Analysis of radix selection on Markov sources

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.