An $O(n^2)$ Self-Stabilizing Algorithm for Computing Bridge-Connected Components

Abstract.

This paper presents a self-stabilizing algorithm that finds the bridge-connected components of a connected undirected graph on an asynchronous distributed or network model of computation. An edge of a graph is a bridge if its removal disconnects the graph. The bridge-connected components problem consists of finding the maximal connected subgraphs (components) of the given graph such that none of the components contains a bridge. The output of the algorithm is available in a distributed manner in the sense that, upon termination of the algorithm, each node of the graph knows its component number: the component number is a representative node number which uniquely identifies a bridge-connected component. It has been proved that the alogrithm is correct and requires $O(n^2)$ time if the depth-first search spanning tree of the graph is known, or else it requires $O(n^2+n\cdot d\cdot\Delta)$ time, where n is the number of nodes, d is the graph diameter, and Δ is an upper bound on the degree of a node.