  Let $s$ denote a distinguished source vertex of a non-negatively real
weighted and undirected graph $G$ with $n$ vertices and $m$ edges. In this
paper we present two efficient \emph{single-source approximate-distance
sensitivity oracles}, namely \emph{compact} data structures which are able to
\emph{quickly} report an approximate (by a multiplicative stretch factor)
distance from $s$ to any node of $G$ following the failure of any edge in $G$.
More precisely, we first present a sensitivity oracle of size $O(n)$ which is
able to report 2-approximate distances from the source in $O(1)$ time. Then, we
further develop our construction by building, for any $0<\epsilon<1$, another
sensitivity oracle having size $O\left(n\cdot \frac{1}{\epsilon} \log
\frac{1}{\epsilon}\right)$, and which is able to report a
$(1+\epsilon)$-approximate distance from $s$ to any vertex of $G$ in
$O\left(\log n\cdot \frac{1}{\epsilon} \log \frac{1}{\epsilon}\right)$ time.
Thus, this latter oracle is essentially optimal as far as size and stretch are
concerned, and it only asks for a logarithmic query time. Finally, our results
are complemented with a space lower bound for the related class of
single-source \emph{additively-stretched} sensitivity oracles, which is helpful
to realize the hardness of designing compact oracles of this type.
