An Efficient Preprocessing on the One-dimensional Real-scale Pattern Matching Problem

Given a pattern string P and a text string T , the one-dimensional real-scale pattern matching problem asks for all matched positions in T at which P occurs for some real scales ≥ 1. This problem was first proposed by Amir et al., who also gave an algorithm with O(n + |P |) time for solving it, where |T | = n. Recently, Wang et al. proposed a preprocessing on T with O(n) time and space, with which for constant-sized alphabets, one can determine all matched positions in O(|P |+Ur) time, where Ur denotes the number of matched positions. For large-sized alphabets, with Wang’s preprocessing, it takes O(|P |+Ur + logn) time to report all matched positions. In addition, Wang et al. also proposed a preprocessing on T with O(n) time and space, with which one can answer the decision version of the real-scale matching problem on constant-sized and large-sized alphabets in O(|P |) and O(|P |+ logn) time, respectively. In this paper, we propose an improved preprocessing for this problem. For constant-sized alphabets, our preprocessing takes only O(n) time and space, while reporting all matched positions requires O(|P | + w) time, where w ≤ Ur. For the case of large-sized alphabets, our preprocessing can also be implemented with O(n) time and space, and then all matched positions can be determined in O(|P |+w+log n) time. Compared with Wang’s result, our algorithm is more efficient in both preprocessing and searching phases.