A Note on a Recent Algorithm for Minimum Cut

Given an undirected edge-weighted graph $G=(V,E)$ with $m$ edges and $n$ vertices, the minimum cut problem asks to find a subset of vertices $S$ such that the total weight of all edges between $S$ and $V\setminus S$ is minimized. Karger’s longstanding $O(m{\log }^{3}n)$ time randomized algorithm for this problem was very recently improved in two independent works to $O(m \log^2 n)$[ICAL{P}^{\mathrm{\prime }}20]andto$O(m \log^2 n + n\log^5 n)$ [STOC’20]. These two algorithms use different approaches and techniques. In particular, while the former is faster, the latter has the advantage that it can be used to obtain efficient algorithms in the cut-query and in the streaming models of computation. In this paper, we show how to simplify and improve the algorithm of [STOC’20] to $O(m{\log }^{2}n+n{\log }^{3}n)$. We obtain this by replacing a randomized algorithm that, given a spanning tree $T$ of $G$, finds in $O(m \log n+n\log^4 n)$timeaminimumcutof$G$that2-respects(cutstwoedgesof)$T$withasimple$O(m \log n+n\log^2 n)$ time deterministic algorithm for the same problem.