1 SENSE OF DIRECTION AND COMMUNICATION COMPLEXITY IN DISTRIBUTED NETWORKS Nicola Santoro*, Jorge Urrutia + and Shmuel Zaks ++ *Distributed Computing Group, Carleton University, Ottawa, Canada + Department of Computer Science, Ottawa University, Ottawa, Canada ++ Department of Computer Science, Technion, Haifa, Israel ABSTRACT The connection between sense of direction and communication complexity in distributed complete networks is studied for two basic problems: finding a minimum-weight spanning tree (MST) and finding a spanning tree (SP). Several models of the complete network are defined, the difference being the amount (and type) of sense of direction available, forming a hierarchy which includes the models previously studied in the literature. It is shown that to move up in the hierarchy might require Ω(n 2 ) messages in the worst case. It is shown that the existing O(n) bound for SP can still be achieved at a lower level in the hierarchy; and that the Ω(n 2 ) bound for MST still holds at a higher level in the hierarchy. 1. INTRODUCTION Consider a network of n processors. Each processor has a distinct identity of which it alone is aware, and has available some labeled direct communication lines to other (possibly, all) processors; it also knows the (non-negative) cost associated with each such line. The network can viewed as an undirected graph G=(V,E) where |V|=n. Communication is achieved by sending messages along the communication lines. It is assumed that messages arrive, with no error, after a finite but otherwise arbitrary delay, and are kept in order of arrival in a queue until processed. All processors execute the same algorithm, which involves local processing as well as sending a message to a neighbor and receiving a message from a neighbor. Any non-empty set of processors may start the algorithm. In this context, two basic problems are finding a minimum-weight spanning-tree ( MST ) and distinguishing a unique processor (a leader); the latter is very closely related to the problem of finding a spanning tree (SP). These problems have been mostly studied in the literature for circular networks and complete networks. The interest in the circular network derives from the fact that it is the simplest symmetrical graph. In this graph both problems are equivalent and have been extensively studied; different bounds