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Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived ..."
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
Partial Quicksort and . . .
"... Partial Quicksort sorts the l smallest elements in a list of length n. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time c1l lnl + c2l + n + o(n). The con ..."
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Cited by 7 (0 self)
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Partial Quicksort sorts the l smallest elements in a list of length n. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time c1l lnl + c2l + n + o(n). The constant c1 can be as small as the information theoretic lower bound log 2 e.
Trickledown processes and their boundaries
, 2012
"... It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in onebyone at a distinguished source vertex, successive part ..."
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Cited by 2 (1 self)
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It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in onebyone at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman’s twoparameter Chinese restaurant process, treegrowth models associated with Mallows ’ φ model of random permutations and with Schützenberger’s noncommutative qbinomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their DoobMartin compactifications, Poisson boundaries and tail σfields.
Partitioning schemes for quicksort and quickselect
, 2003
"... We introduce several modifications of the partitioning schemes used in Hoare’s quicksort and quickselect algorithms, including ternary schemes which identify keys less or greater than the pivot. We give estimates for the numbers of swaps made by each scheme. Our computational experiments indicate th ..."
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We introduce several modifications of the partitioning schemes used in Hoare’s quicksort and quickselect algorithms, including ternary schemes which identify keys less or greater than the pivot. We give estimates for the numbers of swaps made by each scheme. Our computational experiments indicate that ternary schemes allow quickselect to identify all keys equal to the selected key at little additional cost. Key words. Sorting, selection, quicksort, quickselect, partitioning. 1
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 be spli ..."
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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,
On the Variance of Quickselect ∗
, 2005
"... Quickselect with medianofthree is routinely used as the method of choice for selection of the mth element out of n in generalpurpose libraries such as the C++ Standard Template Library. Its average behavior is fairly well understood and has been shown to outperform that of the standard variant, w ..."
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Quickselect with medianofthree is routinely used as the method of choice for selection of the mth element out of n in generalpurpose libraries such as the C++ Standard Template Library. Its average behavior is fairly well understood and has been shown to outperform that of the standard variant, which chooses a random pivot on each stage. However, no results were previously known about the variance of the medianofthree variant, other than for the number of comparisons made when the rank m of the sought element is given by a uniform random variable. Here, we consider the variance of the number of comparisons made by quickselect with medianofthree and other quickselect variants when selecting the mth element for m/n → α as n → ∞. We also investigate the behavior of proportionfroms sampling as s → ∞. 1
The Analysis of Find or Perpetuities on Cadlag Functions
 DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE (SUBM.)
"... In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of some stochastic fixed points equation of the form ..."
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In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of some stochastic fixed points equation of the form
K(µ) D = ∑
, 2007
"... In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form ..."
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In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form
Improved randomized selection
, 2004
"... We show that several versions of Floyd and Rivest’s improved algorithm Select for finding the kth smallest of n elements require at most n + min{k,n − k} + O(n 1/2 ln 1/2 n) comparisons on average and with high probability. This rectifies the analysis of Floyd and Rivest, and extends it to the case ..."
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We show that several versions of Floyd and Rivest’s improved algorithm Select for finding the kth smallest of n elements require at most n + min{k,n − k} + O(n 1/2 ln 1/2 n) comparisons on average and with high probability. This rectifies the analysis of Floyd and Rivest, and extends it to the case of nondistinct elements. Encouraging computational results on large medianfinding problems are reported. Key words. Selection, medians, computational complexity. 1