Slowing Down Top Trees for Better Worst-Case Bounds

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 $σ$, for which the size of the top DAG is $Θ(\frac{n}{\log_σn}\log\log_σn)$. Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is $Ω(\frac{n}{\log_σn})$. 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 $σ$.