Fast Distributed Computation of Cuts Via Random Circulations

Abstract

We describe a new circulation-based method to determine cuts in an undirected graph. A circulation is an oriented labeling of edges with integers so that at each vertex, the sum of the in-labels equals the sum of out-labels. For an integer k, our approach is based on simple algorithms for sampling a circulation (mod k) uniformly at random. We prove that with high probability, certain dependencies in the random circulation correspond to cuts in the 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, and hence 2-edge-connected and 3-edge-connected components.

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 ${\mathcal{D}}$. Previously, Thurimella [J. Algorithms, 1997] gave a $O({\mathcal{D}}+\sqrt{|V|}\log^* |V|)$-time algorithm to identify all cut vertices, 2-edge-connected components, and cut edges, and Tsin [Int. J. Found. Comput. Sci., 2006] gave a $O(|V|+{\mathcal{D}}^2)$-time algorithm to identify all cut pairs and 3-edge-connected components.

We obtain simple $O({\mathcal{D}})$-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving previous time bounds on all graphs. Under certain assumptions these new algorithms are universally optimal, due to a $\Omega({\mathcal{D}})$-time lower bound on every graph. These results yield the first distributed algorithms with sub-linear time for cut pairs and 3-edge-connected components. Let Δ denote the maximum degree. We obtain a $O({\mathcal{D}}+\Delta/\log|V|)$-time distributed algorithm for finding cut vertices; this is faster than Thurimella’s algorithm on all graphs with $\Delta, {\mathcal{D}} = O(\sqrt{|V|})$. The basic distributed algorithms are Monte Carlo, but can be made Las Vegas without increasing the asymptotic complexity.