On-line algorithms for edge-disjoint paths in trees of rings

Abstract

A tree of rings is a graph that can be constructed by starting with a ring and then repeatedly adding a new disjoint ring to the graph and identifying one vertex of the new ring with a vertex of the existing graph. Trees of rings are a common topology for communication networks. We give randomized on-line algorithms for the problem of deciding for a sequence of requests (terminal pairs) in a tree of rings, which requests to accept and which to reject. Accepted requests must be routed along edge-disjoint paths. It is not allowed to reroute or preempt a request once it is accepted. The objective is to maximize the number of accepted requests. For the case that the paths for accepted requests can be chosen by the algorithm, we obtain competitive ratio O(log d), where d is the minimum possible diameter of a tree resulting from the tree of rings by deleting one edge from every ring. For the case where paths are pre-specified as part of the input, our algorithm achieves competitive ratio O(log ℓ), where ℓ is the maximum length of a simple path in the given tree of rings.