  Let $G$ be a unit disk graph in the plane defined by $n$ disks whose
positions are known. For the case when $G$ is unweighted, we give a simple
algorithm to compute a shortest path tree from a given source in $O(n\log n)$
time. For the case when $G$ is weighted, we show that a shortest path tree from
a given source can be computed in $O(n^{1+\varepsilon})$ time, improving the
previous best time bound of $O(n^{4/3+\varepsilon})$.
