Distributed Algorithms For Unidirectional Networks

Abstract

This paper addresses the question of distributively computing over a strongly connected unidirectional data communication network. In unidirectional networks the existence of a communication link from one node to another does not imply the existence of a link in the opposite direction. The strong connectivity means that from every node there is a directed path to any other node. The authors assume an arbitrary topology network in which the strong connectivity is the only restriction. Four models are considered, synchronous and asynchronous, and for each node space availability, which grows as either $O(1)$ bits or $O(\log n)$ bits per incident link, where n is the total number of nodes in the network, is considered. First algorithms for two basic problems in distributed computing in data communication networks, traversal, and election, are provided. Each of these basic protocols produces two directed spanning trees rooted at a distinguished node in the network, one called in-tree, leading to the root, and the other, out-tree, leading from the root. Given these trees, the authors efficiently transform bidirectional algorithms to run on unidirectional networks, and in particular solve other problems such as the broadcast and echo [E. J. CHANG, Decentralized Algorithms in Distributed Systems, Ph.D. thesis, University of Toronto. October 19791 in a way that is more efficient $O(n^2 )$ messages) than direct transformation (which yields $O(nm)$ messages algorithm). The communication cost of the traversal and election algorithms is $O(nm + n^2 \log n)$ bits ($O(nm)$ messages and time), where m is the total number of links in the network. The traversal algorithms for unidirectional networks of finite automata achieve the same cost $O(nm + n^2 \log n)$ bits ($O(nm)$ messages and time) bits) in the asynchronous case, while in the synchronous case the communication cost of the algorithm is ($O(nm)$ bits.