Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS
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We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size |Σ|. For the problem of deciding whether the LCS of strings x,y has length at least L, we obtain a sketch size and streaming space usage of O(L^|Σ| - 1 L). We also prove matching unconditional lower bounds. As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an O(min{nm, n + m^Σ})-time algorithm for this problem, on strings x,y of length n,m, with n > m. We prove optimality of this running time up to lower order factors, assuming the Strong Exponential Time Hypothesis.
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