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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 de ..."
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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...
Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
Forest fires on Z+ with ignition only at 0
, 907
"... We consider a version of the forest fire model on graph G, where each vertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the tim ..."
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We consider a version of the forest fire model on graph G, where each vertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the times between consecutive burnouts at vertex n, divided by log n, converge weakly as n → ∞ to a random variable which distribution is 1 −ρ(x) where ρ(x) is the Dickman function. We also show that on transitive graphs with a nontrivial site percolation threshold and one infinite cluster at most, the distributions of the time till the first burnout of any vertex have exponential tails. Finally, we give an elementary proof of an interesting limit: limn→∞ log log n = γ. ∑ n
Asymptotic Semismoothness Probabilities
"... Abstract We call an integer semismooth with respect to y and z if each of its prime factors is ^ y, and all but one are ^ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(ff; fi) be the asymptotic probability that a random integer n is semismooth with ..."
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Abstract We call an integer semismooth with respect to y and z if each of its prime factors is ^ y, and all but one are ^ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(ff; fi) be the asymptotic probability that a random integer n is semismooth with respect to nfi and nff. We present new recurrence relations for G and related functions. We then give numerical methods for computing G, tables of G, and estimates for the error incurred by this asymptotic approximation.
Alea 6, 399–414 (2009) Forest fires on Z+ with ignition only at 0
"... Abstract. We consider a version of the forest fire model on graph G, where each vertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z ..."
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Abstract. We consider a version of the forest fire model on graph G, where each vertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the times between consecutive burnouts at vertex n, divided by log n, converge weakly as n → ∞ to a random variable which distribution is 1 −ρ(x) where ρ(x) is the Dickman function. We also show that on transitive graphs with a nontrivial site percolation threshold and one infinite cluster at most, the distributions of the time till the first burnout of any vertex have exponential tails. Finally, we give an elementary proof of an interesting limit: lim n∑ n→∞ k=1