二进制反射格雷码内部：2-格雷码顺序的翻转交换语言
Inside the Binary Reflected Gray Code: Flip-Swap Languages in 2-Gray Code Order
摘要：
翻转交换语言是长度为n的二进制字符串的集合S，使得$S \cup 0^n$在两个操作（当适用时）下闭合：（1）翻转最左边的1;以及（2）将最左边的1与其右边的位交换。翻转交换语言对许多组合对象进行建模，包括项链、Lyndon词、前缀正规词、k元Dyck词的左因子以及0-1背包问题的可行解。我们证明，任何翻转交换语言形成一个循环的2-格雷码时，列出二进制反射格雷码（BRGC）的顺序。此外，当提供成员资格测试器时，通用后继规则计算下一个字符串。该规则在上述翻转交换语言中以O（n）-摊销每字符串生成每个字符串，除了长度为n的前缀正常字需要O（$n^{1.864}$）-摊销每字符串。我们的工作概括了Vajnovski [Inf. Process.]关于项链和林登词的结果。Lett. 106（3）：96 $-$ 99，2008]。
英文摘要：
A flip-swap language is a set S of binary strings of length n such that $S \cup 0^n$ is closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal words, left factors of k-ary Dyck words, and feasible solutions to 0-1 knapsack problems. We prove that any flip-swap language forms a cyclic 2-Gray code when listed in binary reflected Gray code (BRGC) order. Furthermore, a generic successor rule computes the next string when provided with a membership tester. The rule generates each string in the aforementioned flip-swap languages in O(n)-amortized per string, except for prefix normal words of length n which require O($n^{1.864}$)-amortized per string. Our work generalizes results on necklaces and Lyndon words by Vajnovski [Inf. Process. Lett. 106(3):96$-$99, 2008].
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