  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 $\tilde{\mathcal{O}}(n^{2/3})$ 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 $\log^{\mathcal{O}(1)} n$ 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 $\Omega(\log n/ \log\log n)$ for the
update time of any polynomial-size data structure that maintains the LCS of two
dynamic strings, and the same lower bound for the update time of any data
structure of size $\tilde{\mathcal{O}}(n)$ that maintains the LCS of a static
and a dynamic string. Both lower bounds hold even allowing amortization and
randomization.
