3-Partition Order-Preserving Pattern Matching

Two strings of equal length are called order-isomorphic if their relative orders are identical at every position. The classical order-preserving pattern matching (OPPM) problem finds all substrings in a text T that are order-isomorphic to a pattern P. However, measurement errors can cause data loss or inaccuracies, making exact pattern detection difficult and motivating the active study of approximate OPPM variants, such as 2-partition OPPM. In this paper, we extend the existing partition-based relaxation of order-isomorphism and define the 3-partition OPPM problem. The 3-partition OPPM problem is to find all substrings in a text T that can be divided into three segments such that each partitioned segment is order-isomorphic to the corresponding segment of a pattern P. We propose an efficient algorithm to solve the problem in $O(nm+m^{2}logm)$ time, where $n=\left|T\right|$ and $m=\left|P\right|$. We conduct experiments on various time series datasets, comparing the number of occurrences and the runtime efficiency among the OPPM, 2-partition OPPM, and proposed 3-partition OPPM algorithms. Our experimental evaluation shows that the proposed algorithm becomes increasingly cost-effective for longer patterns.