A randomized algorithm for two servers in cross polytope spaces

It has been a long-standing open problem to determine the exact randomized competitiveness of the 2-server problem, that is, the minimum competitiveness of any randomized online algorithm for the 2-server problem. For deterministic algorithms the best competitive ratio that can be obtained is 2 and no randomized algorithm is known that improves this ratio for general spaces. For the line, Bartal et al. (1998) [2] give a 15578 competitive algorithm, but their algorithm is specific to the geometry of the line. We consider here the 2-server problem over Cross Polytope Spaces M"2"4. We obtain an algorithm with competitive ratio of 1912, and show that this ratio is best possible. This algorithm gives the second non-trivial example of metric spaces with better than2-competitive ratio. The algorithm uses a design technique called the knowledge state technique - a method not specific to M"2"4.

[1]  Marek Chrobak,et al.  New results on server problems , 1991, SODA '90.

[2]  Christos H. Papadimitriou,et al.  Beyond competitive analysis [on-line algorithms] , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[3]  Marek Chrobak,et al.  A Randomized Algorithm for Two Servers on the Line , 2000, Inf. Comput..

[4]  Marek Chrobak,et al.  Metrical Task Systems, the Server Problem and the Work Function Algorithm , 1996, Online Algorithms.

[5]  Gerhard J. Woeginger,et al.  Online Algorithms , 1998, Lecture Notes in Computer Science.

[6]  Carsten Lund,et al.  Linear programs for randomized on-line algorithms , 1994, SODA '94.

[7]  L. Schläfli.  Theorie der vielfachen Kontinuität , 1901 .

[8]  Marek Chrobak,et al.  An Optimal On-Line Algorithm for k-Servers on Trees , 1991, SIAM J. Comput..

[9]  Lyle A. McGeoch,et al.  Competitive Algorithms for Server Problems , 1990, J. Algorithms.

[10]  Christos H. Papadimitriou,et al.  On the k-server conjecture , 1995, JACM.

[11]  Rüdiger Reischuk,et al.  Knowledge States: A Tool for Randomized Online Algorithms , 2008, HICSS.

[12]  Marek Chrobak,et al.  A Better Lower Bound on the Competitive Ratio of the Randomized 2-Server Problem , 1997, Inf. Process. Lett..

[13]  Marek Chrobak,et al.  Competitive analysis of randomized paging algorithms , 2000, Theor. Comput. Sci..

[14]  Rüdiger Reischuk,et al.  Knowledge states for the caching problem in shared memory multiprocessor systems , 2004, 7th International Symposium on Parallel Architectures, Algorithms and Networks, 2004. Proceedings..

[15]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[16]  Lawrence L. Larmore,et al.  Equitable Revisited , 2007, ESA.

[17]  Rüdiger Reischuk,et al.  Knowledge State Algorithms: Randomization with Limited Information , 2007, ArXiv.

[18]  Marek Chrobak,et al.  A Randomized Algorithm for Two Servers on the Line (Extended Abstract) , 1998, ESA.

[19]  Rüdiger Reischuk,et al.  Knowledge State Algorithms , 2011, Algorithmica.