Efficient Algorithms to Compute Closed Substrings

TL;DR

This paper introduces efficient algorithms with $ ext{O}(n ext{log} n)$ complexity to compute all closed substrings of a string, providing a compact representation and analyzing their performance on different string classes.

Contribution

The paper presents the first space-efficient $ ext{O}(n ext{log} n)$ algorithms for computing all closed substrings and maximal closed strings, along with performance comparisons and a formula for MCSs in Fibonacci words.

Findings

Algorithms run in $ ext{O}(n ext{log} n)$ time.

Compact representation uses $ ext{O}(n ext{log} n)$ space.

Number of MCSs in Fibonacci words approximates $1.382 F_n$.

Abstract

A closed string $u$ is either of length one or contains a border that occurs only as a prefix and as a suffix in $u$ and nowhere else within $u$. In this paper, we present fast $\mathcal{O}(n\log n)$ time algorithms to compute all $\mathcal{O}(n^2)$ closed substrings by introducing a compact representation for all closed substrings of a string $ w[1..n]$, using only $\mathcal{O}(n \log n)$ space. These simple and space-efficient algorithms also compute maximal closed strings. Furthermore, we compare the performance of these algorithms and identify classes of strings where each performs best. Finally, we show that the exact number of MCSs ($M(f_n)$) in a Fibonacci word $ f_n $, for $n \geq 5$, is $\approx \left(1 + \frac{1}{\phi^2}\right) F_n \approx 1.382 F_n$, where $ \phi $ is the golden ratio.