Radix sort trees in the large

TL;DR

This paper studies the asymptotic behavior of radix sort trees built from infinite binary strings, characterizing the boundary conditions and possible limits of the associated Markov chains.

Contribution

It establishes a bijective correspondence between the Doob–Martin boundary of radix sort tree chains with symmetric Bernoulli sources and chains with diffuse distributions, characterizing large-scale behavior.

Findings

Characterization of the Doob–Martin boundary for the radix sort tree chains.

Identification of conditions under which the chains can be conditioned on their large-scale behavior.

Connection between symmetric Bernoulli sources and diffuse probability measures on infinite binary sequences.

Abstract

The trie-based radix sort algorithm stores pairwise different infinite binary strings in the leaves of a binary tree in a way that the Ulam-Harris coding of each leaf equals a prefix (that is, an initial segment) of the corresponding string, with the prefixes being of minimal length so that they are pairwise different. We investigate the {\em radix sort tree chains} -- the tree-valued Markov chains that arise when successively storing infinite binary strings $Z_1,\ldots, Z_n$, $n=1,2,\ldots$ according to the trie-based radix sort algorithm, where the source strings $Z_1, Z_2,\ldots$ are independent and identically distributed. We establish a bijective correspondence between the full Doob--Martin boundary of the radix sort tree chain with a {\em symmetric Bernoulli source} (that is, each $Z_k$ is a fair coin-tossing sequence) and the family of radix sort tree chains for which the common distribution of the $Z_k$ is a diffuse probability measure on $\{0,1\}^\infty$. In essence, our result characterizes all the ways that it is possible to condition such a chain of radix sort trees consistently on its behavior "in the large".