Dynamic Longest Common Substring in Polylogarithmic Time

TL;DR

This paper presents a new algorithm for the dynamic longest common substring problem that achieves polylogarithmic update time, significantly improving over previous solutions, and establishes lower bounds for related data structures.

Contribution

The authors develop an exponentially faster algorithm with amortized O(log^7 n) update time for dynamic LCS, and prove lower bounds for data structure update times.

Findings

New algorithm processes each edit in amortized O(log^7 n) time with high probability.

Established lower bounds of Ω(log n / log log n) for update time of dynamic LCS data structures.

Extended existing reductions to prove lower bounds even with randomization and amortization.

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].