Brief Annoucement: Distributed Swap Edges Computation for Minimum Routing Cost Spanning Trees

Given a weighted graph G(VG, EG) representing a communication network, with n nodes and m edges where the weights are positive integers, its Spanning Tree is typically used to route messages. In [1] the routing cost of a spanning tree is defined as the sum of the distances over all pairs of vertices of this tree. Hence, the most suitable spanning tree for the routing problem is the one minimizing the routing cost: the Minimum Routing Cost Spanning Tree (MRCST). Since trees have the minimal possible number of edges, a single edge failure is enough to interrupt the communication. For this reason, it is very important that it is fault-tolerant at least for the failure of one edge. Two approaches can be followed after the failure of an edge e: one can compute from scratch a new optimal spanning tree for G− e; or re-connect the two disconnected component of the spanning tree with the best possible edge. For temporary network failures, this second approach is clearly preferable to the first one: in fact, it is more efficient to use a swap edge for the duration of the failure, and to quickly revert to the original tree after the failure has been fixed; in this way, the original routing tables do not need major changes. In addition, we also note that finding the spanning tree with minimum routing cost is known to be NP-hard [3]. For these reasons, the ”swap-edge” approach has been followed to solve similar problems in sequential and distributed fashion; see for instance [2,4,6]. In our case, the best possible edge to select is the one that re-connects the two components disconnected by the fault of edge e, and leading to the new spanning tree for G− e having the minimum routing cost. This edge can be pre-computed for any possible edge fault and stored to recover the routing process in case of fault. The same problem has been solved for the sequential model in [6] inO(nm) time complexity. To compute the cost of the spanning tree, the authors make use of the concept of the ”routing load” of one edge, expressing the load of the different routes passing through this edge. The total cost is then computed by multiplying the routing load by the edge weight. We use here a different technique that computes the cost of a tree from the cost of its subtrees. Different is also the information we keep at each tree node in order to rebuild, in a distributed fashion, the tree after an edge fault. Our approach appears more suitable for the distributed solution, because allows us to treat contemporarely any possible edge fault.