  We present an algorithm for the extensively studied Long Path and Long Cycle
problems on unit disk graphs that runs in time $2^{O(\sqrt{k})}(n+m)$. Under
the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs
cannot be solved in time $2^{o(\sqrt{k})}(n+m)^{O(1)}$ [de Berg et al., STOC
2018], hence our algorithm is optimal. Besides the
$2^{O(\sqrt{k})}(n+m)^{O(1)}$-time algorithm for the (arguably) much simpler
Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from
the existence of a $2k$-vertex kernel for the problem), this is the only known
ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path
and Long Cycle on unit disk graphs were only known to be solvable in time
$2^{O(\sqrt{k}\log k)}(n+m)$. This algorithm involved the introduction of a new
type of a tree decomposition, entailing the design of a very tedious dynamic
programming procedure. Our algorithm is substantially simpler: we completely
avoid the use of this new type of tree decomposition. Instead, we use a marking
procedure to reduce the problem to (a weighted version of) itself on a standard
tree decomposition of width $O(\sqrt{k})$.
