Online and Offline Algorithms for Counting Distinct Closed Factors via Sliding Suffix Trees

TL;DR

This paper introduces two algorithms, one online and one offline, that efficiently count distinct closed factors in a string using sliding suffix trees, improving computational performance.

Contribution

It presents novel online and offline algorithms for counting distinct closed factors in strings, leveraging sliding suffix trees for improved efficiency.

Findings

Online algorithm runs in O(n log σ) time with O(n) space.

Offline algorithm achieves O(n) time complexity with O(n) space.

Both algorithms effectively utilize suffix trees for sliding window processing.

Abstract

A string is said to be closed if its length is one, or if it has a non-empty factor that occurs both as a prefix and as a suffix of the string, but does not occur elsewhere. The notion of closed words was introduced by [Fici, WORDS 2011]. Recently, the maximum number of distinct closed factors occurring in a string was investigated by [Parshina and Puzynina, Theor. Comput. Sci. 2024], and an asymptotic tight bound was proved. In this paper, we propose two algorithms to count the distinct closed factors in a string T of length n over an alphabet of size \sigma. The first algorithm runs in O(n log \sigma) time using O(n) space for string T given in an online manner. The second algorithm runs in O(n) time using O(n) space for string T given in an offline manner. Both algorithms utilize suffix trees for sliding windows.