The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm.

中文翻译：

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在多项式时间中对手链进行排名

本文的主要结果是第一个用于对手镯进行排名的多项式时间算法。该算法的时间复杂度为O（k ^ 2 n ^ 4），其中k是字母的大小，n是所考虑的手镯的长度。该算法的关键部分是通过找到其他三个等级来计算相对于手镯组的任何单词的等级：所有项链的等级，回文项链的等级以及封闭的回文项链的等级。本文介绍了最后两个概念。这些等级是我们算法的关键组成部分，以便将问题分解为多个部分。此外，此排序过程用于构建多项式时间取消排序算法。