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A $2$-branching construction for the $χ\leq 2r$ bound

Vinicius Tikara Venturi Date, Leandro Miranda Zatesko

Abstract

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~$1.91$.

A $2$-branching construction for the $χ\leq 2r$ bound

Abstract

The string repetitiveness measures (the size of a smallest suffixient set of a string) and (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound , proved by Navarro et al., is asymptotically tight as the size of the alphabet increases, but achieving near-tight ratios for fixed remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~ if every -length substring admits exactly two -length extensions. We show that 2-branching strings of order~ yield closed-form ratios . For order~, we give an explicit construction for every , narrowing the gap to~ from to . For , we additionally present order- instances with ratios exceeding~.
Paper Structure (12 sections, 13 theorems, 10 equations, 1 figure, 4 tables)

This paper contains 12 sections, 13 theorems, 10 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Burrows--Wheeler Transform
  4. Suffixient sets via super-maximal right-extensions
  5. The construction
  6. Structure details
  7. Measuring the string family
  8. Higher-order constructions
  9. The family $\mathcal{B}^{(k)}$
  10. Explicit order-5 constructions
  11. Comparison
  12. Open Problems

Key Result

Lemma 6

Consider $S_\sigma$. For every $a < \sigma-1$ and every $b \in \Sigma$, the $2$-gram $ab$ occurs exactly twice in $S_\sigma$, and Equivalently, the only $3$-grams of $S_\sigma$ with prefix $ab$ are $aba$ and $ab(a+1)$.

Figures (1)

  • Figure 1: Explicit order-5 strings for $\sigma \in \{3, 4\}$. Both satisfy the 2-branching property with extension pattern $R(u) = \{u[0], (u[0]+1) \bmod \sigma\}$.

Theorems & Definitions (24)

  • Example 1
  • Example 2
  • Example 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Example 9
  • Corollary 10
  • ...and 14 more