The decompressed tree size of $k$-ary chains

TL;DR

This paper analyzes the expected size of trees associated with specific DAGs called chains, revealing a stretched exponential growth pattern in their decompressed size as the chain length increases.

Contribution

It provides the first asymptotic analysis of the expected decompressed tree size for k-ary chains, highlighting a novel exponential growth behavior.

Findings

Expected decompressed tree size grows as e^{c * sqrt(n)} for large n.

Results have implications for the distribution of Brauer chains.

Analyzes chains with fixed out-degree k ≥ 2.

Abstract

A chain is defined as a directed acyclic graph (DAG) with one source and one sink, where the children are ordered and the spanning tree computed using a depth-first search is a path. Such DAGs emerge in the context of tree compression and are therefore uniquely associated with a tree. The tree size of a DAG is defined as the size of the associated tree. For fixed out-degree $k \geq 2$, we compute the asymptotic expected decompressed tree size of a chain of size $n$ chosen uniformly at random, and we show that it contains a stretched exponential term of the form $e^{c \, \sqrt{n}}$. This result also has implications for the limit distribution of Brauer chains of fixed length.