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Optimal Decremental Connectivity in Non-Sparse Graphs

Authors Anders Aamand, Adam Karczmarz , Jakub Łącki , Nikos Parotsidis , Peter M. R. Rasmussen , Mikkel Thorup

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Author Details

Anders Aamand
  • MIT, Cambridge, MA, USA
Adam Karczmarz
  • University of Warsaw, Poland
  • IDEAS NCBR, Warsaw, Poland
Jakub Łącki
  • Google Research, New York, NY,USA
Nikos Parotsidis
  • Google Research, Zürich, Switzerland
Peter M. R. Rasmussen
  • BARC, University of Copenhagen, Denmark
Mikkel Thorup
  • BARC, University of Copenhagen, Denmark

Funding

  • Aamand, Anders: Supported by Thorup’s VILLUM Investigator Grant 16582 and by a DFF-International Postdoc Grant 0164-00022B from the Independent Research Fund Denmark.
  • Karczmarz, Adam: Partially supported by the ERC CoG grant TUgbOAT no 772346.
  • Rasmussen, Peter M. R.: Supported by Thorup’s VILLUM Investigator Grant 16582.
  • Thorup, Mikkel: Supported by Investigator Grant 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation.

Cite As Get BibTex

Anders Aamand, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal Decremental Connectivity in Non-Sparse Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.6

Abstract

We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in O(m + n poly log n) total time. Interspersed with the deletions, it can answer queries whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental c-edge-connectivity to a variant of fully-dynamic c-edge-connectivity on a sparse graph.
For non-sparse graphs with Ω(n poly log n) edges, our connectivity and 2-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an O(m log (n² / m) + n poly log n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n²) edges.
Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/log log n) worst-case time lower bound in the decremental setting [Alstrup, Husfeldt, and Rauhe FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
Keywords
  • decremental connectivity
  • dynamic connectivity

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