String Representation Based on Substring Equation Systems

Abstract

Repetitiveness measures quantify how much repetitive structure a string contains and serve as parameters for compressed representations and indexing data structures. Many compression schemes represent strings by recording equalities between identical substrings. We introduce the substring equation system (SES), a general compression scheme that represents a string as the unique solution to substring-equality and character-assignment constraints. We show that every string $w$ has an SES of size $O(\chi (w))$, where $\chi (w)$ is the size of its smallest suffixient set. This result establishes the reachability of $\chi$, which had been an open problem. We also prove that computing the size $s(w)$ of the smallest SES that represents $w$ is NP-hard and $(1+\epsilon )$-inapproximable for some fixed constant $\epsilon >0$. Finally, we prove that the size $b(w)$ of the smallest bidirectional macro scheme (BMS) representing $w$ satisfies $s(w)\le b(w)\le 4s(w)$. Hence, SES and BMS are equivalent up to a constant factor, and this equivalence gives the new bound $b(w)\in O(\chi (w))$.