Document Open Access Logo

Online Paging with Heterogeneous Cache Slots

Authors Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, Neal E. Young

PDF
Thumbnail PDF

File

LIPIcs.STACS.2023.23.pdf
  • Filesize: 1.58 MB
  • 24 pages

Document Identifiers

Author Details

Marek Chrobak
  • University of California at Riverside, CA, USA
Samuel Haney
  • Tumult Labs, Durham, NC, USA
Mehraneh Liaee
  • Northeastern University, Boston, MA, USA
Debmalya Panigrahi
  • Duke University, Durham, NC, USA
Rajmohan Rajaraman
  • Northeastern University, Boston, MA, USA
Ravi Sundaram
  • Northeastern University, Boston, MA, USA
Neal E. Young
  • University of California at Riverside, CA, USA

Funding

  • Chrobak, Marek: Supported by NSF grants CCF-1536026 and CCF-2153723.
  • Haney, Samuel: Supported by NSF grants CCF-1527084 and CCF-1535972.
  • Liaee, Mehraneh: Supported by NSF grants CCF-1535929 and CCF-1909363.
  • Panigrahi, Debmalya: Supported by NSF grants CCF-1527084, CCF-1535972, CCF-1750140, CCF-1955703, ARO grant W911NF2110230, and Indo-US Joint Center for Algorithms under Uncertainty.
  • Rajaraman, Rajmohan: Supported by NSF grants CCF-1535929 and CCF-1909363.
  • Sundaram, Ravi: Supported by NSF grants CCF-1535929 and IIS-2039945.
  • Young, Neal E.: Supported by NSF grant CCF-1619463.

Cite AsGet BibTex

Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, and Neal E. Young. Online Paging with Heterogeneous Cache Slots. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.23

Abstract

It is natural to generalize the online k-Server problem by allowing each request to specify not only a point p, but also a subset S of servers that may serve it. To initiate a systematic study of this generalization, we focus on uniform and star metrics. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page p, but also a subset S of cache slots, and is satisfied by having a copy of p in some slot in S. We call this problem Slot-Heterogenous Paging. In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore we parameterize the problem by specifying a family 𝒮 ⊆ 2^[k] of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size k and family 𝒮. If all request sets are allowed (𝒮 = 2^[k]), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (𝒮 = {[k]}). As a function of |𝒮| and k, the optimal deterministic ratio is polynomial: at most O(k²|𝒮|) and at least Ω(√{|𝒮|}). For any laminar family {𝒮} of height h, the optimal ratios are O(hk) (deterministic) and O(h²log k) (randomized). The special case that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. For All-or-One Paging the optimal competitive ratios are Θ(k) (deterministic) and Θ(log k) (randomized), while the offline problem is NP-hard. We extend the deterministic upper bound to the weighted variant of All-or-One Paging (a generalization of standard Weighted Paging), showing that it is also Θ(k). Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set P of pages, and is satisfied by fetching any page from P into the cache. The optimal ratios for the latter problem (with laminar family of height h) are at most hk (deterministic) and hH_k (randomized).

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Caching and paging algorithms
  • k-server
  • weighted paging
  • laminar family

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Dimitris Achlioptas, Marek Chrobak, and John Noga. Competitive analysis of randomized paging algorithms. Theor. Comput. Sci., 234(1-2):203-218, 2000. URL: https://doi.org/10.1016/S0304-3975(98)00116-9.
  2. C. J. Argue, Anupam Gupta, Ziye Tang, and Guru Guruganesh. Chasing convex bodies with linear competitive ratio. Journal of the ACM, 68(5):32:1-32:10, August 2021. URL: https://doi.org/10.1145/3450349.
  3. Nikhil Ayyadevara and Ashish Chiplunkar. The randomized competitive ratio of weighted k-server is at least exponential. CoRR, abs/2102.11119, 2021. URL: http://arxiv.org/abs/2102.11119.
  4. Nikhil Bansal, Niv Buchbinder, Aleksander Mądry, and Joseph Naor. A polylogarithmic-competitive algorithm for the k-server problem. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 267-276. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.63.
  5. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. A primal-dual randomized algorithm for weighted paging. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 507-517. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/FOCS.2007.7.
  6. Nikhil Bansal, Marek Eliás, and Grigorios Koumoutsos. Weighted k-server bounds via combinatorial dichotomies. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 493-504. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.52.
  7. Nikhil Bansal, Marek Eliáš, Grigorios Koumoutsos, and Jesper Nederlof. Competitive algorithms for generalized k-server in uniform metrics. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 992-1001, 2018. URL: https://doi.org/10.1137/1.9781611975031.64.
  8. Nikhil Bansal, Joseph (Seffi) Naor, and Ohad Talmon. Efficient online weighted multi-level paging. In Kunal Agrawal and Yossi Azar, editors, SPAA '21: 33rd ACM Symposium on Parallelism in Algorithms and Architectures, Virtual Event, USA, 6-8 July, 2021, pages 94-104. ACM, 2021. URL: https://doi.org/10.1145/3409964.3461801.
  9. Nathan Beckmann, Phillip B. Gibbons, Bernhard Haeupler, and Charles McGuffey. Writeback-aware caching. In Bruce M. Maggs, editor, 1st Symposium on Algorithmic Principles of Computer Systems, APOCS 2020, Salt Lake City, UT, USA, January 8, 2020, pages 1-15. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611976021.1.
  10. L. A. Belady. A study of replacement algorithms for a virtual-storage computer. IBM Systems Journal, 5(2):78-101, 1966. URL: https://doi.org/10.1147/sj.52.0078.
  11. Marcin Bienkowski, Łukasz Jeż, and Pawel Schmidt. Slaying Hydrae: Improved bounds for generalized k-server in uniform metrics. In Pinyan Lu and Guochuan Zhang, editors, 30th International Symposium on Algorithms and Computation, ISAAC 2019, volume 149 of LIPIcs, pages 14:1-14:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2019.14.
  12. Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998. Google Scholar
  13. Allan Borodin, Nathan Linial, and Michael E. Saks. An optimal on-line algorithm for metrical task system. J. ACM, 39(4):745-763, 1992. URL: https://doi.org/10.1145/146585.146588.
  14. Mark Brehob, Richard J. Enbody, Eric Torng, and Stephen Wagner. On-line restricted caching. J. Sched., 6(2):149-166, 2003. URL: https://doi.org/10.1023/A:1022989909868.
  15. Mark Brehob, Stephen Wagner, Eric Torng, and Richard J. Enbody. Optimal replacement is NP-hard for nonstandard caches. IEEE Trans. Computers, 53(1):73-76, 2004. URL: https://doi.org/10.1109/TC.2004.1255792.
  16. Sébastien Bubeck, Michael B. Cohen, Yin Tat Lee, James R. Lee, and Aleksander Mądry. K-server via multiscale entropic regularization. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 3-16. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188798.
  17. Sébastien Bubeck, Yuval Rabani, and Mark Sellke. Online multiserver convex chasing and optimization. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2093-2104. SIAM, 2021. Google Scholar
  18. Niv Buchbinder, Shahar Chen, and Joseph Naor. Competitive algorithms for restricted caching and matroid caching. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 209-221. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_18.
  19. Niv Buchbinder, Christian Coester, and Joseph (Seffi) Naor. Online k-taxi via double coverage and time-reverse primal-dual. In Mohit Singh and David P. Williamson, editors, Integer Programming and Combinatorial Optimization - 22nd International Conference, IPCO 2021, Atlanta, GA, USA, May 19-21, 2021, Proceedings, volume 12707 of Lecture Notes in Computer Science, pages 15-29. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-73879-2_2.
  20. Jannik Castenow, Björn Feldkord, Till Knollmann, Manuel Malatyali, and Friedhelm Meyer auf der Heide. The k-server with preferences problem. In SPAA '22: 34rd ACM Symposium on Parallelism in Algorithms and Architectures, 2022. To appear. URL: https://arxiv.org/abs/2205.11102.
  21. Ashish Chiplunkar and Sundar Vishwanathan. Metrical service systems with multiple servers. Algorithmica, 71(1):219-231, 2015. URL: https://doi.org/10.1007/s00453-014-9903-7.
  22. Ashish Chiplunkar and Sundar Vishwanathan. Randomized memoryless algorithms for the weighted and the generalized k-server problems. ACM Trans. Algorithms, 16(1), December 2019. URL: https://doi.org/10.1145/3365002.
  23. Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, and Neal E. Young. Online paging with heterogeneous cache slots, 2022. URL: http://arxiv.org/abs/2206.05579.
  24. Marek Chrobak and John Noga. Competitive algorithms for relaxed list update and multilevel caching. J. Algorithms, 34(2):282-308, 2000. URL: https://doi.org/10.1006/jagm.1999.1061.
  25. Christian Coester and Elias Koutsoupias. The online k-taxi problem. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 1136-1147. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316370.
  26. Christian Coester and Elias Koutsoupias. Towards the k-server conjecture: A unifying potential, pushing the frontier to the circle. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 57:1-57:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.57.
  27. Leonid Domnitser, Aamer Jaleel, Jason Loew, Nael Abu-Ghazaleh, and Dmitry Ponomarev. Non-monopolizable caches: Low-complexity mitigation of cache side channel attacks. ACM Trans. Archit. Code Optim., 8(4), January 2012. Google Scholar
  28. Esteban Feuerstein. Uniform service systems with k servers. In Gerhard Goos, Juris Hartmanis, Jan van Leeuwen, Cláudio L. Lucchesi, and Arnaldo V. Moura, editors, LATIN'98: Theoretical Informatics, volume 1380, pages 23-32, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/BFb0054307.
  29. Amos Fiat, Richard M. Karp, Michael Luby, Lyle A. McGeoch, Daniel Dominic Sleator, and Neal E. Young. Competitive paging algorithms. J. Algorithms, 12(4):685-699, 1991. URL: https://doi.org/10.1016/0196-6774(91)90041-V.
  30. Amos Fiat, Manor Mendel, and Steven S. Seiden. Online companion caching. In Rolf Möhring and Rajeev Raman, editors, Algorithms - ESA 2002, pages 499-511, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. Google Scholar
  31. Amos Fiat and Moty Ricklin. Competitive algorithms for the weighted server problem. Theor. Comput. Sci., 130(1):85-99, 1994. URL: https://doi.org/10.1016/0304-3975(94)90154-6.
  32. Samuel Haney. Algorithms for Networks With Uncertainty. PhD thesis, Duke University, 2019. URL: https://dukespace.lib.duke.edu/dspace/handle/10161/18661.
  33. Shahin Kamali and Helen Xu. Multicore paging algorithms cannot be competitive. In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures, pages 547-549, 2020. Google Scholar
  34. Anna R. Karlin, Mark S. Manasse, Larry Rudolph, and Daniel Dominic Sleator. Competitive snoopy caching. Algorithmica, 3:77-119, 1988. URL: https://doi.org/10.1007/BF01762111.
  35. Elias Koutsoupias and Christos H. Papadimitriou. On the k-server conjecture. J. ACM, 42(5):971-983, 1995. URL: https://doi.org/10.1145/210118.210128.
  36. Elias Koutsoupias and David Scot Taylor. The CNN problem and other k-server variants. Theor. Comput. Sci., 324(2-3):347-359, 2004. URL: https://doi.org/10.1016/j.tcs.2004.06.002.
  37. Fangfei Liu, Qian Ge, Yuval Yarom, Frank Mckeen, Carlos Rozas, Gernot Heiser, and Ruby B. Lee. CATalyst: Defeating last-level cache side channel attacks in cloud computing. In 2016 IEEE International Symposium on High Performance Computer Architecture (HPCA), pages 406-418, 2016. URL: https://doi.org/10.1109/HPCA.2016.7446082.
  38. Mark S. Manasse, Lyle A. McGeoch, and Daniel Dominic Sleator. Competitive algorithms for server problems. J. Algorithms, 11(2):208-230, 1990. URL: https://doi.org/10.1016/0196-6774(90)90003-W.
  39. Lyle A. McGeoch and Daniel Dominic Sleator. A strongly competitive randomized paging algorithm. Algorithmica, 6(6):816-825, 1991. URL: https://doi.org/10.1007/BF01759073.
  40. M. Mendel and Steven S. Seiden. Online companion caching. Theoretical Computer Science, 324(2):183-200, 2004. Online Algorithms: In Memoriam, Steve Seiden. URL: https://doi.org/10.1016/j.tcs.2004.05.015.
  41. Jignesh Patel. Restricted k-server problem. Master’s thesis, Michigan State University, 2004. URL: https://d.lib.msu.edu/etd/32678.
  42. Mark Sellke. Chasing convex bodies optimally. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, pages 1509-1518. Society for Industrial and Applied Mathematics, 2020. URL: https://doi.org/10.1137/1.9781611975994.92.
  43. Daniel Dominic Sleator and Robert Endre Tarjan. Amortized efficiency of list update and paging rules. Commun. ACM, 28(2):202-208, 1985. URL: https://doi.org/10.1145/2786.2793.
  44. Zhenghong Wang and Ruby B. Lee. New cache designs for thwarting software cache-based side channel attacks. In Proceedings of the 34th Annual International Symposium on Computer Architecture, ISCA '07, pages 494-505, New York, NY, USA, 2007. URL: https://doi.org/10.1145/1250662.1250723.
  45. Ying Ye, Richard West, Zhuoqun Cheng, and Ye Li. COLORIS: A dynamic cache partitioning system using page coloring. In 2014 23rd International Conference on Parallel Architecture and Compilation Techniques (PACT), pages 381-392, 2014. URL: https://doi.org/10.1145/2628071.2628104.
  46. Wei Zang and Ann Gordon-Ross. CaPPS: cache partitioning with partial sharing for multi-core embedded systems. Des. Autom. Embed. Syst., 20(1):65-92, 2016. URL: https://doi.org/10.1007/s10617-015-9168-7.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact