Uncommon Suffix Tries

TL;DR

This paper constructs specific suffix tries from VLMC sources to demonstrate that their height and saturation levels can significantly deviate from typical assumptions, revealing new structural properties.

Contribution

It introduces suffix trie examples with atypical height and saturation behaviors derived from VLMC sources, challenging standard assumptions.

Findings

Height can grow faster than any power of n.

Saturation level can be negligible compared to log n.

Examples include logarithmic and factorial infinite combs.

Abstract

Common assumptions on the source producing the words inserted in a suffix trie with $n$ leaves lead to a $\log n$ height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of $n$ and another one whose saturation level is negligible with respect to $\log n$. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources; they are easily extended to families of sources having the same properties. The first example corresponds to a "logarithmic infinite comb" and enjoys a non uniform polynomial mixing. The second one corresponds to a "factorial infinite comb" for which mixing is uniform and exponential.