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DOI: 10.4230/LIPIcs.STACS.2016.18
URN: urn:nbn:de:0030-drops-57196
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Bilò, Davide ; Gualà, Luciano ; Leucci, Stefano ; Proietti, Guido

Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

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Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f >= 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) sigma-approximate single-source shortest-path tree (sigma-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched at most by a factor of sigma. To this respect, we provide an algorithm that efficiently computes an f-EFT (2|F|+1)-ASPT of size O(f n). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1), plus an additive term of (f+1)*log(n).

Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in ~{O}(f m) time, has size O(fn *log^2(n)), and is able to report, after the failure of the edge set F, in O(|F|^2 * log^2(n)) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m * log^3(n)) time a sensitivity oracle of size O(m * log^2(n)), that reports in O(k^2 * log^2(n)) time the (at most 2k) edges either exiting from or entering into the MSF. As a result of independent interest, it is worth noticing that our MSF oracle can be employed to handle arbitrary sequences of o(sqrt[4]{n}/log(n)) (non-simultaneous) updates with a worst-case time per update of o(sqrt{n}). Thus, for relatively short sequences of updates, our oracle should be preferred w.r.t. the best-known (in a worst-case sense) MSF fully-dynamic algorithm, requiring O(sqrt{n}) time per update.

BibTeX - Entry

  author =	{Davide Bil{\`o} and Luciano Gual{\`a} and Stefano Leucci and Guido Proietti},
  title =	{{Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-57196},
  doi =		{10.4230/LIPIcs.STACS.2016.18},
  annote =	{Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree}

Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree
Collection: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016

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