Process convergence for the complexity of Radix Selection on Markov sources

TL;DR

This paper analyzes the complexity of Radix Selection when data is generated by Markov sources, deriving limit theorems and exploring the process convergence for different models of data ranks.

Contribution

It introduces a stochastic process model for Radix Selection complexity on Markov sources and establishes new limit theorems, including Gaussian process convergence.

Findings

Mean and variance of complexity are derived.

Limit theorems with Gaussian process limits are established.

Weak convergence results are obtained for specific models.

Abstract

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 set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations.