On the Smallest Size of Internal Collage Systems
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866915751762329600 |
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| author | Migita, Soichiro Uehata, Kyotaro I, Tomohiro |
| author_facet | Migita, Soichiro Uehata, Kyotaro I, Tomohiro |
| contents | A Straight-Line Program (SLP) for a string $T$ 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 $c(T) \in O(z(T))$ and there is a string family with $c(T) \in Ω(b(T) \log |T|)$, where $z(T)$ is the number of phrases in the 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 $\hat{c}(T)$ for $T$ is at least $b(T)$. While $c(T) \le \hat{c}(T)$ is obvious, it is unknown how large $\hat{c}(T)$ is compared to $c(T)$. In this paper, we prove that $\hat{c}(T) = Θ(c(T))$ by showing that any collage system of size $m$ can be transformed into an internal collage system of size $O(m)$ in $O(m^2)$ 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 $\hat{c}(T)$ for a given $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Smallest Size of Internal Collage Systems Migita, Soichiro Uehata, Kyotaro I, Tomohiro Data Structures and Algorithms A Straight-Line Program (SLP) for a string $T$ 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 $c(T) \in O(z(T))$ and there is a string family with $c(T) \in Ω(b(T) \log |T|)$, where $z(T)$ is the number of phrases in the 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 $\hat{c}(T)$ for $T$ is at least $b(T)$. While $c(T) \le \hat{c}(T)$ is obvious, it is unknown how large $\hat{c}(T)$ is compared to $c(T)$. In this paper, we prove that $\hat{c}(T) = Θ(c(T))$ by showing that any collage system of size $m$ can be transformed into an internal collage system of size $O(m)$ in $O(m^2)$ 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 $\hat{c}(T)$ for a given $T$. |
| title | On the Smallest Size of Internal Collage Systems |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.11602 |