An optimal online algorithm for metrical task systems

  • Authors:
  • A. Borodin;N. Linial;M. Saks

  • Affiliations:
  • Dept. of Computer Science, University of Toronto Toronto, Canada;Institute of Computer Science, Hebrew University, Jerusalem, Israel;Dept. of Mathematics and RUTCOR, Rutgers University, New Brunswick, New Jersey and Bell Communications Research, Morristown, New Jersey

  • Venue:
  • STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
  • Year:
  • 1987

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Abstract

In practice, almost all dynamic systems require decisions to be made online, without full knowledge of their future impact on the system. We introduce a general model for the processing of sequences of tasks and develop a general online decision algorithm. We show that, for an important class of special cases, this algorithm is optimal among all online algorithms.Specifically, a task system (S, d) for processing sequences of tasks consists of a set S of states and a cost matrix d where d(i, j) is the cost of changing from state i to state j (we assume that d satisfies the triangle inequality and all diagonal entries are O.) The cost of processing a given task depends on the state of the system. A schedule for a sequence T1, T2 … Tk of tasks is a sequence s1, s2 … sk of states where si is the state in which Ti is processed; the cost of a schedule is the sum of all task processing costs and state transition costs incurred.An online scheduling algorithm is one that chooses si only knowing T1 T2 … Ti. Such an algorithm operates within waste factor w if, on any input task sequence, its costs is within an additive constant of w times the optimal offline schedule cost. The online waste factor w(S, d) is the infirm waste factor of any online scheduling algorithm for (S, d). We show that w(S, d) = 2|S| - 1 for every task system in which d symmetric, and w(S, d) = &Ogr;(|S|2) for every task system.