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Calculating the length of a longest common subsequence (LCS) of two strings$A$ and $B$ of length $n$ and $m$ is a classic research topic, with manyworst-case oriented results known. We present two algorithms for LCS lengthcalculation with respectively $O(mn \log\log n / \log^2 n)$ and $O(mn / \log^2n + r)$ time complexity, the latter working for $r = o(mn / (\log n \log\logn))$, where $r$ is the number of matches in the dynamic programming matrix. Wealso describe conditions for a given problem sufficient to apply ourtechniques, with several concrete examples presented, namely the edit distance,LCTS and MerLCS problems.

New tabulation and sparse dynamic programming based techniques for sequence similarity problems