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...

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.

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 one-byone 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 two-parameter Chinese restaurant process, tree-growth models associated with Mallows ’ φ model of random permutations and with Schützenberger’s non-commutative q-binomial 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 Doob-Martin compactifications, Poisson boundaries and tail σ-fields.

Quickselect with median-of-three is routinely used as the method of choice for selection of the mth element out of n in general-purpose 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 median-of-three 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 median-of-three and other quickselect variants when selecting the mth element for m/n → α as n → ∞. We also investigate the behavior of proportion-from-s sampling as s → ∞. 1

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