Longest (Sub-)Periodic Subsequence

TL;DR

This paper introduces algorithms for finding the longest (sub-)periodic subsequence in a string, achieving significant improvements in time and space complexity under various restrictions.

Contribution

It presents the first algorithms with polynomial time and space bounds for computing the longest (sub-)periodic subsequence, with optimizations for specific cases.

Findings

Longest periodic subsequence can be computed in $O(n^7)$ time.

Restrictions on exponents reduce complexity to $O(n^3)$ time.

Space complexity is reduced to $O(n^2)$ words in optimized cases.

Abstract

We present an algorithm computing the longest periodic subsequence of a string of length $n$ in $O(n^7)$ time with $O(n^4)$ words of space. We obtain improvements when restricting the exponents or extending the search allowing the reported subsequence to be subperiodic down to $O(n^3)$ time and $O(n^2)$ words of space.