# A polylog(n)-competitive algorithm for metrical task systems

@inproceedings{Bartal1997APA, title={A polylog(n)-competitive algorithm for metrical task systems}, author={Yair Bartal and Avrim Blum and Carl Burch and Andrew Tomkins}, booktitle={Symposium on the Theory of Computing}, year={1997} }

We present a randomized on-line algorithm for the Metrical Task System problem that achieves a competitive ratio of O(log6 n) for arbitrary metric spaces, against an oblivious adversary. This is the first algorithm to achieve a sublinear competitive ratio for all metric spaces. Our algorit hm uses a recent result of Bartal [Bar96] that an arbitrary metr ic space can be probabilistically approximated by a set of metric spaces called “ k-hierarchical well-separated trees” ( kHST’s). Indeed, the…

## 98 Citations

### Towards the randomized k-server conjecture: a primal-dual approach

- Computer Science, MathematicsSODA '10
- 2010

Recently, Coté et al. [10] proposed an approach for solving the k-server problem on Hierchically Separated Trees (HSTs). In particular, they define a problem on a uniform metric, and show that if an…

### Metrical Task Systems and the k-Server Problem on HSTs

- Mathematics, Computer ScienceICALP
- 2010

The main technical contribution here is to extend many of these techniques to work directly on HSTs to obtain a refined guarantee for the unfair metrical task systems problem on an HST.

### Finely-competitive paging

- Computer Science40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

We construct an online algorithm for paging that achieves an O(r+log k) competitive ratio when compared to an offline strategy that is allowed the additional ability to "rent" pages at a cost of 1/r.…

### A Polylogarithmic-Competitive Algorithm for the k-Server Problem

- Computer Science, Mathematics2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

The first polylogarithmic-competitive randomized algorithm for the k-server problem on an arbitrary finite metric space is given, which improves upon the (2k-1)-competitive algorithm of Koutsoupias and Papadimitriou whenever n is sub-exponential in k.

### Approximating a finite metric by a small number of tree metrics

- Computer Science, MathematicsProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

This paper derandomizes the use of Bartal's algorithm and obtains the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing and a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming.

### A Decomposition Theorem for Task Systems and Bounds for Randomized Server Problems

- Computer Science, MathematicsSIAM J. Comput.
- 2000

A lower bound of $\Omega(\sqrt{\log k / \log k})$ is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary and the bound holds for arbitrary metric spaces and provides a new lower bound for the metrical task system problem as well.

### A Randomized Algorithm for Two Servers on the Line (Extended Abstract)

- Mathematics, Computer ScienceESA
- 1998

It is shown that the randomized 2-server problem can be reduced to the deterministic (2l; l) problem, and a lower bound of 2 is proved on the competitive ratio of the (4; 2)-server problem.

### The k-Server Problem and Fractional Analysis

- Mathematics, Computer Science
- 2005

It is shown that on the line and circle, the randomized version of the k-server problem is equivalent to the fractional version, and the cases for which these versions are not equivalent are classified by presenting a fractional algorithm which cannot be simulated by any randomized algorithm.

### A General Decomposition Theorem for the k-Server Problem

- MathematicsInf. Comput.
- 2001

The first general decomposition theorem for the k-server problem is presented, which implies O(polylog(k)-competitive randomized algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated sub-spaces, and takes a first step towards a general O- competitive algorithm.

### Ramsey-type theorems for metric spaces with applications to online problems

- MathematicsJ. Comput. Syst. Sci.
- 2006

## References

SHOWING 1-10 OF 17 REFERENCES

### Randomized Algorithms for Metrical Task Systems

- Computer Science, MathematicsTheor. Comput. Sci.
- 1995

### Probabilistic approximation of metric spaces and its algorithmic applications

- Computer Science, MathematicsProceedings of 37th Conference on Foundations of Computer Science
- 1996

It is proved that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion.

### A decomposition theorem and bounds for randomized server problems

- Mathematics, Computer ScienceProceedings., 33rd Annual Symposium on Foundations of Computer Science
- 1992

The authors prove a lower bound of Omega ( square root logk/loglogk) for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary and provides a new lower bound for the metrical task system problem as well.

### Lower bounds for randomized k-server and motion-planning algorithms

- Computer Science, MathematicsSTOC '91
- 1991

Lower bounds on the competitive ratio of randomized algorithms for two on-line problems: the k-server problem, suggested by [MMS], and an on- line motion-planning problem due to [PY], are proved.

### An optimal online algorithm for metrical task systems

- Computer ScienceSTOC
- 1987

A general model for the processing of sequences of tasks and a general online decision algorithm are introduced and it is shown that this algorithm is optimal among all online algorithms.

### Competitive algorithms for on-line problems

- Computer ScienceSTOC '88
- 1988

This paper presents several general results concerning competitive algorithms, as well as results on specific on-line problems.

### An optimal on-line algorithm for metrical task system

- PhysicsJACM
- 1992

A general model for the processing of sequences of tasks is introduced, and a general on-line decision algorithm is developed, which is shown that, for an important class of special cases, this algorithm is optimal among all on- line algorithms.

### On-line Learning and the Metrical Task System Problem

- Computer ScienceCOLT '97
- 1997

An experimental comparison of how these algorithms perform on a process migration problem, a problem that combines aspects of both the experts-tracking and MTS formalisms, is presented.