Process convergence for the complexity of Radix Selection on Markov sources

A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the ranks to be selected. The orders of mean and variance of the complexity and limit theorems are derived. For uniform data and the asymmetric Bernoulli model, we find weak convergence of the appropriately normalized complexity towards a Gaussian process with explicit mean and covariance functions in the space of càdlàg functions with the Skorokhod metric. For all other Markov sources we show that such a convergence does not hold. We also study two further models for the ranks: uniformly chosen ranks and the worst case rank complexities which are of interest in computer science. AMS 2010 subject classifications. Primary 60F17, 60G15 secondary 68P10, 60C05, 68Q25.