Minimum Cut in $$O(m\log ^2 n)$$ Time

Abstract

We give a randomized algorithm that finds a minimum cut in an undirected weighted m-edge n-vertex graph G with high probability in $$O(m \log ^2 n)$$ O ( m log 2 n ) time. This is the first improvement to Karger’s celebrated $$O(m \log ^3 n)$$ O ( m log 3 n ) time algorithm from 1996. Our main technical contribution is a deterministic $$O(m \log n)$$ 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.