Dynamic LCS maintained under edits in amortized O(log^7 n) time whp, with Omega(log n / log log n) lower bound.
Landau, Moshe Lewenstein, and Dina Sokol
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Dynamic Longest Common Substring in Polylogarithmic Time
Dynamic LCS maintained under edits in amortized O(log^7 n) time whp, with Omega(log n / log log n) lower bound.