  We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot
n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$
and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least
$k$ vertices, (2) a cycle on exactly $k$ vertices, (3) a cycle on at least $k$
vertices, (4) a feedback vertex set of size at most $k$, and (5) a set of $k$
pairwise vertex-disjoint cycles. For the first three problems, no
subexponential time parameterized algorithms were previously known. For the
remaining two problems, our algorithms significantly outperform the previously
best known parameterized algorithms that run in time $2^{O(k^{0.75}\log{k})}
\cdot n^{O(1)}$. Our algorithms are based on a new kind of tree decompositions
of unit disk graphs where the separators can have size up to $k^{O(1)}$ and
there exists a solution that crosses every separator at most $O(\sqrt{k})$
times. The running times of our algorithms are optimal up to the $\log{k}$
factor in the exponent, assuming the Exponential Time Hypothesis.
