Analysis of the Harmonic Algorithm for Three Servers

HARMONIC is a randomized algorithm for the k-server problem that, at each step, given a request point r, chooses the server to be moved to r with probability inversely proportional to the distance to r. For general k, it is known that the competitive ratio of HARMONIC is at least 1/2k(k + 1), while the best upper bound on this ratio is exponential in k. It has been conjectured that Harmonic is 1/2k(k + 1)-competitive for all k. This conjecture has been proven in a number of special cases, including k = 2 and for the so-called lazy adversary.In this paper we provide further evidence for this conjecture, by proving that HARMONIC is 6-competitive for k = 3. Our approach is based on the random walk techniques and their relationship to the electrical network theory. We propose a new potential function F and reduce the proof of the validity of ? to several inequalities involving hitting costs. Then we show that these inequalities hold for k = 3.

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