Fault-Tolerant Distance Oracles Below the $n \cdot f$ Barrier

Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with $O(k^2 f^{\frac{1}{2}-\frac{1}{2k}} n^{1+\frac{1}{k}} + k f n)$ edges for any odd $k$. For any $k = \tilde{O}(1)$, this bound is essentially optimal for deterministic spanners in part due to a known folklore lower bound that \emph{any} $f$-fault-tolerant spanner requires $\Omega(nf)$ edges in the worst case. For $f \geq n$, this $\Omega(nf)$ barrier means that any $f$-fault tolerant spanners are trivial in size. Crucially however, this folklore lower bound exploits that the spanner \emph{is itself a subgraph}. It does not rule out distance-reporting data structures that may not be subgraphs. This leads to our central question: can one beat the $n \cdot f$ barrier with fault-tolerant distance oracles? We give a strong affirmative answer to this question. As our first contribution, we construct $f$-fault-tolerant distance oracles with stretch $O(\log(n)\log\log(n))$ that require only $\widetilde{O}(n\sqrt{f})$ bits of space; substantially below the spanner barrier of $n \cdot f$. Beyond this, in the regime $n \leq f \leq n^{3/2}$ we show that by using our new \emph{high-degree, low-diameter} decomposition in combination with tools from sparse recovery, we can even obtain stretch $7$ distance oracles in space $\widetilde{O}(n^{3/2}f^{1/3})$ bits. We also show that our techniques are sufficiently general to yield randomized sketches for fault-tolerant ``oblivious'' spanners and fault-tolerant deterministic distance oracles in bounded-deletion streams, with space below the $nf$ barrier in both settings.