Competitiveness and response time in on-line algorithms
The study of competitive algorithms has concentrated on competitiveness — comparing online algorithms to optimal off-line algorithms on sequences of operations. Published algorithms proven (or suggested) to be competitive invariably have pessimal response time i.e. their worst-case single operation time is as bad as possible. We consider whether or not such algorithms can be adapted to improve the response time without sacrificing competitiveness. We consider lists, off-line static search trees, dynamic search trees, and the k-server problem on a line segment of length L. For lists, pessimal response time is unavoidable. For off-line static search trees our algorithm is 2-competitive and has response time 2 log n. For dynamic search trees our algorithm has logarithmic amortized time and is statically optimal (like splay trees), but the response time is O(√n log n) and Ω(log2n). For the k-server problem we prove that any algorithm with O(optimal) response time has a competitive ratio of at least Ω(L/k). This is achieved by a simple on-line algorithm. We also show that even a weak limit on response time for the k-server problem (e.g. response time less than half pessimal) yields an Ω(L/k) separation between on-line and off-line algorithms. Our results apply to high-performance multi-head disk drives where response time is critical.
Unable to display preview. Download preview PDF.
- [CCJF85]A.R. Calderbank, E.G. Coffman Jr, and L. Flatto. Sequence problems in two-server systems. Mathematics of Operations Research, 10(4):585–598, November 1985.Google Scholar
- [CKPV90]M. Chrobak, H. Karloff, T. Payne, and S. Vishwanathan. New results on server problems. In Proc. of the Symp. on Discrete Algorithms, pages 291–300. SIAM, 1990.Google Scholar
- [FRR90]A. Fiat, Y. Rabani, and Y. Ravid. Competitive k-server algorithms. In Proceedings of the Symposium on Foundations of Computer Science, pages 454–463. IEEE, 1990.Google Scholar
- [Hof83]M. Hofri. Should the two-headed disk be greedy? — yes, it should. Information Processing Letters, 16:83–85, February 1983.Google Scholar
- [Meh84]K. Mehlhorn. Data Structures and Algorithms, Vol 1: Sorting and Searching. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1984.Google Scholar
- [MMS88]M. Manasse, L. McGeoch, and D. Sleator. Competitive algorithms for on-line problems. In Proceedings of the Symposium on Theory of Computing, pages 322–333, New York, 1988. ACM.Google Scholar
- [She90]M. Sherk. Self-Adjusting k-ary Search Trees and Self-Adjusting Balanced Search Trees. PhD thesis, U. of Toronto, 1990. Available as Comp. Sci. Technical Report 234/90.Google Scholar
- [ST83]D. Sleator and R. Tarjan. Self-adjusting binary trees. In Proceedings of the Symposium on Theory of Computing, pages 235–245, New York, 1983. ACM.Google Scholar
- [ST85a]D. Sleator and R. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202–208, February 1985.Google Scholar
- [ST85b]D. Sleator and R. Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32:652–686, July 1985.Google Scholar
- [Tar85]R. Tarjan. Amortized computational complexity. SIAM Journal of Alg. Disc. Meth., 6(2):306–318, April 1985.Google Scholar