SpaceEfficient FaultTolerant Diameter Oracles
Abstract
We design $f$edge faulttolerant 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 $GF$. 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 informationtheoretic 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 nonnegative edge weights, we give an $f$FDO with stretch $(f+2)$, query time $O(f^2\log^2{n})$, $\widetilde{O}(fn)$ 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 $f = o(\log n/ \log\log n)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.03485
 Bibcode:
 2021arXiv210703485B
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 Full version of a paper to appear at MFCS'21. Abstract shortened to meet ArXiv requirements