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

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 many worst-case oriented results known. We present two algorithms for LCS length calculation with respectively $O(mn \log\log n / \log^2 n)$ and $O(mn / \log^2 n + r)$ time complexity, the latter working for $r = o(mn / (\log n \log\log n))$, where $r$ is the number of matches in the dynamic programming matrix. We also describe conditions for a given problem sufficient to apply our techniques, with several concrete examples presented, namely the edit distance, LCTS and MerLCS problems.

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