Relating Left and Right Extensions of Maximal Repeats

TL;DR

This paper investigates the relationship between left and right extensions of maximal repeats in strings, establishing tight bounds on their ratio and analyzing the stability of the CDAWG index under reversal.

Contribution

It proves tight bounds on the ratio of left to right extensions of maximal repeats and analyzes the stability of CDAWG size under string reversal.

Findings

The ratio of left to right extensions can be as large as O(√n).

The established bounds are asymptotically tight.

The ratio depends on the alphabet size, with specific bounds given.

Abstract

The compact directed acyclic word graph (CDAWG) of a string $T$ is an index occupying $O(\mathsf{e})$ space, where $\mathsf{e}$ is the number of right extensions of maximal repeats in $T$. For highly repetitive datasets, the measure $\mathsf{e}$ typically is small compared to the length $n$ of $T$ and, thus, the CDAWG serves as a compressed index. Unlike other compressibility measures (as LZ77, string attractors, BWT runs, etc.), $\mathsf{e}$ is very unstable with respect to reversals: the CDAWG of the reversed string $\overset{{}_{\leftarrow}}{T} = T[n] \cdots T[2] T[1]$ has size $O(\overset{{}_{\leftarrow}}{\mathsf{e}})$, where $\overset{{}_{\leftarrow}}{\mathsf{e}}$ is the number of left extensions of maximal repeats in $T$, and there are strings $T$ with $\frac{\overset{{}_{\leftarrow}}{\mathsf{e}}}{\mathsf{e}} \in \Omega(\sqrt{n})$. In this note, we prove that this lower bound is tight: $\frac{\overset{{}_{\leftarrow}}{\mathsf{e}}}{\mathsf{e}} \in O(\sqrt{n})$. Furthermore, given the alphabet size $\sigma$, we establish the alphabet-dependent bound $\frac{\overset{{}_{\leftarrow}}{\mathsf{e}}}{\mathsf{e}} \le \min\{\frac{2n}{\sigma}, \sigma\}$ and we show that it is asymptotically tight.