# Order-Preserving Pattern Matching Indeterminate Strings

## TL;DR

This paper introduces the first polynomial-time algorithm for order-preserving pattern matching with indeterminate strings, enabling analysis of noisy time series and patterns with uncertain data, which was previously infeasible.

## Contribution

It presents a novel polynomial algorithm for the $$OPPM problem with indeterminate strings, extending exact OPPM to handle uncertainty in pattern and text.

## Key findings

- Algorithm runs in $O(mr\u2212lg r)$ time for one indeterminate string
- Mappings to satisfiability problems for both pattern and text cases
- Proves $$OPPM is NP-hard in the general case

## Abstract

Given an indeterminate string pattern $p$ and an indeterminate string text $t$, the problem of order-preserving 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 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 $\mu$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 NP-hard in general.