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

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

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Author Details

Kaito Harada
  • Osaka University, Suita, Japan
Naoki Kitamura
  • Osaka University, Suita, Japan
Taisuke Izumi
  • Osaka University, Suita, Japan
Toshimitsu Masuzawa
  • Osaka University, Suita, Japan

Funding

This work was supported by MEXT/JSPS KAKENHI Grant Number 22K21277, 23K16838, 23K24825, and 23H04385.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ESA.2024.65

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Data structures design and analysis
Keywords
  • data structure
  • distance sensitivity oracle
  • replacement path problem
  • graph algorithm

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