LIMITING DISTRIBUTION FOR THE NUMBER OF SYMBOL COMPARISONS USED BY QUICKSORT IS NONDEGENERATE (EXTENDED ABSTRACT)
"... In a continuoustime setting, Fill [2] proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable ..."
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Cited by 3 (2 self)
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In a continuoustime setting, Fill [2] proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable Y —not even that it is nondegenerate. We establish the nondegeneracy of Y. The proof is perhaps surprisingly difficult. 1. The number of symbol comparisons used by QuickSort: Brief review of a limitingdistribution result In this section we briefly review the main theorem of [2]. An infinite sequence of independent and identically distributed keys is generated; each key is a random word (w1, w2,...) = w1w2 · · · , that is, an infinite sequence, or “string”, of symbols wi drawn from a totally ordered finite alphabet Σ. The common distribution µ of the keys (called a probabilistic source) is allowed to be any distribution over words, i.e., the distribution of any stochastic process with time parameter set {1, 2,...} and state space Σ. We know thanks to Kolmogorov’s consistency criterion (e.g.,
A Gaussian limit process for optimal FIND algorithms
, 2013
"... We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to b ..."
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Cited by 1 (1 self)
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We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as n → ∞, which depends on α. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties. AMS 2010 subject classifications. Primary 60F17, 68P10; secondary 60G15, 60C05, 68Q25. Key words. FIND algorithm, Quickselect, complexity, key comparisons, functional limit theorem,
A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms
 STACS'13
, 2013
"... ..."
The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is
, 2012
"... In a continuoustime setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random varia ..."
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In a continuoustime setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable Y —not even that it is nondegenerate. We establish the nondegeneracy of Y. The proof is perhaps surprisingly difficult.
Dependence and phase changes in randommary search trees
, 2015
"... We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all rootkey distances or over all rootnode distances) in random mary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when 3 ..."
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We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all rootkey distances or over all rootnode distances) in random mary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when 3 6 m 6 13 but becomes of higher order when m> 14. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when 3 6 m 6 26 but is periodically oscillating for largerm. Such a less anticipated phenomenon is not exceptional and we extend the results in two directions: one for more general shape parameters, and the other for other classes of random logtrees such as fringebalanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method.
Sample Based Curve Fitting Computation on the Performance of Quicksort in Personal Computer
"... Abstract — In this paper we have used curve fitting technique for analyzing the classical Quicksort algorithm and its performance in worst case on personal computer. The proposed generic model can be viewed as: Time ~ f (Number of Elements). We fit the data points (Time vs Number of elements) in twe ..."
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Abstract — In this paper we have used curve fitting technique for analyzing the classical Quicksort algorithm and its performance in worst case on personal computer. The proposed generic model can be viewed as: Time ~ f (Number of Elements). We fit the data points (Time vs Number of elements) in twenty one different models from various types of fit such as Polynomial, Exponential, Power, Gaussian and Fourier. This analysis leads us to identify the best model amongst these models. We found that a model of Fourier series is the best fit.
THE NUMBER OF BIT COMPARISONS USED BY QUICKSORT: AN AVERAGECASE ANALYSIS
"... Abstract. The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (su ..."
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Abstract. The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed in terms of the number of key comparisons. We introduce the prospect of a fair comparison between algorithms of the two types by providing an averagecase analysis of the number of bit comparisons required by Quicksort. Counting bit comparisons rather than key comparisons introduces an extra logarithmic factor to the asymptotic average total. We also provide a new algorithm, “BitsQuick”, that reduces this factor to constant order by eliminating needless bit comparisons. 1. Introduction and
Symbol Comparisons Used by QuickSort
"... 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 sourc ..."
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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.