Lai, WenfengLiyanage, AdieshaZhu, BinhaiZou, Peng2024-10-292024-07Lai, W., Liyanage, A., Zhu, B. et al. The longest letter-duplicated subsequence and related problems. Acta Informatica 61, 315–329 (2024). https://doi.org/10.1007/s00236-024-00459-71432-0525https://scholarworks.montana.edu/handle/1/18902Motivated by computing duplication patterns in sequences, a new problem called the longest letter-duplicated subsequence (LLDS) is proposed. Given a sequence S of length n, a letter- duplicated subsequence is a subsequence of S in the form of x d1 1 x d2 2 . . . x d k k with x i ∈ , x j = x j+1 and di ≥ 2 for all i in [k] and j in [k − 1]. A linear time algorithm for computing a longest letter-duplicated subsequence (LLDS) of S can be easily obtained. In this paper, we focus on two variants of this problem: (1) ‘all-appearance’ version, i.e., all letters in must appear in the solution, and (2) the weighted version. For the former, we obtain dichotomous results: We prove that, when each letter appears in S at least 4 times, the problem and a relaxed version on feasibility testing (FT) are both NP-hard. The reduction is from (3+, 1, 2−)- SAT, where all 3-clauses (i.e., containing 3 lals) are monotone (i.e., containing only positive literals) and all 2-clauses contain only negative literals. We then show that when each letter appears in S at most 3 times, then the problem admits an O(n) time algorithm. Finally, we consider the weighted version, where the weight of a block x di i (di ≥ 2) could be any positive function which might not grow with di . We give a non-trivial O(n2) time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of S whose weight is maximized.en-UScc-byhttps://creativecommons.org/licenses/by/4.0/longest letter-duplicated subsequencecomputing duplication patternsArticle10.1007/s00236-024-00459-7