A closed string contains a proper factor occurring as both a prefix and a suffix but not elsewhere in the string. Closed strings were introduced by Fici (WORDS 2011) as objects of combinatorial interest. This paper addresses a new problem by extending the closed string problem to the [Formula: see text]-closed string problem, for which a level of approximation is permitted up to a number of Hamming distance errors, set by the parameter [Formula: see text].We address the problem of deciding whether or not a given string of length [Formula: see text] over an integer alphabet is [Formula: see text]-closed and additionally specifying the border resulting in the string being [Formula: see text]-closed. Specifically, we present an [Formula: see text]-time and [Formula: see text]-space algorithm to achieve this along with the pseudocode of an implementation and proof-of-concept experimental results.

中文翻译：

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k-闭合字符串的有效识别

一个封闭的字符串包含一个适当的因子，既作为前缀又作为后缀出现，但在字符串的其他地方不存在。Fici (WORDS 2011) 引入了闭合字符串作为组合兴趣的对象。本文通过将闭串问题扩展到 [Formula: see text]-闭串问题来解决一个新问题，对于该问题，允许一个近似级别达到多个汉明距离误差，由参数 [Formula: see text]。我们解决了确定长度为 [Formula: see text] 在整数字母表上的给定字符串是否是 [Formula: see text] 封闭的问题，并另外指定导致字符串为 [Formula: see text] 的边界文本]-关闭。具体来说，我们提出了一个[公式：见正文]-时间和[公式：