OrderPreserving Pattern Matching Indeterminate Strings
Abstract
Given an indeterminate string pattern $p$ and an indeterminate string text $t$, the problem of orderpreserving pattern matching with character uncertainties ($\mu$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 wellstudied exact orderpreserving 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 $\mu$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 $\mu$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 $\mu$OPPM problem proved to be NPhard in general.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.02589
 Bibcode:
 2019arXiv190502589C
 Keywords:

 Computer Science  Data Structures and Algorithms