Abstract | ||
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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 $\Omega(\log n/ \log\log n)$ 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. |
Year | DOI | Venue |
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2020 | 10.4230/LIPIcs.ICALP.2020.27 | ICALP |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 3 |
Name | Order | Citations | PageRank |
---|---|---|---|
Charalampopoulos Panagiotis | 1 | 0 | 0.68 |
Pawel Gawrychowski | 2 | 226 | 46.74 |
Pokorski Karol | 3 | 0 | 0.34 |