In the k-server problem we wish to minimize, in an online fashion, the movement cost of k servers in response to a sequence of requests. For two servers, it is known that the optimal deterministic algorithm has competitive ratio 2, and it has been a long-standing open problem whether it is possible to improve this ratio using randomization. We give a positive answer to this problem when the underlying metric space is a real line, by providing a randomized online algorithm for this case with competitive ratio at most 155/78 ≈ 1.987. This is the first algorithm for two servers that achieves a competitive ratio smaller than 2 in a nonuniform metric space with more than three points. We consider a more general problem called the (k, l)-server problem, in which a request is served using l out of k available servers. We show that the randomized 2-server problem can be reduced to the deterministic (2l,l)-server problem. We prove a lower bound of 2 on the competitive ratio of the (4, 2)-server problem. This implies that one unbiased random bit is not sufficient to improve the ratio of 2 for the two-server problem. Then we give a 155/78-competitive algorithm for the (6, 3)-server problem on the real line. Our algorithm is simple and memoryless. The solution has been obtained using linear programming techniques that may have applications for other online problems.
Bibliographical noteFunding Information:
1This research was partially conducted while the author was at International Computer Science Institute, University of California, Berkeley. 2Research supported by NSF Grant CCR-9503498. This research was partially conducted when the author was visiting International Computer Science Institute, University of California, Berkeley. 3Research supported by NSF Grant CCR-9503441.