Asymptotic distribution theory for Hoare's selection algorithm

Abstract

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds thelth smallest in a set ofnnumbers. Lettingntend to infinity and considering the valuesl =1, ···,nsimultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Multivariate linear recursions with Markov-dependent coefficients

“…The equation has a remarkable variety of both theoretical as well as real-world applications. For examples in theoretical probability, see for instance [21,22,24,28,32]. For examples of applications in economics, see for instance [4,34,8,17,26].…”

Analysis of the Expected Number of Bit Comparisons Required by Quickselect

“…Similarly, if k < m, then the algorithm recursively operates on the set of keys larger than the pivot and returns the (k − m)-th smallest key from the subset. Although previous studies (e.g., Knuth [11], Mahmoud et al [15], Prodinger [18], Grübel and U. Rösler [7], Lent and Mahmoud [14], Mahmoud and Smythe [16], Devroye [1], Hwang and Tsai [9]) examined Quickselect with regard to key comparisons, this study is the first to analyze the bit complexity of the algorithm.…”

On the number of iterations required by Von Neumann addition

“…Using martingale methods Regnier [15] showed that (Y n − n log n)/n converges in distribution as n → ∞, Rösler [17] obtained the same result with a completely different and somewhat more constructive method. For Find see Grübel and Rösler [10] and Grübel [8,9], where the convergence in distribution of Z n /n was established for the number Z n of comparisons needed to find a specific quantile such as the median. It is an immediate consequence of these results that there is a concentration of mass phenomenon in the first, but not in the second case: Y n /EY n converges to a fixed value as n → ∞, Z n /EZ n does not.…”