Modeling Locality: A Probabilistic Analysis of LRU and FWF

  • Luca Becchetti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3221)

Abstract

In this paper we explore the effects of locality on the performance of paging algorithms. Traditional competitive analysis fails to explain important properties of paging assessed by practical experience. In particular, the competitive ratios of paging algorithms that are known to be efficient in practice (e.g. LRU) are as poor as those of naive heuristics (e.g. FWF). It has been recognized that the main reason for these discrepancies lies in an unsatisfactory modelling of locality of reference exhibited by real request sequences.

Following [13], we explicitely address this issue, proposing an adversarial model in which the probability of requesting a page is also a function of the page’s age. In this way, our approach allows to capture the effects of locality of reference. We consider several families of distributions and we prove that the competitive ratio of LRU becomes constant as locality increases, as expected. This result is strengthened when the distribution satisfies a further concavity/convexity property: in this case, the competitive ratio of LRU is always constant.

We also propose a family of distributions parametrized by locality of reference and we prove that the performance of FWF rapidly degrades as locality increases, while the converse happens for LRU.

We think, our results provide one contribution to explaining the behaviours of these algorithms in practice.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albers, S., Favrholdt, L.M., Giel, O.: On paging with locality of reference. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 258–267 (2002)Google Scholar
  2. 2.
    Belady, L.A.: A study of replacement algorithms for a virtual-storage computer. IBM Systems Journal 5(2), 78–101 (1966)CrossRefGoogle Scholar
  3. 3.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  4. 4.
    Borodin, A., Irani, S., Raghavan, P., Schieber, B.: Competitive paging with locality of reference. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 249–259 (1991)Google Scholar
  5. 5.
    Boyar, J., Favrholdt, L.M., Larsen, K.S.: The relative worst case order ratio applied to paging. In: Proceedings of the 5th Italian Conference on Algorithms and Complexity, pp. 58–69 (2003)Google Scholar
  6. 6.
    Boyar, J., Favrholdt, L.M., Larsen, K.S.: The relative worst order ratio applied to paging. Tech. report ALCOMFT-TR-03-32, Future and Emerging Technologies program under the EU, contract number IST-1999-14186 (2003)Google Scholar
  7. 7.
    Denning, P.J.: The working set model for program behaviour. Communications of the ACM 11(5), 323–333 (1968)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fiat, A., Karlin, A.R.: Randomized and multipointer paging with locality of reference. In: Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, pp. 626–634 (1995)Google Scholar
  9. 9.
    Fiat, A., Karp, R., Luby, M., McGeoch, L., Sleator, D., Young, N.E.: Competitive paging algorithms. Journal of Algorithms 12(4), 685–699 (1991)MATHCrossRefGoogle Scholar
  10. 10.
    Fiat, A., Mendel, M.: Truly online paging with locality of reference. In: 38th IEEE Annual Symposium on Foundations of Computer Science, pp. 326–335 (1997)Google Scholar
  11. 11.
    Irani, S., Karlin, A.R., Phillips, S.: Strongly competitive algorithms for paging with locality of reference. SIAM Journal on Computing 25(3), 477–497 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Karlin, R., Phillips, S.J., Raghavan, P.: Markov paging. In: Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pp. 208–217 (1992)Google Scholar
  13. 13.
    Koutsoupias, E., Papadimitriou, C.H.: Beyond competitive analysis. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 394–400 (1994)Google Scholar
  14. 14.
    Chrobak, M., Noga, J.: LRU is better than FIFO. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 78–81 (1998)Google Scholar
  15. 15.
    Sleator, D., Tarjan, R.E.: Amortized Efficiency of List Update and Paging Rules. Communications of the ACM 28, 202–208 (1985)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tanenbaum, A.S.: Modern Operating Systems. Prentice-Hall, Englewood Cliffs (1992)MATHGoogle Scholar
  17. 17.
    Torng, E.: A unified analysis of paging and caching. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pp. 194–203 (1995)Google Scholar
  18. 18.
    Young, N.: The k-server dual and loose competitiveness for paging. Algorithmica 11(6), 525–541 (1994)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Young, N.E.: On-line file caching. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 82–86 (1998)Google Scholar
  20. 20.
    Young, N.E.: On-line paging against adversarially biased random inputs. Journal of Algorithms 37(1), 218–235 (2000)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Young, N.: Competitive paging and dual-guided on-line weighted caching and matching algorithms. PhD thesis, Department of Computer Science, Princeton University (1991)Google Scholar
  22. 22.
    Young, N.E.: Bounding the diffuse adversary. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 420–425 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Luca Becchetti
    • 1
  1. 1.University of Rome “La Sapienza” 

Personalised recommendations