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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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Cited by 24 (1 self)
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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...
Power laws for family sizes in a duplication model
, 2004
"... Qian, Luscombe, and Gerstein (2001) introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate one. Wh ..."
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Cited by 3 (1 self)
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Qian, Luscombe, and Gerstein (2001) introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate one. When a new individual is born, it has the same type as its parent with probability 1 − r and is a new type, different from all previously observed types, with probability r. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches N. We also show that if 1 ≪ S ≪ N 1−r, then the number of families of size at least S is approximately CNS −1/(1−r) , while if N 1−r ≪ S the distribution decays more rapidly than any power. Running head: Power laws for gene family sizes. ∗ Partially supported by NSF grants from the probability program (0202935) and from a joint DMS/NIGMS initiative to support research in mathematical biology (0201037). † Supported by an NSF Postdoctoral Fellowship.
Strong Convergence on Weakly Logarithmic Combinatorial Assemblies
, 2009
"... We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the author’s analytic approach, we generalize the socalled Fundamen ..."
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We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the author’s analytic approach, we generalize the socalled Fundamental Lemma giving independent process approximation in the total variation distance of the component structure of an assembly. To evaluate the influence of strongly dependent large components, we obtain estimates of the appropriate conditional probabilities by unconditioned ones. These estimates are applied to examine additive functions defined on such a class of structures. Some analogs of Major’s and Feller’s theorems which concern almost sure behavior of sums of independent random variables are proved. 1
Random Partitions With Restricted Part Sizes*
"... ABSTRACT: For a subset S of positive integers let �(n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ �(n, S), let Mn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on �(n, S), and regard Mn as a random variable. ..."
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ABSTRACT: For a subset S of positive integers let �(n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ �(n, S), let Mn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on �(n, S), and regard Mn as a random variable. In this paper the limiting density of the (suitably normalized) random variable Mn is determined for sets that are sufficiently regular. In particular, our results cover the case S ={Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman’s coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes. © 2007 Wiley Periodicals, Inc. Random
Competition between Discrete Random Variables, with Applications to Occupancy Problems
, 2008
"... Consider n players whose “scores ” are independent and identically distributed values {Xi} n i=1 from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p → 0 and (ii) F is uniform on {1,2,...,N}; the latter case clearly corresponds to the cla ..."
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Consider n players whose “scores ” are independent and identically distributed values {Xi} n i=1 from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p → 0 and (ii) F is uniform on {1,2,...,N}; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the Ustatistic W which counts the number of “ties” between pairs i,j; second, the univariate statistic Yr, which counts the number of strict rway ties between contestants, i.e., episodes of the 1 form Xi1 = Xi2 =... = Xir; Xj = Xi1;j = i1,i2,...,ir; and, last but not least, the multivariate vector ZAB = (YA,YA+1,...,YB). We provide Poisson approximations for the distributions of W, Yr and ZAB under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary. 1