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

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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)


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
  • decremental connectivity
  • dynamic connectivity


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