文章基本信息

标题：Dynamic Longest Common Substring in Polylogarithmic Time
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作者：Panagiotis Charalampopoulos ; PaweÅ, Gawrychowski ; Karol Pokorski 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：27:1-27:19
DOI：10.4230/LIPIcs.ICALP.2020.27
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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 ð'ªÌf(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^ð'ª(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 Î©(log n/ log log n) for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, even allowing amortization and randomization.
关键词：string algorithms; dynamic algorithms; longest common substring