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 m-th 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 m-th 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+1-m ) , 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...

Abstract. Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions. 1.

Introduction. Let F q [X] be the semigroup of monic polynomials f over a finite field F q having q elements. There exists a fairly extensive bibliography of papers dealing with the value distribution problems of various maps F q [X] ! R when the polynomials f are taken "at random". Usually, the probability measure n (: : : ) := q \Gamman #ff : ffif = n; : : : g; where ffif := deg f , is applied. We mention here the investigations [1], [5], [7--13], [17--20], [25]. On the other hand, there exists a parallel theory investigating the value distribution of the maps Sn ! R, where Sn denotes the symmetric group of order n, when a permutation oe 2 Sn is taken with the equal probability 1=n! (see, for instance, [3], [6], [10], [12], [14], [21], [23], [26]). Observe that despite the fact that the same analytic or probabilistic methods can be applied, the problems arising in these two theories have been considered separately. To demonstrate a new point of view, we quote a corollary