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• W3033674217 abstract "The longest common substring problem consists in finding a longest string that appears as a (contiguous) substring of two input strings. We consider the dynamic variant of this problem, in which we are to maintain two dynamic strings $S$ and $T$, each of length at most $n$, that undergo substitutions of letters, in order to be able to return a longest common substring after each substitution. Recently, Amir et al. [ESA 2019] presented a solution for this problem that needs only $tilde{mathcal{O}}(n^{2/3})$ time per update. This brought the challenge of determining whether there exists a faster solution with polylogarithmic update time, or (as is the case for other dynamic problems), we should expect a polynomial (conditional) lower bound. We answer this question by designing a significantly faster algorithm that processes each substitution in amortized $log^{mathcal{O}(1)} n$ time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski et al. [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves, so that after each update we can report a pair of nodes, one from each tree, of maximum combined weight, which have at least one common leaf-descendant of each color. We complement this with a lower bound of $Omega(log n/ loglog n)$ for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, and the same lower bound for the update time of any data structure of size $tilde{mathcal{O}}(n)$ that maintains the LCS of a static and a dynamic string. Both lower bounds hold even allowing amortization and randomization." @default.
• W3033674217 created "2020-06-12" @default.
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• W3033674217 date "2020-06-03" @default.
• W3033674217 modified "2023-09-27" @default.
• W3033674217 title "Dynamic Longest Common Substring in Polylogarithmic Time" @default.
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