Randomized Competitive Algorithms for Generalized Caching

We consider online algorithms for the generalized caching problem. Here we are given a cache of size $k$ and pages with arbitrary sizes and fetching costs. Given a request sequence of pages, the goal is to minimize the total cost of fetching the pages into the cache. Our main result is an online algorithm with competitive ratio $O(\log^2k)$, which gives the first $o(k)$ competitive algorithm for the problem. We also give improved $O(\log k)$-competitive algorithms for the special cases of the bit model and fault model, improving upon the previous $O(\log^2k)$ guarantees due to Irani [Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 701-710]. Our algorithms are based on an extension of the online primal-dual framework introduced by Buchbinder and Naor [Math. Oper. Res., 34 (2009), pp. 270-286] and involve two steps. First, we obtain an $O(\log k)$-competitive fractional algorithm based on solving online an LP formulation strengthened with exponentially many knapsack cover constraints. Second, we design a suitable online rounding procedure to convert this online fractional algorithm into a randomized algorithm. Our techniques provide a unified framework for caching algorithms and are substantially simpler than those previously used.

[1]  Sandy Irani,et al.  Cost-Aware WWW Proxy Caching Algorithms , 1997, USENIX Symposium on Internet Technologies and Systems.

[2]  Joseph Naor,et al.  A Primal-Dual Randomized Algorithm for Weighted Paging , 2007, FOCS.

[3]  Marek Chrobak,et al.  New results on server problems , 1991, SODA '90.

[4]  Amos Fiat,et al.  Competitive Paging Algorithms , 1991, J. Algorithms.

[5]  Robert D. Carr,et al.  Strengthening integrality gaps for capacitated network design and covering problems , 2000, SODA '00.

[6]  Neal E. Young,et al.  On-line caching as cache size varies , 1991, SODA '91.

[7]  Haim Kaplan.  LP-based analysis of greedy-dual-size , 1999, SODA '99.

[8]  Sandy Irani,et al.  Page replacement with multi-size pages and applications to Web caching , 1997, STOC '97.

[9]  Sandy Irani,et al.  Randomized Weighted Caching with Two Page Weights , 2002, Algorithmica.

[10]  Neal E. Young,et al.  Thek-server dual and loose competitiveness for paging , 1994, Algorithmica.

[11]  Neal E. Young,et al.  On-Line File Caching , 2002, SODA '98.

[12]  Robert E. Tarjan,et al.  Amortized efficiency of list update and paging rules , 1985, CACM.

[13]  Dror Rawitz,et al.  On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique , 2005, SIAM J. Discret. Math..

[14]  Joseph Naor,et al.  Online Primal-Dual Algorithms for Covering and Packing Problems , 2005, ESA.

[15]  Marek Chrobak,et al.  Competitive analysis of randomized paging algorithms , 2000, Theor. Comput. Sci..

[16]  Reuven Bar-Yehuda,et al.  A unified approach to approximating resource allocation and scheduling , 2001, JACM.

[17]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[18]  Sanjeev Khanna,et al.  Page replacement for general caching problems , 1999, SODA '99.