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Problems of finding distributively, the centers and medians of asynchronous communication networks are considered, and efficient distributed algorithms for determining these parameters are presented. The princip81 results of this paper are: 1. We show that for certain tree structures, the alg~ rithms of Korsch et.al.[7] fsil to anive at a correct solution, for both synchronous and asyrtchmnous tree networks. Furthermore, their algorithms which are centralized, fail to terminate for such uee structures. We present modi&ations to their basic algorithm to ensure correctness and termination. 2. A new decentralized distributed algorithm is presented to determine these topological parameters. This is the first decentralized algorithm to find centers and medians reported in literature. The centers are determined in h.JS) + D(T)/2 time using at most 3(n-1) + D(T’)/2 messages where haX(S) is the maximum height of a subtree rooted at any start node. The time sttd message complexities for median finding algorithms, are also linear in the number of nodes in the tree. 3. Finally, we present extensions of ~ese algorithms for weighted trees. keywords and @rases: Communication networks, Distributed algorithms, Centers, Medians INTRODUCTION Topological information available at each node in a disuibuted system is crucial to the design of efficient distributed algorithms for various needs. If the entire topology of a network is available at -h node in the network, then several computations such as finding shortest path, minimum spanning trees, leader, etc., could be done at each individual nodes, without requiring to coordinate Permission to oopy without fee all or part of thk material ie granted providad that the oopiea are not made or distributed for direct commercial sdvantaga, tha ACM copyright notica and the title of the publication and ite data appaar, and notice ie givan that copying is by parmiaaion of the Aaaociation for Computing Machinery. To oopy otharwiae, or to rapublish, raquires a fsa endlor specific permission. 01992 ACM 0.89791 .50z-x~z/000z~808m..$ 1.50 such computations with other nodes in the network. However, at least two factors restrict the availability of such irtformation at each node. FmL the dynamic nature of a network renders rhe static storage of topology, obsolete with time, thus requiring repeated updates erwh time a node or a link fails/comes up. ‘fltis might cause increased traffic in the network and lead to congestion. The second limiting factor is due to the storage space requirements. Hence, the global structure of a network is generally not maintained at each node. In most of the standard network models, the initial topological information is restricted to each node knowing its neighbors in the network. Each node then is provided with a set of distributed algorithms which would enable them to find the topology information whenever necessary. These algorithms are tesrned learning a@Jrirhrns [9]. Algorithms for learning various network parameters such as neighbors, global topology, centess, medians, diameter, leader, shortest paths, minimum spanning trees, etc. [14,7-1 1], have been reported in literature. In this paper, we present efficient distributed algorithms for finding centers, and medians of the graph representing the network, and in panicular , we consider asynchronous tree networks. The center and median finding algorithms are initiated by one or many nodes whenever the centers and medians have to be located in the network. Handler et.al [5] discuss the basic properties, and applications of centers and medians for networks. Earlier algorithms of Korach et.rd.[7] provide solutions for finding centers and medians for communication networim, and it is shown thst their algorithm for a tree network is optit@ in time. However, we show that the protocols described in their paper fail to find proper solution, and to terminate for certain mee structures. The problems with their algorithm are discussed and corrections are presented in this paper. We present a decentralized algorithm to determine the centers and medians of asynchronous tree networks. Furtkrnore, this is the fust decentralized algorirhm for finding centers and medians reported in literature. The algcrithm in [7] incorporates decentralization in two phases. In the 6rst phase, the leader in the network is determined using a

Distributed algorithms for locating centers and medians in communication networks