Extending Karger's randomized min-cut Algorithm for a Synchronous Distributed setting

TL;DR

This paper adapts Karger's randomized min-cut algorithm for a distributed environment, enabling local computations at each node while maintaining probabilistic correctness and analyzing its message and time complexities.

Contribution

It introduces a distributed version of Karger's min-cut algorithm, addressing technical challenges and providing complexity analysis in a distributed setting.

Findings

Achieves the same success probability as the original algorithm.

Operates with O(mn^{2}) message complexity.

Runs in O(n^{2}) time complexity.

Abstract

A min-cut that seperates vertices s and t in a network is an edge set of minimum weight whose removal will disconnect s and t. This problem is the dual of the well known s-t max-flow problem. Several algorithms for the min-cut problem are based on max-flow computation although the fastest known min-cut algorithms are not flow based. The well known Karger's randomized algorithm for min-cut is a non-flow based method for solving the (global) min-cut problem of finding the min s-t cut over all pair of vertices s,t in a weighted undirected graph. This paper presents an adaptation of Karger's algorithm for a synchronous distributed setting where each node is allowed to perform only local computations. The paper essentially addresses the technicalities involved in circumventing the limitations imposed by a distributed setting to the working of Karger's algorithm. While the correctness proof follows directly from Karger's algorithm, the complexity analysis differs significantly. The algorithm achieves the same probability of success as the original algorithm with O(mn^{2}) message complexity and O(n^{2}) time complexity, where n and m denote the number of vertices and edges in the graph.