The harmonic online K-server algorithm is competitive

We show that the Harmonic algorithm for the online K-server problem is ( ~11 x 2K – 2K)competitive against an adaptive online adversary for K 22. From a result of [BBKTW90], we get a deterministic algorithm which is ( ~K x 2K – 2102competitive. These are the best competitive ratios that have been published so far for online algorithms over a general metric space when K >2. 1 Competitive Algorithms There are many applications that might be described as online because it is necessary to make decisions without access to future inputs. Associated with each decision is a cost, and the goal is to minimize the total cost. An operating system’s algorithm for paging memory is a good example of an online algorithm. It must decide which page to swap out before it sees the next memory access. In contrast, an ofline procedure is allowed to know the entire sequence of inputs in advance, before it makes arty decisions. We want online algorithms whose cost compares favorably to the cost of an optimal offline algorithm. *Supported by a fellowship from Lockheed Corp. Permissionto copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the pubhcatlonand its dateappear,and notice N given that copying is by permissionof theAssociationfor computing Machinery.To copyotherwise, or to republish, requiresa fee and/or specificpermission. @ 1991 ACM 089791-397-3/91/0004/0260 $1.50 Sleator and Tarjan ([ ST85]) founded the study of competitive algorithms by introducing the idea that the performance of an online algorithm should be measured by the ratio of the cost it incurs on a sequence of inputs to the minimum offline cost for processing the sequence. Formally, a deterministic online algorithm is C-competitive if there exists a function 1 of the initial configuration so that for every finite input sequence the cost incurred by the algorithm is bounded by 1 plus C’ times the minimum cost of processing the input sequence. C’ is called the competitive ratio of the algorithm. 2 The Harmonic Online KServer Algorithm The online K-server problem was introduced by Martasse, McGeoch and Sleator ([MMS88]). We are given initial locations of K servers in a metric space. Requests for service at points {xt } come in over time. Immediately after the tih request is received, one of the servers must be moved from its current location to St. The choice of which server is moved must be made without reference to Zu for any u > t. Moving a server costs the distance the server moves. Our goal is to find an algorithm whose competitive ratio is as low as possible. The framework of K servers moving about in a metric space is quite general. For example, paging is a special case of the online K-server problem. We interpret a server at point F’ to mean that the page P is in main memory. K is the size of main memory, zt is the page containing the tth memory