The CNN Problem and Other k-Server Variants

Abstract

We study several interesting variants of the k-server problem. In the cnn problem, one server services requests in the Euclidean plane. The difference from the k-server problem is that the server does not have to move to a request, but it has only to move to a point that lies in the same horizontal or vertical line with the request. This, for example, models the problem faced by a crew of a certain news network trying to shoot scenes on the streets of Manhattan from a distance; the crew has only to be on a matching street or avenue. The CNN problem contains as special cases two important problems: the bridge problem, also known as the cow-path problem, and the weighted 2-server problem in which the 2 servers may have different speeds. We show that any deterministic on-line algorithm has competitive ratio at least \( 6 + \sqrt {17} \). We also show that some successful algorithms for the k-server problem fail to be competitive. In particular, we show that no natural lazy memoryless randomized algorithm can be competitive.The CNN problem also motivates another variant of the k-server problem, in which servers can move simultaneously, and we wish to minimize the time spent waiting for service. This is equivalent to the regular k-server problem under the \( \mathcal{L}_\infty \) norm for movement costs. We give a 1/2k(k + 1) upper bound for the competitive ratio on trees.