An Optimal Distributed Algorithm for Computing Bridge-Connected

Consider a connected undirected graph G =< N , E >, where N is the set of n nodes, and E is the set of e edges. Without loss of generality, I assume N to be {1, 2, . . . , n}. An edge (i, j) ∈ E is a bridge of G if the removal of (i, j) disconnects G. Clearly, the removal of a bridge increases the number of connected components of a graph by one. The bridge-connected components problem consists of "nding the maximal connected subgraphs of G such that none of the subgraphs contains a bridge. The best sequential algorithm for the bridge-"nding problem, due to Tarjan [1], has time complexity O(n + e). The bridge-connected components problem can also be solved in O(n + e) time, initially, by applying Tarjan's algorithm on G and "nally computing the connected components of G ′, where G ′ =< N , E ′ > and E ′ = E − {(i, j)|(i, j) is a bridge of G}. The bridge-"nding problem has been extensively investigated in the context of parallel processing (see for example [2, 3, 4, 5, 6]). A parallel algorithm for the bridge-connected components problem is presented in [3]. In this paper, I present an optimal distributed algorithm for the bridge-connected components problem. In general, starightforward adaptation of a parallel algorithm on a distributed model of computation seems to be dif"cult, since the distributed models are inherently asynchronous. In this case, the graph is embedded in a communication network and each processor is assigned with a control algorithm such that it allows the processor to identify the bridge-connected component to which it belongs. The output of the algorithm is available in a distributed manner in the sense that each node knows only its component number: the component number is a representative node number which is considered to be the component number for the connected component in question. The distributed algorithm presented in this paper uses O(n) messages and O(n) units of time. 2. COMPUTATIONAL MODEL