Creating Optimal Distributed Algorithms for Minimum Spanning Trees

This paper examines the complexity of distributed algorithms for nding a Minimum Spanning Tree in undirected graphs; the goal is to create algorithms optimal with respect to both communication O(E + N logN) and time O(N), where E; N is the number of edges and nodes respectively. A fundamental bad case that leads to non-optimal performance and the proposed techniques to overcome this problem are presented. We introduce new techniques based on the the idea that we call Distributed Information; nodes store information that summarizes properties of groups of nodes. The techniques (and the corresponding algorithms) are classiied in communication optimal and time optimal ones. Finally, the structure of the algorithm proposed in Awe87] and the above classiication can lead to a pattern for creating optimal algorithms. In addition, a simple O(E) messages and O(N) time algorithm for counting the nodes of the network is introduced; It can be used as part of the optimal algorithm or it may be of independent interest.