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Space-Efficient Fault-Tolerant Diameter Oracles

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Abstract

We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F| ⩽ f, returns the diameter of G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F).
For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1+ε), constant query time, space O(m), and a combinatorial preprocessing time of Õ(mn + n^{1.5} √{Dm/ε}), where D is the diameter.
We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of Õ(mn + n²/ε), which is better for constant ε > 0. 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 Õ(n^{2.5794} + n²/ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1+ε)-approximate FDO is near-optimal in all parameters.
In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f+2), query time O(f²log²{n}), Õ(fn) space, and preprocessing time Õ(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space.
Many real-world networks have polylogarithmic diameter. We show that for those graphs 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^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.

BibTeX - Entry

@InProceedings{bilo_et_al:LIPIcs.MFCS.2021.18,
  author =	{Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin},
  title =	{{Space-Efficient Fault-Tolerant Diameter Oracles}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14458},
  URN =		{urn:nbn:de:0030-drops-144581},
  doi =		{10.4230/LIPIcs.MFCS.2021.18},
  annote =	{Keywords: derandomization, diameter, distance sensitivity oracle, fault-tolerant data structure, space lower bound}
}

Keywords: derandomization, diameter, distance sensitivity oracle, fault-tolerant data structure, space lower bound
Seminar: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue date: 2021
Date of publication: 18.08.2021


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