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On Fully Dynamic Strongly Connected Components

Authors Adam Karczmarz , Marcin Smulewicz

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Adam Karczmarz
  • University of Warsaw, Poland
  • IDEAS NCBR, Warsaw, Poland
Marcin Smulewicz
  • University of Warsaw, Poland

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Adam Karczmarz and Marcin Smulewicz. On Fully Dynamic Strongly Connected Components. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 68:1-68:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We consider maintaining strongly connected components (SCCs) of a directed graph subject to edge insertions and deletions. For this problem, we show a randomized algebraic data structure with conditionally tight O(n^1.529) worst-case update time. The only previously described subquadratic update bound for this problem [Karczmarz, Mukherjee, and Sankowski, STOC'22] holds exclusively in the amortized sense. For the less general dynamic strong connectivity problem, where one is only interested in maintaining whether the graph is strongly connected, we give an efficient deterministic black-box reduction to (arbitrary-pair) dynamic reachability. Consequently, for dynamic strong connectivity we match the best-known O(n^1.407) worst-case upper bound for dynamic reachability [van den Brand, Nanongkai, and Saranurak FOCS'19]. This is also conditionally optimal and improves upon the previous O(n^1.529) bound. Our reduction also yields the first fully dynamic algorithms for maintaining the minimum strong connectivity augmentation of a digraph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • dynamic strongly connected components
  • dynamic strong connectivity
  • dynamic reachability


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