保序模式匹配不确定字符串
Order-Preserving Pattern Matching Indeterminate Strings
摘要：
给定一个不确定的字符串模式$p$和一个不确定的字符串文本$t$，具有字符不确定性的保序模式匹配（$μ$ OPPM）问题是寻找$t$中满足$p$定义的可能排序之一的所有子串.当文本和模式都是确定的字符串时，我们面临的问题是精确保序模式匹配（OPPM）问题，它在时间序列分析中有着广泛的应用。尽管具有相关性，但确切的OPPM问题存在两个主要缺点：1）无法处理文本中的不确定性，从而无法分析有噪时间序列; 2）无法处理模式中的不确定性，从而强制要求严格满足所有模式位置之间的顺序。本文首次提出了一种多项式算法来解决在模式或文本上观察到不确定性时的$μ$ OPPM问题。给定两个字符串的长度为$m$和$O(r)$不确定字符串的位置，我们证明了$μ$ OPPM问题可以解决的$O(mr\lg r)$时间，当一个字符串是不确定的和$r\in\mathbb{N}^+$。当模式和文本都存在不确定性时，给出了可满足性问题的映射，并给出了一般问题复杂性的结果，证明了$μ$ OPPM问题一般是NP-难的.
英文摘要：
Given an indeterminate string pattern $p$ and an indeterminate string text $t$, the problem of order-preserving pattern matching with character uncertainties ($μ$OPPM) is to find all substrings of $t$ that satisfy one of the possible orderings defined by $p$. When the text and pattern are determinate strings, we are in the presence of the well-studied exact order-preserving pattern matching (OPPM) problem with diverse applications on time series analysis. Despite its relevance, the exact OPPM problem suffers from two major drawbacks: 1) the inability to deal with indetermination in the text, thus preventing the analysis of noisy time series; and 2) the inability to deal with indetermination in the pattern, thus imposing the strict satisfaction of the orders among all pattern positions. This paper provides the first polynomial algorithm to answer the $μ$OPPM problem when indetermination is observed on the pattern or text. Given two strings with length $m$ and $O(r)$ uncertain characters per string position, we show that the $μ$OPPM problem can be solved in $O(mr\lg r)$ time when one string is indeterminate and $r\in\mathbb{N}^+$. Mappings into satisfiability problems are provided when indetermination is observed on both the pattern and the text, and results concerning the general problem complexity are presented as well, with $μ$OPPM problem proved to be NP-hard in general.
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