Settling the Randomized k-sever Conjecture on Some Special Metrics

The k-server problem is one of the most fundamental online problems, which is introduced by Manasse, McGeoch and Sleator [27, 28]. The problem is to schedule k mobile servers to serve a sequence of requests in a metric space with the minimum possible movement distance. The randomized k-sever conjecture states that there exists O(log k)-competitive randomized algorithms for the k-sever problem. The conjectures has been open for over 24 years. In this paper, we settle the randomized k-sever conjecture for the following metric spaces: line, circle, Hierarchically well-separated tree (HST), if k = 2 or n = k + 1 for arbitrary metric spaces. Specially, we show that there are O(log k)-competitive randomized k-sever algorithms for above metric spaces. For any general metric space with n points, we show that there is an O(log k log n)-competitive randomized k-sever algorithm, which improved the previous best competitive ratio O(log k log n log log n) by Nikhil Bansal et al. (FOCS 2011, pages 267276). Above algorithms refer to lazy algorithms, i.e., algorithms move only one sever to serve the requested point only if the requested point is not served. In addition, we still show that there exists a O(log k)-competitive randomized non-lazy algorithm for the k-sever problem.