Computing Longest (Common) Lyndon Subsequences

TL;DR

This paper introduces algorithms for computing the longest Lyndon subsequence and longest common Lyndon subsequence in strings, with complexities ranging from cubic to quartic time depending on the problem and model.

Contribution

It presents the first algorithms for finding the longest Lyndon subsequence and extends to the longest common Lyndon subsequence problem with specific complexity bounds.

Findings

Algorithms for longest Lyndon subsequence in O(n^3) time

Online algorithm with O(n^3 σ) complexity

Extension to longest common Lyndon subsequence in O(n^4 σ) time

Abstract

Given a string $T$ with length $n$ whose characters are drawn from an ordered alphabet of size $\sigma$, its longest Lyndon subsequence is a longest subsequence of $T$ that is a Lyndon word. We propose algorithms for finding such a subsequence in $O(n^3)$ time with $O(n)$ space, or online in $O(n^3 \sigma)$ space and time. Our first result can be extended to find the longest common Lyndon subsequence of two strings of length $n$ in $O(n^4 \sigma)$ time using $O(n^3)$ space.