Understanding Search Trees via Statistical Physics

TL;DR

This paper applies statistical physics techniques to analyze the asymptotic behavior of random m-ary search trees, revealing phase transitions in their properties and extending results to vector data elements.

Contribution

It introduces a novel application of statistical physics to derive exact asymptotic results and identify phase transitions in search tree models, including for vector data.

Findings

Height and balanced height follow a traveling front structure.

Variance of node count exhibits a phase transition at m_c=26.

Phase transition also occurs at a critical dimension D_c≈8.69.

Abstract

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a traveling front structure. In addition, the variance of the number of nodes needed to store a data string of a given size N is shown to undergo a striking phase transition at a critical value of the branching ratio m_c=26. We identify the mechanism of this phase transition, show that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...