Longest Common Subsequence on Weighted Sequences

Abstract

We consider the general problem of the Longest Common Subsequence (LCS) on weighted sequences. Weighted sequences are an extension of classical strings, where in each position every letter of the alphabet may occur with some probability. In this paper we provide faster algorithms and prove a series of hardness results for more general variants of the problem. In particular, we provide an NP-Completeness result on the general variant of the problem instead of the log-probability version used in earlier papers, already for alphabets of size 2. Furthermore, we design an EP T AS for bounded alphabets, which is also an improved, compared to previous results, P T AS for unbounded alphabets. These are in a sense optimal, since it is known that there is no F P T AS for bounded alphabets, while we prove that there is no EP T AS for unbounded alphabets. Finally, we provide a matching conditional (under the Exponential Time Hypothesis) lower bound for any P T AS. As a side note, we prove that it is sufficient to work with only one threshold in the general variant of the problem.

Weighted Shortest Common Supersequence Problem Revisited

“…A related problem is the Weighted Longest Common Subsequence problem (the WLCS problem, in short). It was introduced by Amir et al [4] and further studied in [14] and, very recently, in [20]. In the WLCS problem, we are also given two weighted strings W 1 and W 2 and a threshold 1 z on probability, but the task is to compute the longest (standard) string S such that S matches a subsequence of W 1 with probability at least 1 z and S matches a subsequence of W 2 with probability at least 1 z .…”