  We introduce an algorithm for a search of extremal fractal curves in large
curve classes. It heavily uses SAT-solvers -- heuristic algorithms that find
models for CNF boolean formulas. Our algorithm was implemented and applied to
the search of fractal surjective curves $\gamma\colon[0,1]\to[0,1]^d$ with
minimal dilation
$$\sup_{t_1<t_2}\frac{\|\gamma(t_2)-\gamma(t_1)\|^d}{t_2-t_1}.$$
  We report new results of that search in the case of Euclidean norm.
  One of the results is a new curve that we call "YE", a self-similar
(monofractal) plain curve of genus $5\times 5$ with dilation
$5\frac{43}{73}=5.5890\ldots$, which is best-known among plain monofractals.
Moreover, the YE-curve is the unique minimal curve among monofractals of genus
$\le 6\times 6$. We give a proof of minimality, which relies both on
computations and theoretical results.
  We notice that the classes of facet-gated multifractals are rigid enough to
allow an efficient search, and contain many curves with small dilation. In
dimension $3$ we have found facet-gated bifractals (that we call "Spring") of
genus $2\times2\times 2$ with dilation $<17$. In dimension $4$ we obtained that
there is a curve with dilation $<62$.
  Some lower bounds on the dilation for wider classes of cubically decomposable
curves are proved.}
