Longest Common Substring Made Fully Dynamic
Amihood Amir, Panagiotis Charalampopoulos, Solon P. Pissis, Jakub, Radoszewski

TL;DR
This paper introduces the first fully dynamic algorithm for the Longest Common Substring problem that operates in sublinear time per edit, significantly advancing dynamic string processing capabilities.
Contribution
The authors develop the first sublinear-time fully dynamic algorithm for LCS, also applying their techniques to related string problems like longest repeat, palindrome, and Lyndon substring.
Findings
Achieves rac{ ilde{O}(n)}{ ext{space}} time per edit for LCS
Extends techniques to longest repeat, palindrome, and Lyndon substring problems
Builds on previous restricted dynamic variants to handle fully dynamic scenarios
Abstract
In the longest common substring (LCS) problem, we are given two strings and , each of length at most , and we are asked to find a longest string occurring as a fragment of both and . This is a classical and well-studied problem in computer science with a known -time solution. In the fully dynamic version of the problem, edit operations are allowed in either of the two strings, and we are asked to report an LCS after each such operation. We present the first solution to this problem that requires sublinear time per edit operation. In particular, we show how to return an LCS in time (or time if edits are allowed in only one of the two strings) after each operation using space. This line of research was recently initiated by the authors [SPIRE 2017] in a somewhat…
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