Abstract
Borodin, Linial, and Saks introduced a general model for online systems called metrical task systems (1992, J. Assoc. Comput. Mach. 39 (4), 745–763). In this paper, the unfair two state problem, a natural generalization of the two state metrical task system problem, is studied. A randomized algorithm for this problem is presented, and it is shown that this algorithm is optimal. Using the analysis of the unfair two state problem, a proof of a decomposition theorem similar to that of Blum, Karloff, Rabani, and Saks (1992, “Proc. 33rd Symposium on Foundations of Computer Science,” pp. 197–207) is presented. This theorem allows one to design divide and conquer algorithms for specific metrical task systems. Our theorem gives the same bounds asymptotically, but it has less restrictive boundary conditions.
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