  Let $V$ be a set of $n$ points in the plane. The unit-disk graph $G = (V, E)$
has vertex set $V$ and an edge $e_{uv} \in E$ between vertices $u, v \in V$ if
the Euclidean distance between $u$ and $v$ is at most 1. The weight of each
edge $e_{uv}$ is the Euclidean distance between $u$ and $v$. Given $V$ and a
source point $s\in V$, we consider the problem of computing shortest paths in
$G$ from $s$ to all other vertices. The previously best algorithm for this
problem runs in $O(n \log^2 n)$ time [Wang and Xue, SoCG'19]. The problem has
an $\Omega(n\log n)$ lower bound under the algebraic decision tree model. In
this paper, we present an improved algorithm of $O(n \log^2 n / \log \log n)$
time (under the standard real RAM model). Furthermore, we show that the problem
can be solved using $O(n\log n)$ comparisons under the algebraic decision tree
model, matching the $\Omega(n\log n)$ lower bound.
