Analysis of radix selection on Markov sources

TL;DR

This paper analyzes the complexity of Radix Selection when data is generated from a Markov source, deriving limit theorems and showing different asymptotic behaviors depending on data distribution.

Contribution

It provides the first detailed stochastic process analysis of Radix Selection complexity for Markov sources, including limit theorems and convergence results.

Findings

Normalized complexity converges to a Gaussian process for uniform data.

Complexity distribution is less concentrated and non-normal for general Markov sources.

Explicit mean and covariance functions are derived for the limiting Gaussian process.

Abstract

The complexity of the algorithm Radix Selection is considered for independent data generated from a Markov source. The complexity is measured by the number of bucket operations required and studied as a stochastic process indexed by the ranks; also the case of a uniformly chosen rank is considered. The orders of mean and variance of the complexity and limit theorems are derived. We find weak convergence of the appropriately normalized complexity towards a Gaussian process with explicit mean and covariance functions (in the space D[0,1] of cadlag functions on [0,1] with the Skorokhod metric) for uniform data and the asymmetric Bernoulli model. For uniformly chosen ranks and uniformly distributed data the normalized complexity was known to be asymptotically normal. For a general Markov source (excluding the uniform case) we find that this complexity is less concentrated and admits a limit law with non-normal limit distribution.