The generalized constrained longest common subsequence in the run-length encoded format

Abstract

The longest common subsequence (LCS) is commonly used as a similarity measurement between two given sequences (strings). Several variants of the LCS problem have been developed. The generalized constrained LCS (CLCS) problems have four variants: substring exclusion (STR-EC-LCS), substring inclusion, subsequence exclusion and subsequence inclusion. In the STR-EC-LCS problem, a given constraint string is excluded as a substring from the answer. For the STR-EC-LCS problem with run-length encoded (RLE) strings, we are given two strings X and Y, along with a constraint string P, consisting of M, N and R runs, respectively. In their plain (non-RLE) representations, the lengths of these strings are $\left|X\right|=m$, $\left|Y\right|=n$ and $\left|P\right|=r$. We combine the state transition with the dynamic programming approach to solve the STR-EC-LCS problem with RLE strings. The time complexity of our algorithm is O$\left(r\right(Mn+mN\left)\right)$. The previous algorithm for STR-EC-LCS without RLE has a time complexity of O$\left(mnr\right)$. Since $M\le m$ and $N\le n$, our algorithm improves the previous algorithm in time complexity. Additionally, our algorithm, which is based on state transitions, is universal and can be applied to the other three variants of the generalized constrained LCS problem with the same complexity.