A $2$-branching construction for the $χ\leq 2r$ bound
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910031551660032 |
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| author | Date, Vinicius Tikara Venturi Zatesko, Leandro Miranda |
| author_facet | Date, Vinicius Tikara Venturi Zatesko, Leandro Miranda |
| contents | The string repetitiveness measures $χ$ (the size of a smallest suffixient set of a string) and $r$ (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound $χ\leq 2r$, proved by Navarro et al., is asymptotically tight as the size $σ$ of the alphabet increases, but achieving near-tight ratios for fixed $σ> 2$ remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~$k$ if every $(k{-}1)$-length substring admits exactly two $k$-length extensions. We show that 2-branching strings of order~$k$ yield closed-form ratios $χ/r = (2σ^{k-1}+1)/(σ^{k-1}+4)$. For order~$3$, we give an explicit construction for every $σ\geq 2$, narrowing the gap to~$2$ from $O(1/σ)$ to $O(1/σ^2)$. For $σ\in \{3,4\}$, we additionally present order-$5$ instances with ratios exceeding~$1.91$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $2$-branching construction for the $χ\leq 2r$ bound Date, Vinicius Tikara Venturi Zatesko, Leandro Miranda Data Structures and Algorithms F.2.2 The string repetitiveness measures $χ$ (the size of a smallest suffixient set of a string) and $r$ (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound $χ\leq 2r$, proved by Navarro et al., is asymptotically tight as the size $σ$ of the alphabet increases, but achieving near-tight ratios for fixed $σ> 2$ remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~$k$ if every $(k{-}1)$-length substring admits exactly two $k$-length extensions. We show that 2-branching strings of order~$k$ yield closed-form ratios $χ/r = (2σ^{k-1}+1)/(σ^{k-1}+4)$. For order~$3$, we give an explicit construction for every $σ\geq 2$, narrowing the gap to~$2$ from $O(1/σ)$ to $O(1/σ^2)$. For $σ\in \{3,4\}$, we additionally present order-$5$ instances with ratios exceeding~$1.91$. |
| title | A $2$-branching construction for the $χ\leq 2r$ bound |
| topic | Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2602.20949 |