The optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors

Finding spanning trees became a central research issue in distributed networks and communication protocols, as most algorithms for these environments, in a general network structure , assume an underlying structure of a spanning tree, on whose edges all communication activities take place. Using such a tree it is easy to distinguish one processor as a leader, pass a token in the network {a technique widely used in these environments to assure mutual exclusion, to check termination, etc.), broadcast messages, and other related applications. In this work we present two lower bounds on the amount of communication activities needed in order to construct spanning trees with special properties in a complete network. In [1] a distributed algorithm for constructing a minimum spanning tree is constructed, and the question is raised there whether finding any spanning tree is easier than finding a minimum weight one. One of the lower bounds presented here answers this question. In certain applications it is desired that the nodes of the spanning tree will have bounded degrees, in order to prevent bottleneck effects and overcome physical restrictions in the communication network. Our second bound is related to this problem.