Slowing Down Top Trees for Better Worst-Case Bounds

TL;DR

This paper demonstrates that the top tree compression scheme can be slowed down to match the information-theoretic lower bound by simple modifications, exposing a weakness in the original method and improving understanding of its limitations.

Contribution

It constructs trees that show the top tree scheme's inefficiency and proposes a minor modification to achieve optimal compression bounds, simplifying previous approaches.

Findings

Constructed trees with top DAG size matching upper bounds

Identified a weakness in the original top tree scheme

Proposed a simple modification to achieve optimal bounds

Abstract

We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on $n$ nodes labeled from an alphabet of size $\sigma$, for which the size of the top DAG is $\Theta(\frac{n}{\log_\sigma n}\log\log_\sigma n)$. Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is $\Omega(\frac{n}{\log_\sigma n})$. This settles an open problem stated by Lohrey et al. [arXiv 2017], who designed a more involved version achieving the lower bound. We show that this can be also guaranteed by a very minor modification of the original scheme: informally, one only needs to ensure that different parts of the tree are not compressed too quickly. Arguably, our version is more uniform, and in particular, the compression procedure is oblivious to the value of $\sigma$.