Minimum Cut in O(m log² n) Time

Authors Paweł Gawrychowski , Shay Mozes , Oren Weimann

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Author Details

Paweł Gawrychowski
  • University of Wrocław, Poland
Shay Mozes
  • The Interdisciplinary Center Herzliya, Israel
Oren Weimann
  • University of Haifa, Israel


We thank Daniel Anderson and Guy Blelloch for drawing our attention to an inaccuracy in a prior version of Section 3.1.

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Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Minimum Cut in O(m log² n) Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We give a randomized algorithm that finds a minimum cut in an undirected weighted m-edge n-vertex graph G with high probability in O(m log² n) time. This is the first improvement to Karger’s celebrated O(m log³ n) time algorithm from 1996. Our main technical contribution is a deterministic O(m log n) time algorithm that, given a spanning tree T of G, finds a minimum cut of G that 2-respects (cuts two edges of) T.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Minimum cut
  • Minimum 2-respecting cut


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