An algorithm for detecting termination of distributed computation in arbitrary network topologies within linear time

To determine if a distributed process has terminated is very important. Several algorithms have been proposed for this purpose since early 1980s. Unfortunately, most of these algorithms assume a logical ring structure on network topology. Thinking that in order to form a logical ring, one may need either to include some channels in a logical ring for more than once or physically add some extra channels to a network if the network itself does not physically form a ring, it is obvious to see that these algorithms may increase network expenses. This paper proposes a two-phase algorithm which does not require a logical ring topology, thus eliminating both multiple use of some channels in a logical ring and introduction of extra channels to a network. The algorithm can be finished in O(D+M) time in the worst case, here D denotes the distance of the network, M denotes the number of computation messages occurring during the execution of the algorithm. The message complexity of this algorithm for each successful detection is O(E+M), here E denotes the number of edges in the network.