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| '''Task systems''' are mathematical objects used to model the set of possible configuration of [[online algorithm]]s. They were introduced by [[Allan Borodin|Borodin]], [[Nati Linial|Linial]] and [[Michael Saks (mathematician)|Saks]] (1992) to model a variety of online problems. A task system determines a set of states and costs to change states. Task systems obtain as input a sequence of requests such that each request assigns processing times to the states. The objective of an online algorithm for task systems is to create a schedule that minimizes the overall cost incurred due to processing the tasks with respect to the states and due to the cost to change states.
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| If the cost function to change states is a [[Metric (mathematics)|metric]], the task system is a '''metrical task system''' (MTS). This is the most common type of task systems.
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| Metrical task systems generalize online problems such as [[Page replacement algorithm|paging]], [[List accessing problem|list accessing]], and the [[k-server problem]] (in finite spaces).
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| ==Formal Definition==
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| A task system is a pair <math>(S,d)</math> where <math>S=\{s_1,s_2,\dotsc,s_n\}</math> is a set of '''states''' and <math> d:S \times S \rightarrow \mathbb{R}</math> is a distance function. If <math>d</math> is a metric, <math>(S,d)</math> is a metrical task system. An input to the task system is a sequence <math>\sigma = T_1,T_2,\dotsc,T_l</math> such that for each <math>i</math>, <math>T_i</math> is a vector of <math>n</math> non-negative entries that determine the processing costs for the <math>n</math> states when processing the <math>i</math>th task.
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| An algorithm for the task system produces a schedule <math>\pi</math> that determines the sequence of states. For instance, <math> \pi(i)=s_j</math> means that the <math>i</math>th task <math>T_i</math> is run in state <math>s_j</math>. The processing cost of a schedule is
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| <math> \mathrm{cost}(\pi,\sigma) = \sum_{i=1}^l d(\pi(i-1),\pi(i)) + T_i(\pi(i)).</math>
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| The objective of the algorithm is to find a schedule such that the cost is minimized.
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| ==Known Results==
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| As usual for online problems, the most common measure to analyze algorithms for metrical task systems is the [[Competitive analysis (online algorithm)|competitive analysis]], where the performance of an online algorithm is compared to the performance of an optimal offline algorithm. For deterministic online algorithms, there is a tight bound <math> 2n-1</math> on the competitive ratio due to Borodin et al. (1992).
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| For randomized online algorithms, the competitive ratio is lower bounded by <math> \Omega(\log n / \log\log n) </math> and upper bounded by <math> O(\log^2 n \log\log n)</math>. The lower bound is due to Bartal et al. (2006,2005). The upper bound is due to Fiat and Mendel (2003) who improved upon a result of Bartal et al. (1997).
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| There are many results for various types of restricted metrics.
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| ==See also==
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| * [[Adversary (online algorithm)|Adversary model]]
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| * [[Competitive analysis (online algorithm)|Competitive analysis]]
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| * [[K-server problem]]
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| * [[Online algorithm]]
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| * [[Page replacement algorithm]]
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| * [[Real-time computing]]
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| ==References==
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| * {{cite conference
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| | author = Yair Bartal, Avrim Blum, Carl Burch, and Andrew Tomkins
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| | title = A polylog(n)-Competitive Algorithm for Metrical Task Systems
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| | booktitle = Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing
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| | year = 1997
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| | pages = 711–719
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| | doi = 10.1145/258533.258667}}
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| * {{cite journal
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| | author = Yair Bartal, Béla Bollobás, Manor Mendel
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| | title = Ramsey-type theorems for metric spaces with applications to online problems
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| | journal= Journal of Computer and System Sciences
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| | year = 2006
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| | volume= 72
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| | pages = 890–921
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| }}
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| *{{cite journal
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| | author=Yair Bartal, Nathan Linial, Manor Mendel, [[Assaf Naor]]
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| | title= On metric Ramsey-type phenomena
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| | journal= Annals of Mathematics
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| | volume= 162
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| | year=2005
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| | pages= 643–709
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| }}
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| * {{cite book
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| | author= [[Allan Borodin]] and Ran El-Yaniv
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| | title= ''Online Computation and Competitive Analysis''
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| | url= http://www.cs.technion.ac.il/~rani/book.html
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| | publisher= Cambridge University Press
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| | year= 1998
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| | pages= 123–149
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| }}
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| * {{cite journal
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| | author = [[Allan Borodin]], [[Nati Linial]], and [[Michael Saks (mathematician)|Michael Saks]]
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| | title = An optimal online algorithm for metrical task systems
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| | journal = Journal of the ACM
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| | volume = 39
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| | pages = 745–763
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| | year = 1992
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| | doi = 10.1145/146585.146588}}
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| * {{cite journal
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| | author = Amos Fiat and Manor Mendel
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| | title = Better Algorithms for Unfair Metrical Task Systems and Applications
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| | journal = SIAM J. Comput.
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| | volume = 32
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| | pages = 1403–1422
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| | year = 2003
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| | doi = 10.1137/S0097539700376159}}
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| [[Category:Online algorithms]]
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