Order-preserving pattern matching indeterminate strings

Given an indeterminate string pattern p and an indeterminate string text t, the problem of orderpreserving pattern matching with character uncertainties (μOPPM) 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. In this paper, we provide the first polynomial algorithms to answer the μOPPM problem when: 1) indetermination is observed on the pattern or text; and 2) indetermination is observed on both the pattern and the text and given by uncertainties between pairs of characters. First, given two strings with the same length m and O(r) uncertain characters per string position, we show that the μOPPM problem can be solved in O(mr lg r) time when one string is indeterminate and r ∈ N⁺ and in O(m²) time when both strings are indeterminate and r=2. Second, given an indeterminate text string of length n, we show that μOPPM can be efficiently solved in polynomial time and linear space.