Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the all-best-swaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where |V | = n and |E | = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1

Abstract We consider the problem of computing the optimal swap edges of a shortest-path tree. This theoretical problem arises in practice in systems that offer point-offailure shortest-path rerouting service in presence of a single link failure: if the shortest path is not affected by the failed link, then the message will be delivered through that path; otherwise, the system will guarantee that, when the message reaches the node where the failure has occurred, the message will then be rerouted through the shortest-path to its destination. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. We present two simple and efficient distributed algorithms for computing the optimal swap edges of a shortest-path tree. One algorithm uses messages containing a constant amount of information, while the other is tailored for systems that allow long messages. The amount of data transferred by the protocols is the same and depends on on the structure of the shortest-path spanning-tree; it is no more, and sometimes significantly less, than the cost of constructing the shortest-path tree.

Recently great attention has been given to point-of-failure swap rerouting, an efficient technique for routing in presence of transient failures. According to this technique, a message follows the normal routing table information unless the next hop has failed; in this case, it is redirected towards a precomputed link, called swap; once this link has been crossed, normal routing is resumed. The amount of precomputed information required in addition to the routing table is rather small: a single link per each destination. Several efficient serial algorithms have been presented to compute this information; none of them can unfortunately be efficiently implemented in a distributed environment. In this paper we present protocols, based on a new strategy, that allow the efficient computation of all the optimal swap edges under several optimization criteria.

Abstract. Given a 2-node connected, undirected graph G = (V,E), with n nodes and m edges with real weights, and given a minimum spanning tree (MST) T =(V, ET)ofG, we studythe problem of finding, for everynode v ∈ V,theMSTofG − v =(V \{v},E \ Ev), where Ev is the set of edges incident to v in G. We show that this problem can be solved in O(min(m · α(n, n),m+ n log n)) time and O(m) space. Our solution improves on the previouslyknown O(m log n) time bound. 1

We consider the problem of computing the optimal swap edges of a shortest-path tree. This problem arises in designing systems that offer point-of-failure shortest-path rerouting service in presence of a single link failure: if the shortest path is not affected by the failed link, then the message will be delivered through that path; otherwise, the system will guarantee that, when the message reaches the node where the failure has occurred, the message will then be re-routed through the shortest-path to its destination. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. We present two simple and efficient distributed algorithms for computing the optimal swap edges of a shortest-path tree. One algorithm uses messages containing a constant amount of information, while the other is tailored for systems that allow long messages. The amount of data transferred by the protocols is the same and depends on the structure of the shortest-path spanning-tree; it is no more, and sometimes significantly less, than the cost of constructing the shortest-path tree.

Abstract. We consider the problem of computing the best swap edges of a shortest-path tree Tr rooted in r. That is, given a single link failure: if the path is not affected by the failed link, then the message will be delivered through that path; otherwise, we want to guarantee that, when the message reaches the edge (u, v) where the failure has occurred, the message will then be re-routed using the computed swap edge. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. In [6], distributed solutions to compute the swap edge that minimizes the distance from u to r have been presented. In contrast, in this paper we focus on selecting, efficiently and distributively, the best swap edge according to an objective function suggested in [13]: we choose the swap edge that minimizes the distance from u to v.

In systems using shortest-path routing tables, a single link failure is enough to interrupt the message transmission by disconnecting one or more shortestpath spanning trees. The on-line recomputation of an alternative path or of the entire new shortest path trees, rebuilding the routing tables accordingly, is rather

Keywords: We consider the problem of computing the optimal swap edges of a shortest-path tree. This theoretical problem arises in practice in systems that offer point-offailure shortest-path rerouting service in presence of a single link failure: if the shortest path is not affected by the failed link, then the message will be delivered through that path; otherwise, the system will guarantee that, when the message reaches the node where the failure has occurred, the message will then be rerouted through the shortest-path to its destination. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. We present two simple and efficient distributed algorithms for computing the optimal swap edges of a shortest-path tree. One algorithm uses messages containing a constant amount of information, while the other is tailored for systems that allow long messages. The amount of data transferred by the protocols is the same and depends on on the structure of the shortest-path spanning-tree; it is no more, and sometimes significantly less, than the cost of constructing the shortest-path tree.

2 Scientific Work

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,