Faster and simpler online computation of string net frequency

Abstract

An occurrence of a repeated substring u in a string S is called a net occurrence if extending the occurrence to the left or to the right decreases the number of occurrences to 1. The net frequency (NF) of a repeated substring u in a string S is the number of net occurrences of u in S. Very recently, Guo et al. [SPIRE 2024] proposed an online O(nlog σ)-time algorithm that maintains a data structure of O(n) space which answers Single-NF queries in $O(m\log \sigma +{\sigma }^2)$ time and reports all answers of the All-NF problem in O(nσ²) time. Here, n is the length of the input string S, m is the query pattern length, and σ is the alphabet size. The σ² term is a major drawback of their method since computing string net frequencies is originally motivated for Chinese language text processing where σ can be thousands large. This paper presents an improved online O(nlog σ)-time algorithm, which answers Single-NF queries in O(mlog σ) time and reports all answers to the All-NF problem in output-optimal $O\left(\right|{\mathrm{NF}}^{+}\left(S\right)\left|\right)$ time, where ${\mathrm{NF}}^{+}\left(S\right)$ is the set of substrings of S paired with their positive NF values. We note that $|{\mathrm{NF}}^{+}\left(S\right)|=O\left(n\right)$ always holds. In contrast to Guo et al.’s algorithm that is based on Ukkonen’s suffix tree construction, our algorithm is based on Weiner’s suffix tree construction.