A border of a word w is a word that is both a non-empty proper prefix and suffix of w. If w has a border, then it is said to be bordered; otherwise, it is said to be unbordered. The main results of this paper are the first algorithms to rank and unrank length-n bordered and unbordered words over a k-letter alphabet. We show that, under the unit-cost RAM model, ranking bordered and unbordered words can be done in $O(k{n}^3)$ time using $O(n)$ space, and unranking them can be done in $O(n^{4}k\log k)$ time using $O(n)$ space.

中文翻译：

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对有界词和无界词进行排名和取消排名

单词w的边界是既是w的非空真前缀又是后缀的单词。如果w有边界，则称其为有边界的；否则，就说它是无边界的。本文的主要成果是第一个对k个字母字母表上的长度为n 个有边界和无边界单词进行排序和取消排序的算法。我们表明，在单位成本 RAM 模型下，可以对有界词和无界词进行排序$氧（k{n}^3）$时间使用$氧（n）$空间，并且取消它们的排名可以在$氧（n^{4}k\mathrm{日志}k）$时间使用$氧（n）$空间。