A Distributed Algorithm for Finding All Best Swap Edges Of a Minimum Diameter Spanning Tree

ABSTRACT Communication in networks suffers if a link fails. When the links are edge of a tree that has been chose

Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures

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 a new spanning tree. Such an optimal replacement is called a best swap. As a natural extension, the allbest -swaps (ABS) problem is the problem of finding a best swap for every edge of the MDST. Given a weighted graph G = (V, E), where |V | = n and |E| = m, we solve the ABS problem in O(n # m) time and O(m) space, thus improving previous bounds for m = o(n 2 ). We also show that the diameter of the tree resulting from a best swap is at most 5/2 as long as the diameter of a MDST recomputed from scratch. Communicated by Takao Nishizeki, Roberto Tamassia, and Dorothea Wagner: submitted January 1999; revised February 2000 and December 2000. Work supported by the EU TMR Grant CHOROCHRONOS and by the Swiss National Science Foundation. A preliminary version of this paper was presented to the 6th European Symposium on Algorithms (ESA'98), Venice, Italy, 1998. E. Nardelli et al., All Best Swaps of a MDST , JGAA, 5(5) 39--57 (2001) 40

Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures

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 a new spanning tree. Such an optimal replacement is called a best swap. As a natural extension, the allbest-swaps (ABS) problem is the problem of finding a best swap for every edge of the MDST. Given a weighted graph G =(V,E), where |V | = n and |E | = m, we solve the ABS problem in O(n √ m)timeandO(m) space, thus improving previous bounds for m = o(n 2). We also show that the diameter of the tree resulting from a best swap is at most 5/2 aslong as the diameter of a MDST recomputed from scratch

Dynamic distributed programming and applications to swap edge problem

Link failure is a common reason for disruption in communication networks. If communication between processes of a weighted distributed network is maintained by a spanning tree T, and if one edge e of T fails, communication can be restored by finding a new spanning tree, T’. If the network is 2-edge connected, T’ can always be constructed by replacing e by a single edge, e’, of the network. We refer to e’ as a swap edge of e. The best swap edge problem is to find the best choice of e’, that is, that e which causes the new spanning tree T’ to have the least cost, where cost is measured in a way that is determined by the application. Two examples of such measures are total weight of T‘ and diameter of T’. The all best swap edges problem is the problem of determining, in advance of any failure, the best swap edge for every edge in T. The justification for this problem is that we wish to be ready, when a failure occurs, to quickly activate a replacement for the failed edge. In this thesis, we give algorithms for the all best swap edges problem for six different cost measures. We first present an algorithm which can be adapted to all six measures, and which takes O (d2) time, where d is the diameter of T. This algorithm is essentially a form of distributed dynamic programming, since we compute the answers to sub problems at each node of T. We then present a novel paradigm for speeding up distributed computations under certain conditions. We apply this paradigm to find O(d)-time distributed algorithms for the all best swap edge problem for all but one of our cost measures. Formal algorithms and their correctness proofs will be given