Hoare's Selection Algorithm: A Markov Chain Approach

Abstract

We obtain bounds for the distribution of the number of comparisons needed by Hoare's randomized selection algorithm FIND and give a new proof for Grübel and Rösler's (1996) result on the convergence of this distribution. Our approach is based on the construction and analysis of a suitable associated Markov chain. Some numerical results for the quantiles of the limit distributions are included, leading for example to the statement that, for a set S with n elements and n large, FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of S.