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• W3182800350 abstract "We design $f$-edge fault-tolerant diameter oracles ($f$-FDOs). We preprocess a given graph $G$ on $n$ vertices and $m$ edges, and a positive integer $f$, to construct a data structure that, when queried with a set $F$ of $|F| leq f$ edges, returns the diameter of $G-F$. For a single failure ($f=1$) in an unweighted directed graph of diameter $D$, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch $(1+varepsilon)$, constant query time, space $O(m)$, and a combinatorial preprocessing time of $widetilde{O}(mn + n^{1.5} sqrt{Dm/varepsilon})$.We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of $widetilde{O}(mn + n^2/varepsilon)$. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of $widetilde{O}(n^{2.5794} + n^2/varepsilon)$. We further prove an information-theoretic lower bound showing that any FDO with stretch better than $3/2$ requires $Omega(m)$ bits of space. For multiple failures ($f>1$) in undirected graphs with non-negative edge weights, we give an $f$-FDO with stretch $(f+2)$, query time $O(f^2log^2{n})$, $widetilde{O}(fn)$ space, and preprocessing time $widetilde{O}(fm)$. We complement this with a lower bound excluding any finite stretch in $o(fn)$ space. We show that for unweighted graphs with polylogarithmic diameter and up to $f = o(log n/ loglog n)$ failures, one can swap approximation for query time and space. We present an exact combinatorial $f$-FDO with preprocessing time $mn^{1+o(1)}$, query time $n^{o(1)}$, and space $n^{2+o(1)}$. When using fast matrix multiplication instead, the preprocessing time can be improved to $n^{omega+o(1)}$, where $omega < 2.373$ is the matrix multiplication exponent." @default.
• W3182800350 created "2021-07-19" @default.
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• W3182800350 date "2021-07-07" @default.
• W3182800350 modified "2023-09-27" @default.
• W3182800350 title "Space-Efficient Fault-Tolerant Diameter Oracles" @default.
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