, Volume 71, Issue 2, pp 279–306 | Cite as

Online File Caching with Rejection Penalties

  • Leah EpsteinEmail author
  • Csanád Imreh
  • Asaf Levin
  • Judit Nagy-György


In the file caching problem, the input is a sequence of requests for files out of a slow memory. A file has two attributes, a positive retrieval cost and an integer size. An algorithm is required to maintain a cache of size k such that the total size of files stored in the cache never exceeds k. Given a request for a file that is not present in the cache at the time of request, the file must be brought from the slow memory into the cache, possibly evicting other files from the cache. This incurs a cost equal to the retrieval cost of the requested file. Well-known special cases include paging (all costs and sizes are equal to 1), the cost model, which is also known as weighted paging, (all sizes are equal to 1), the fault model (all costs are equal to 1), and the bit model (the cost of a file is equal to its size). If bypassing is allowed, a miss for a file still results in an access to this file in the slow memory, but its subsequent insertion into the cache is optional.

We study a new online variant of caching, called caching with rejection. In this variant, each request for a file has a rejection penalty associated with the request. The penalty of a request is given to the algorithm together with the request. When a file that is not present in the cache is requested, the algorithm must either bring the file into the cache, paying the retrieval cost of the file, or reject the file, paying the rejection penalty of the request. The objective function is the sum of total rejection penalty and the total retrieval cost. This problem generalizes both caching and caching with bypassing.

We design deterministic and randomized algorithms for this problem. The competitive ratio of the randomized algorithm is O(logk), and this is optimal up to a constant factor. In the deterministic case, a k-competitive algorithm for caching, and a (k+1)-competitive algorithm for caching with bypassing are known. Moreover, these are the best possible competitive ratios. In contrast, we present a lower bound of 2k+1 on the competitive ratio of any deterministic algorithm for the variant with rejection. The lower bound is valid already for paging. We design a (2k+2)-competitive algorithm for caching with rejection. We also design a different (2k+1)-competitive algorithm, that can be used for paging and for caching in the bit and fault models.


Caching Paging Online algorithms 



The authors would like to thank an anonymous reviewer who suggested the reduction presented in Theorem 6 below, and thus allowed us to simplify the randomized result significantly, making the paper much more elegant.


  1. 1.
    Achlioptas, D., Chrobak, M., Noga, J.: Competitive analysis of randomized paging algorithms. Theor. Comput. Sci. 234(1–2), 203–218 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Adamaszek, A., Czumaj, A., Englert, M., Räcke, H.: An O(logk)-competitive algorithm for generalized caching. In: Proc. of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2012), pp. 1681–1689 (2012) CrossRefGoogle Scholar
  3. 3.
    Albers, S.: New results on web caching with request reordering. Algorithmica 58(2), 461–477 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Albers, S., Arora, S., Khanna, S.: Page replacement for general caching problems. In: Proc. of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA1999), pp. 31–40 (1999) Google Scholar
  5. 5.
    Bansal, N., Buchbinder, N., Naor, J.: Towards the randomized k-server conjecture: a primal-dual approach. In: Proc. of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2010), pp. 40–55 (2010) CrossRefGoogle Scholar
  6. 6.
    Bansal, N., Buchbinder, N., Naor, S.: A primal-dual randomized algorithm for weighted paging. J. ACM 59(4), 19 (2012), 24 pp. MathSciNetGoogle Scholar
  7. 7.
    Bansal, N., Buchbinder, N., Naor, S.: Randomized competitive algorithms for generalized caching. SIAM J. Comput. 41(2), 391–414 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. J. ACM 48(5), 1069–1090 (2001) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., Stougie, L.: Multiprocessor scheduling with rejection. SIAM J. Discrete Math. 13(1), 64–78 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bein, W.W., Larmore, L.L., Noga, J.: Equitable revisited. In: Proc. of the 15th Annual European Symposium on Algorithms (ESA2007), pp. 419–426 (2007) Google Scholar
  11. 11.
    Belady, L.A.: A study of replacement algorithms for a virtual-storage computer. IBM Systems Journal 5(2), 78–101 (1966) CrossRefGoogle Scholar
  12. 12.
    Blum, A., Burch, C., Kalai, A.: Finely-competitive paging. In: Proc. of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS1999), pp. 450–458 (1999) Google Scholar
  13. 13.
    Brehob, M., Enbody, R.J., Torng, E., Wagner, S.: On-line restricted caching. J. Sched. 6(2), 149–166 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Brodal, G.S., Moruz, G., Negoescu, A.: OnlineMin: a fast strongly competitive randomized paging algorithm. In: Proc. of the 9th International Workshop on Approximation and Online Algorithms (WAOA2011), pp. 164–175 (2011) Google Scholar
  15. 15.
    Cao, P., Irani, S.: Cost-aware www proxy caching algorithms. In: Proc. of the USENIX Symposium on Internet Technologies and Systems, pp. 193–206 (1997) Google Scholar
  16. 16.
    Chrobak, M., Karloff, H.J., Payne, T.H., Vishwanathan, S.: New results on server problems. SIAM J. Discrete Math. 4(2), 172–181 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Chrobak, M., Noga, J.: Competitive algorithms for relaxed list update and multilevel caching. J. Algorithms 34(2), 282–308 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Chrobak, M., Woeginger, G.J., Makino, K., Xu, H.: Caching is hard—even in the fault model. Algorithmica 63(4), 781–794 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Cohen, E., Kaplan, H.: Caching documents with variable sizes and fetching costs: an LP-based approach. Algorithmica 32(3), 459–466 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Dósa, G., He, Y.: Bin packing problems with rejection penalties and their dual problems. Inf. Comput. 204(5), 795–815 (2006) CrossRefzbMATHGoogle Scholar
  21. 21.
    Epstein, L.: Bin packing with rejection revisited. Algorithmica 56(4), 505–528 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Epstein, L., Levin, A., Woeginger, G.J.: Graph coloring with rejection. J. Comput. Syst. Sci. 77(2), 439–447 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Epstein, L., Noga, J., Woeginger, G.J.: On-line scheduling of unit time jobs with rejection: minimizing the total completion time. Oper. Res. Lett. 30(6), 415–420 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Fiat, A., Karp, R.M., Luby, M., McGeoch, L.A., Sleator, D.D., Young, N.: Competitive paging algorithms. J. Algorithms 12(4), 685–699 (1991) CrossRefzbMATHGoogle Scholar
  25. 25.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Irani, S.: Page replacement with multi-size pages and applications to web caching. Algorithmica 33(3), 384–409 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Karlin, A., Manasse, M., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3(1–4), 79–119 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Karp, R.M.: On-line algorithms versus off-line algorithms: how much is it worth to know the future? In: van Leeuwen, J. (ed.) IFIP Congress (1). IFIP Transactions, vol. A-12, pp. 416–429. North-Holland, Amsterdam (1992) Google Scholar
  29. 29.
    Koufogiannakis, C., Young, N.E.: Greedy Δ-approximation algorithm for covering with arbitrary constraints and submodular cost. Algorithmica 66(1), 113–152 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Manasse, M.S., McGeoch, L.A., Sleator, D.D.: Competitive algorithms for server problems. J. Algorithms 11(2), 208–230 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    McGeoch, L.A., Sleator, D.D.: A strongly competitive randomized paging algorithm. Algorithmica 6(6), 816–825 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Mendel, M., Seiden, S.S.: Online companion caching. Theor. Comput. Sci. 324(2–3), 183–200 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Nagy-György, J., Imreh, C.: Online scheduling with machine cost and rejection. Discrete Appl. Math. 155(18), 2546–2554 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Seiden, S.S.: Preemptive multiprocessor scheduling with rejection. Theor. Comput. Sci. 262(1), 437–458 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Sleator, D.D., Tarjan, R.E.: Amoritzed efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985) CrossRefMathSciNetGoogle Scholar
  36. 36.
    Young, N.E.: The k-server dual and loose competitiveness for paging. Algorithmica 11(6), 525–541 (1994) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Young, N.E.: On-line file caching. Algorithmica 33(3), 371–383 (2002) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Leah Epstein
    • 1
    Email author
  • Csanád Imreh
    • 2
  • Asaf Levin
    • 3
  • Judit Nagy-György
    • 4
  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Department of InformaticsUniversity of SzegedSzegedHungary
  3. 3.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael
  4. 4.Department of MathematicsUniversity of SzegedSzegedHungary

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