A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in $\mathcal{O}({\log }^{2}n)$ amortized time per word, while the best generation algorithm hitherto has $\mathcal{O}(n)$ running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.

中文翻译：

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生成格雷码以在每个单词的摊余对数时间内为普通单词添加前缀

前缀普通字是一个二进制字，其属性是子字符串的长度不能比相同长度的前缀多1。通过证明前缀普通词集合是冒泡语言，我们可以穷尽所有长度为n的前缀普通词作为组合格雷码列出，其中连续的字符串最多相差两次交换或位翻转。此格雷码可以在$\mathcal{Ø}（{\mathrm{日志}}^2ñ）$ 摊分每个单词的时间，而迄今为止最好的生成算法 $\mathcal{Ø}（ñ）$每个单词的运行时间。我们还介绍了一种用于普通单词前缀的成员资格测试器，以及一种泡泡语言的新颖表征。