  An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch
$\sigma \ge 1$ is a data structure that preprocesses a given undirected,
unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer
$f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$
edges, it returns a $\sigma$-approximation of the $s$-$t$-distance in $G-F$.
  We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005]
showed that this is only possible for $\sigma \ge 3$. We present, for any
constant $f \ge 1$ and $\alpha \in (0, \frac{1}{2})$, and any $\varepsilon >
0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes
$\widetilde{O}(n^{2-\frac{\alpha}{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$
space and has an $O(n^\alpha/\varepsilon^2)$ query time. The time to build the
oracle is $\widetilde{O}(mn^{2-\frac{\alpha}{f+1}}) \cdot O(\log
n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with
diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with
stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has
$\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of
$O(D^{f+o(1)} mn^{1/k})$.
  Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch
$1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit
with a super-quadratic space requirement. We show how to reduce their
preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$.
