Order-preserving pattern matching with scaling

Given a text T and a pattern P, the order-preserving pattern matching (OPPM for short) problem is to find all substrings of T which have the same relative orders as P. Recently, approximate OPPM that allows errors have been studied such as OPPM with k-mismatches. In this paper we define the OPPM with scaling which is a novel criteria for approximate OPPM by considering the relative orders of cusps and the scale of lengths of strings between them. Also we present an algorithm to solve the OPPM problem with scaling in O(n+mlog⁡m) time, which is the same time bound as the best known exact OPPM algorithm.