On convex body chasing
A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setFi is specified, and the player has to move to a point inFi. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the “off-line” cost (i.e., the least possible cost when allFi are known in advance). We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces. The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line/off-line problems such as thek-server problem; we make some remarks on these more general problems.
Unable to display preview. Download preview PDF.
- 1.S. Ben-David, A. Borodin, R. M. Karp, G. Tardos, and A. Wigderson. On the Power of Randomization in On-line Algorithms. InProc. of the 22nd Ann. ACM Symp. on Theory of Computing, pages 379–386, May 1990.Google Scholar
- 2.P. Berman, H. J. Karloff, and G. Tardos. A Competitive Three-Server Algorithm. InProc. of the 1st Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 280–290, January 1990.Google Scholar
- 3.A. Bordoin, N. Linial, and M. Saks. An Optimal On-Line Algorithm for Metrical Task Systems. InProc. of the 19th Ann. ACM Symp. on Theory of Computing, pages 373–382, May 1987. Final version to appear inJ. Assoc. Comput. Mach. Google Scholar
- 4.M. Chrobak, H. J. Karloff, T. Payne, and S. Vishwanathan. New Results on Server Problems. InProc. of the 1st Ann. ACM-SIAM Symp. on Discrete Algorithms, pates 291–300, January 1990.Google Scholar
- 5.D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random Walks on Weighted Graphs and Applications to On-line Algorithms. InProc. of the 22nd Ann. ACM Symp. on Theory of Computing, pages 369–378, May 1990.Google Scholar
- 6.A. Fiat, D. P. Foster, H. Karloff, Y. Rabani, Y. Ravid, and S. Vishwanathan. Competitive Algorithms for Layered Graph Traversal. Manuscript, 1991.Google Scholar
- 7.A. Fiat, Y. Rabani, and Y. Ravid. Competitivek-Server Algorithms. InProc. of the 31st Ann. IEEE Symp. on Foundations of Computer Science, pages 454–463, October 1990.Google Scholar
- 8.E. Grove. The Harmonic Onlinek-Server Algorithm Is Competitive. Manuscript, 1990.Google Scholar
- 9.M. S. Manasse, L. A. McGeoch, and D. D. Sleator. Competitive Algorithms for On-Line Problems. InProc. of the 20th Ann. ACM Symp. on Theory of Computing, pages 322–333, May 1988.Google Scholar
- 10.C. H. Papadimitriou and M. Yannakakis. Shortest Paths Without a Map. InProc. 16th ICALP, pages 610–620, July 1989.Google Scholar