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DOI: 10.4230/LIPIcs.ESA.2024.65

A Nearly Linear Time Construction of Approximate Single-Source Distance Sensitivity Oracles

Authors: Kaito Harada, Naoki Kitamura, Taisuke Izumi, and Toshimitsu Masuzawa

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract

An α-approximate vertex fault-tolerant distance sensitivity oracle (α-VSDO) for a weighted input graph G = (V, E, w) and a source vertex s ∈ V is the data structure answering an α-approximate distance from s to t in G-x for any given query (x, t) ∈ V × V. It is a data structure version of the so-called single-source replacement path problem (SSRP). In this paper, we present a new nearly linear-time algorithm of constructing a (1 + ε)-VSDO for any directed input graph with polynomially bounded integer edge weights. More precisely, the presented oracle attains Õ(m log (nW)/ ε + n log² (nW)/ε²) construction time, Õ(n log (nW) / ε) size, and Õ(1/ε) query time, where n is the number of vertices, m is the number of edges, and W is the maximum edge weight. These bounds are all optimal up to polylogarithmic factors. To the best of our knowledge, this is the first non-trivial algorithm for SSRP/VSDO beating Õ(mn) computation time for directed graphs with general edge weight functions, and also the first nearly linear-time construction breaking approximation factor 3. Such a construction has been unknown even for undirected and unweighted graphs. In addition, our result implies that the known conditional lower bounds for the exact SSRP computation does not apply to the case of approximation.

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Kaito Harada, Naoki Kitamura, Taisuke Izumi, and Toshimitsu Masuzawa. A Nearly Linear Time Construction of Approximate Single-Source Distance Sensitivity Oracles. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 65:1-65:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{harada_et_al:LIPIcs.ESA.2024.65,
  author =	{Harada, Kaito and Kitamura, Naoki and Izumi, Taisuke and Masuzawa, Toshimitsu},
  title =	{{A Nearly Linear Time Construction of Approximate Single-Source Distance Sensitivity Oracles}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{65:1--65:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.65},
  URN =		{urn:nbn:de:0030-drops-211367},
  doi =		{10.4230/LIPIcs.ESA.2024.65},
  annote =	{Keywords: data structure, distance sensitivity oracle, replacement path problem, graph algorithm}
}

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A Nearly Linear Time Construction of Approximate Single-Source Distance Sensitivity Oracles

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