具有少量Burrows-Wheeler变换游程的后继右特殊字符串
Successor-bispecial strings with minimum Burrows--Wheeler transform runs
AI总结：
研究字母表上后继右特殊字符串的最小BWT游程数，构造了阶为3的显式族，其游程数为σ²+2，并分析了χ/r比值的界。
摘要：
我们研究字母表 $\Sigma$（大小为 $\sigma$）上的后继右特殊字符串，它是 de Bruijn 字符串的最小分支类似物，并询问可能有多少个 Burrows-Wheeler 变换（BWT）游程。在阶为 $k$ 的 de Bruijn 字符串中，每个 $(k-1)$-上下文具有所有 $\sigma$ 个右扩展；在这里，每个上下文仍然是右特殊的，但恰好有两个右扩展，由后继规则选择。对于阶 $3$，我们为每个 $\sigma \geq 2$ 构造了一个显式族 $B_\sigma^{(3)}$，其循环 BWT 有 $r_c = \sigma^2 + 2$ 个游程。一个合适的终止线性化具有相同的游程数 $r = r_c = \sigma^2 + 2$，而最小后缀集的大小为 $\chi = 2\sigma^2 + 1$。比值 $\chi/r = 2 - 3/(\sigma^2 + 2)$ 量化了这种强制分支在多大程度上饱和了已知界 $\chi/r \leq 2$，我们之前已证明该界是渐近紧的。与我们早期的字母表增长构造相比，这将差距从 $O(1/\sigma)$ 改进到 $O(1/\sigma^2)$。我们还证明了当原始三项式 $x^3 - x + c$ 存在时，阶 $3$ 模式作为归一化线性反馈移位寄存器（LFSR）de Bruijn 序列在 $\mathbb F_q$ 上的块状两行投影出现。对于更高阶，我们使用最后到最先（LF）置换分析自然边界合并候选 $L_{\sigma,k}$：它在 $k = 4$ 且所有 $\sigma \geq 3$ 时失败，而对于 $\sigma \in {3,4}$ 验证的 $k = 5$ 实例产生了超过 $1.96$ 的 $\chi/r$ 比值。
英文摘要：
We study successor-bispecial strings over an alphabet $Σ$ of size $σ$, a minimal-branching analogue of de Bruijn strings, and ask how few Burrows--Wheeler transform (BWT) runs are possible. In a de Bruijn string of order $k$, every $(k-1)$-gram has all $σ$ right-extensions; here, every $(k-1)$-gram has exactly two right-extensions, determined by a successor rule, which also forces two left-extensions. For order $3$, we construct an explicit family $B_σ^{(3)}$, for every $σ\geq 2$, whose cyclic BWT has $r_c = σ^2 + 2$ runs. A suitable terminated linearization has the same run count, $r = r_c = σ^2 + 2$, while the smallest suffixient set has size $χ= 2σ^2 + 1$. The ratio $χ/r = 2 - 3/(σ^2 + 2)$ nearly saturates the known bound $χ/r \leq 2$, which we have previously shown to be asymptotically tight. Compared with our earlier general construction, this improves the gap from $O(1/σ)$ to $O(1/σ^2)$. We also show that the order-$3$ pattern appears as a blockwise two-row projection of normalized linear-feedback shift register (LFSR) de Bruijn sequences over $\mathbb F_q$, when primitive trinomials $x^3 - x + c$ exist. For higher orders, we prove a general lower bound $r_c \geq σ^{k-1} + 2$ for every $σ\geq 3$ in the exact-length regime and analyze the boundary-merged higher-order candidate using the last-to-first (LF) permutation: it fails for $k = 4$ and all $σ\geq 3$, while verified $k = 5$ instances for $σ\in {3,4}$ yield $χ/r$ ratios exceeding $1.96$.
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