Fast Computation of Small Cuts via Cycle Space Sampling

TL;DR

This paper introduces a sampling-based method using cycle space properties to efficiently identify cuts in graphs, leading to faster algorithms in sequential, distributed, and parallel computing models.

Contribution

The paper presents a novel cycle space sampling technique that enables simple, linear-time algorithms for cut detection and improves distributed and parallel algorithms for related problems.

Findings

Linear-time algorithms for cut edges and pairs in sequential models.

Faster distributed algorithms matching or surpassing previous bounds.

Optimal O(log V) parallel algorithm for cut pairs and 3-edge-connected components.

Abstract

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.