Traveling Front Solutions to Directed Diffusion Limited Aggregation, Digital Search Trees and the Lempel-Ziv Data Compression Algorithm

TL;DR

This paper applies the traveling front method to analyze directed diffusion limited aggregation on Cayley trees, revealing connections to digital search trees and the Lempel-Ziv compression algorithm, and deriving exact asymptotic results.

Contribution

It introduces a novel application of the traveling front approach to derive exact asymptotics for aggregation models and links these results to fundamental problems in computer science.

Findings

Derived exact asymptotic statistics for particle counts in aggregation models.

Established connections between aggregation, search trees, and data compression.

Provided implications for computer science problems based on the results.

Abstract

We use the traveling front approach to derive exact asymptotic results for the statistics of the number of particles in a class of directed diffusion limited aggregation models on a Cayley tree. We point out that some aspects of these models are closely connected to two different problems in computer science, namely the digital search tree problem in data structures and the Lempel-Ziv algorithm for data compression. The statistics of the number of particles studied here is related to the statistics of height in digital search trees which, in turn, is related to the statistics of the length of the longest word formed by the Lempel-Ziv algorithm. Implications of our results to these computer science problems are pointed out.