  We study sketching and streaming algorithms for the Longest Common
Subsequence problem (LCS) on strings of small alphabet size $|\Sigma|$. For the
problem of deciding whether the LCS of strings $x,y$ has length at least $L$,
we obtain a sketch size and streaming space usage of $\mathcal{O}(L^{|\Sigma| -
1} \log L)$.
  We also prove matching unconditional lower bounds.
  As an application, we study a variant of LCS where each alphabet symbol is
equipped with a weight that is given as input, and the task is to compute a
common subsequence of maximum total weight. Using our sketching algorithm, we
obtain an $\mathcal{O}(\textrm{min}\{nm, n + m^{{\lvert \Sigma
\rvert}}\})$-time algorithm for this problem, on strings $x,y$ of length $n,m$,
with $n \ge m$. We prove optimality of this running time up to lower order
factors, assuming the Strong Exponential Time Hypothesis.
