Degree profile of $m$-ary search trees: A vehicle for data structure compression

TL;DR

This paper analyzes the degree profile of random m-ary search trees using Pólya urns, revealing a phase transition at m=27, and proposes a compact tree structure for data compression.

Contribution

It provides a detailed probabilistic analysis of node outdegree distributions and introduces a new compact m-ary tree for data structure compression.

Findings

Number of nodes of each outdegree is asymptotically normal for m ≤ 26.

A phase transition occurs at m=27 affecting distribution behavior.

Proposes a space-efficient m-ary tree structure.

Abstract

We revisit the random $m$-ary search tree and study a finer profile of its node outdegrees with the purpose of exploring possibilities of data structure compression. The analysis is done via P\'olya urns. The analysis shows that the number of nodes of each individual node outdegree has a phase transition: Up to $m=26$, the number of nodes of outdegree $k$, for $k=0,1, \ldots, m$, is asymptotically normal; that behavior changes at $m = 27$. Based on the analysis, we propose a compact $m$-ary tree that offers significant space saving.