The 3-server problem in the plane

  • Authors:

  • Wolfgang W. Bein;Marek Chrobak;Lawrence L. Larmore

  • Affiliations:

  • Department of Computer Science, University of Nevada, Las Vegas, NV;Department of Computer Science, University of California, Riverside, CA;Department of Computer Science, University of Nevada, Las Vegas, NV

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2002

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Abstract

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 (we assume that k ≥ 2). The request issued at each step is specified by a point r in a given metric space M. To serve this request, one of the k servers must move to r. It is known that if M has at least k + 1 points then no online algorithm for the k-server problem in M has competitive ratio smaller than k. The best known upper bound on the competitive ratio in arbitrary metric spaces, by Koutsoupias and Papadimitriou (J. ACM 42 (1995) 971), is 2k - 1. There are only a few special cases for which k-competitive algorithms are known: for k = 2, when M is a tree, or when M has at most k + 2 points. We prove that the Work Function Algorithm is 3-competitive for the 3-server problem in the Manhattan plane. As a corollary, we obtain a 4.243-competitive algorithm for 3 servers in the Euclidean plane. The best previously known competitive ratio for 3 servers in these metric spaces was 5.