Distributed Computation of All Node Replacements of a Minimum Spanning Tree

In many network applications the computation takes place on the minimum-cost spanning tree (MST ) of the network G; unfortunately, a single link or node failure disconnects the tree. The All Node Replacements (ANR) problem is the problem of precomputing, for each node u in G, the new MST should u fail. The simpler problem dealing with single edge failures is called All Edge Replacements (AER). Both problems have been extensively investigated for serial and parallel settings, and efficient solutions have been designed for those environments. The situation is surprisingly different in distributed settings. In fact, for either problem, no distributed solution exists to date which performs better than the brute-force repeated application of a MST construction protocol. In other words, currently, the solution of either problems uses O(nm+n log n) messages, where n is the number of nodes and m is the number of edges. In this paper we consider the problem of computing all the replacement minimum-cost spanning trees distributively, and we efficiently solve the problem. We design a solution protocol for the ANR problem and we prove that the total amount of data items communicated during the computation (the data complexity) is O(n). This communication can be achieved transmitting only O(n) long messages between neighbours, if the system so allows; otherwise O(n) standard messages suffice. Even in systems that do not allow long messages, the proposed protocol constitutes a significant improvement over the individual computation of the replacement trees. Indeed, for dense graphs, our protocol constructs all the n replacement MST s of the minimum spanning tree T with the same number of messages required just to compute T .