In the sensitive distance oracle problem, there are three phases. We first preprocess a given directed graph $G$ with $n$ nodes and integer weights from $[-W,W]$. Second, given a single batch of $f$ edge insertions and deletions, we update the data structure. Third, given a query pair of nodes $(u,v)$, return the distance from $u$ to $v$. In the easier problem called sensitive reachability oracle problem, we only ask if there exists a directed path from $u$ to $v$. Our first result is a sensitive distance oracle with $tilde{O}(Wn^{omega+(3-omega)mu})$ preprocessing time, $tilde{O}(Wn^{2-mu}f^{2}+Wnf^{omega})$ update time, and $tilde{O}(Wn^{2-mu}f+Wnf^{2})$ query time where the parameter $muin[0,1]$ can be chosen. The data-structure requires $O(Wn^{2+mu} log n)$ bits of memory. This is the first algorithm that can handle $fgelog n$ updates. Previous results (e.g. [Demetrescu et al. SICOMP08; Bernstein and Karger SODA08 and FOCS09; Duan and Pettie SODA09; Grandoni and Williams FOCS12]) can handle at most 2 updates. When $3le flelog n$, the only non-trivial algorithm was by [Weimann and Yuster FOCS10]. When $W=tilde{O}(1)$, our algorithm simultaneously improves their preprocessing time, update time, and query time. In particular, when $f=omega(1)$, their update and query time is $Omega(n^{2-o(1)})$, while our update and query time are truly subquadratic in $n$, i.e., ours is faster by a polynomial factor of $n$. To highlight the technique, ours is the first graph algorithm that exploits the kernel basis decomposition of polynomial matrices by [Jeannerod and Villard J.Comp05; Zhou, Labahn and Storjohann J.Comp15] developed in the symbolic computation community. [...]
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