On the Smallest Size of Internal Collage Systems

A Straight-Line Program (SLP) for a stirng T is a context-free grammar in Chomsky normal form that derives T only, which can be seen as a compressed form of T. Kida et al.\ introduced collage systems [Theor. Comput. Sci., 2003] to generalize SLPs by adding repetition rules and truncation rules. The smallest size $c(T)$ of collage systems for T has gained attention to see how these generalized rules improve the compression ability of SLPs. Navarro et al. [IEEE Trans. Inf. Theory, 2021] showed that c(T) ∈ O(z(T)) and there is a string family with c(T) ∈ Ω(b(T) log |T|), where $z(T)$ is the number of Lempel-Ziv parsing of T and $b(T)$ is the smallest size of bidirectional schemes for T. They also introduced a subclass of collage systems, called internal collage systems, and proved that its smallest size ĉ(T) for T is at least $b(T)$. While c(T) ≤ ĉ(T) is obvious, it is unknown how large ĉ(T) is compared to $c(T)$. In this paper, we prove that ĉ(T) = Θ(c(T)) by showing that any collage system of size m can be transformed into an internal collage system of size $O(m)$ in O(m²) time. Thanks to this result, we can focus on internal collage systems to study the asymptotic behavior of $c(T)$, which helps to suppress excess use of truncation rules. As a direct application, we get $b(T) = O(c(T))$, which answers an open question posed in [Navarro et al., IEEE Trans. Inf. Theory, 2021]. We also give a MAX-SAT formulation to compute ĉ(T) for a given T.