Data Structures for Node Connectivity Queries

Author Zeev Nutov

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Zeev Nutov
  • The Open University of Israel, Ra'anana, Israel

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Zeev Nutov. Data Structures for Node Connectivity Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 82:1-82:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Let κ(s,t) denote the maximum number of internally disjoint st-paths in an undirected graph G. We consider designing a data structure that includes a list of cuts, and answers the following query: given s,t ∈ V, determine whether κ(s,t) ≤ k, and if so, return a pointer to an st-cut of size ≤ k (or to a minimum st-cut) in the list. A trivial data structure that includes a list of n(n-1)/2 cuts and requires Θ(kn²) space can answer each query in O(1) time. We obtain the following results. - In the case when G is k-connected, we show that 2n cuts suffice, and that these cuts can be partitioned into 2k+1 laminar families. Thus using space O(kn) we can answers each min-cut query in O(1) time, slightly improving and substantially simplifying the proof of a recent result of Pettie and Yin [S. Pettie and L. Yin, 2021]. We then extend this data structure to subset k-connectivity. - In the general case we show that (2k+1)n cuts suffice to return an st-cut of size ≤ k, and a list of size k(k+2)n contains a minimum st-cut for every s,t ∈ V. Combining our subset k-connectivity data structure with the data structure of Hsu and Lu [T-H. Hsu and H-I. Lu, 2009] for checking k-connectivity, we give an O(k² n) space data structure that returns an st-cut of size ≤ k in O(log k) time, while O(k³ n) space enables to return a minimum st-cut.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • node connectivity
  • minimum cuts
  • data structure
  • connectivity queries


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