Dynamic Longest Common Substring in Polylogarithmic Time

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 edit operations, i.e., substitutions, insertions, and deletions of letters, in order to be able to return a longest common substring after each edit. Amir, Charalampopoulos, Pissis, and Radoszewski [Algorithmica 2020] presented a solution for this problem that requires $\tilde{\mathcal{O}}(n^{2/3})$ time per update. This brought the challenge of determining whether there exists a solution with polylogarithmic update time or we should expect a polynomial (conditional) lower bound. We answer this question by designing an exponentially faster algorithm that processes each edit operation in amortized $O(\log^7 n)$ time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski, Karczmarz, Kociumaka, L\k{a}cki, and Sankowski [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves. We complement our solution with a lower bound of $\Omega(\log n/ \log\log n)$ for the update time of any polynomial-size data structure that maintains an LCS of two dynamic strings, and the same lower bound for the update time of any $\tilde{\mathcal{O}}(n)$-size data structure that maintains the LCS of a static and a dynamic string. Both lower bounds hold even allowing amortization and randomization. This requires extending P\u{a}tra\c{s}cu's reduction from the lopsided set disjointness problem to the butterfly reachability problem [SICOMP 2011].