Slowing down top trees for better worst-case compression

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 $\Theta (\frac{n}{{\log }_{\sigma }n}\log {\log }_{\sigma }n)$. Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is $\Omega \left(\frac{n}{{\log }_{\sigma }n}\right)$. Lohrey et al. [IPL 2019] designed a more involved version of the original algorithm 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.