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Theory of Computing Systems

, Volume 62, Issue 8, pp 2006–2034 | Cite as

Online Bin Packing with Advice of Small Size

  • Spyros Angelopoulos
  • Christoph Dürr
  • Shahin Kamali
  • Marc P. Renault
  • Adi Rosén
Article
  • 41 Downloads

Abstract

In this paper, we study the advice complexity of the online bin packing problem. In this well-studied setting, the online algorithm is supplemented with some additional information concerning the input. We improve upon both known upper and lower bounds of online algorithms for this problem. On the positive side, we first provide a relatively simple algorithm that achieves a competitive ratio arbitrarily close to 1.5, using constant-size advice. Our result implies that 16 bits of advice suffice to obtain a competitive ratio better than any online algorithm without advice, thus improving the previously known bound of O(log(n)) bits required to attain this performance. In addition, we introduce a more complex algorithm that still requires only constant-size advice, and has a competitive ratio arbitrarily close to 1.47012. This is the currently best performance of any online bin packing algorithm with sublinear advice. On the negative side, we extend a construction due to Boyar et al. (Algorithmica 74(1), 507–527 2016) so as to show that no online algorithm with sub-linear advice can be 7/6-competitive, improving on the lower bound of 9/8 from Boyar et al.

Keywords

Online bin packing Competitive analysis Advice complexity 

Notes

Acknowledgments

Research supported in part by project ANR-11-BS02-0015 “New Techniques in Online Computation–NeTOC”. A preliminary version of this paper appeared in the Proceedings of the 14th International Symposium on Algorithms and Data Structures (WADS), 2015 [2].

References

  1. 1.
    Adamaszek, A., Renault, M.P., Rosėn, A., van Stee, R.: Reordering buffer management with advice. J. Schedul. (2016)Google Scholar
  2. 2.
    Angelopoulos, S., Dürr, C., Kamali, S., Renault, M.P., Rosén, A.: Online bin packing with advice of small size. In: Dehne, F., Sack, J., Stege, U. (eds.) Algorithms and Data Structures - 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings, Lecture Notes in Computer Science, vol. 9214, pp 40–53. Springer (2015)Google Scholar
  3. 3.
    Asgeirsson, E., Ayesta, U., Coffman, E., Etra, J., Momcilovic, P., Phillips, D., Vokhshoori, V., Wang, Z., Wolfe, J.: Closed on-line bin packing. Acta Cybernet. 15(3), 361–367 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Balogh, J., Békési, J., Galambos, G.: New lower bounds for certain classes of bin packing algorithms. Theoret. Comput. Sci. 440–441, 1–13 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bianchi, M.P., Böckenhauer, H., Brülisauer, T., Komm, D., Palano, B.: Online minimum spanning tree with advice - (extended abstract). In: Freivalds, R.M., Engels, G., Catania, B. (eds.) SOFSEM 2016: Theory and Practice of Computer Science - 42nd International Conference on Current Trends in Theory and Practice of Computer Science, Harrachov, Czech Republic, January 23-28, 2016, Proceedings, Lecture Notes in Computer Science, vol. 9587, pp 195–207. Springer (2016)Google Scholar
  6. 6.
    Bȯckenhauer, H., Komm, D., Krȧlovic, R., Rossmanith, P.: The online knapsack problem: Advice and randomization. Theor. Comput. Sci. 527, 61–72 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Böckenhauer, H.J., Hromkovic, J., Komm, D., Krug, S., Smula, J., Sprock, A.: The string guessing problem as a method to prove lower bounds on the advice complexity. Theoret. Comput. Sci. 554, 95–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Böckenhauer, H.J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the k-server problem. In: Proc. 38th International Colloquium on Automata, Languages, and Programming (ICALP), Lecture Notes in Comput. Sci., vol. 6755, pp. 207–218. Springer (2011)Google Scholar
  9. 9.
    Böckenhauer, H.J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the advice complexity of online problems. In: Proc. 20th International Symp. on Algorithms and Computation (ISAAC), Lecture Notes in Comput. Sci., vol. 5878, pp. 331–340. Springer (2009)Google Scholar
  10. 10.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press (1998)Google Scholar
  11. 11.
    Boyar, J., Favrholdt, L., Kudahl, C., Larsen, K. S., Mikkelsen, J.W.: Online algorithms with advice: A survey. SIGACT News 47(3), 93–129 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boyar, J., Kamali, S., Larsen, K.S., López-Ortiz, A.: On the list update problem with advice. Inf. Comput. (2016)Google Scholar
  13. 13.
    Boyar, J., Kamali, S., Larsen, K.S., Lȯpez-Ortiz, A.: Online bin packing with advice. Algorithmica 74(1), 507–527 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dobrev, S., Kralovic, R., Pardubská, D.: How much information about the future is needed? In: Proc. 34th International Conf. on Current Trends in Theory and Practice of Computer Science (SOFSEM), Lecture Notes in Comput. Sci., vol. 4910, pp. 247–258. Springer (2008)Google Scholar
  15. 15.
    Dobrev, S., Královič, R., Pardubská, D.: Measuring the problem-relevant information in input. RAIRO Inform Theor. Appl. 43(3), 585–613 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theoret. Comput. Sci. 412(24), 2642–2656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Epstein, L., Levin, A.: On bin packing with conflicts. SIAM J. Optim. 19 (3), 1270–1298 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Galambos, G., Woeginger, G.J.: Repacking helps in bounded space online bin packing. Computing 49, 329–338 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gambosi, G., Postiglione, A., Talamo, M.: Algorithms for the relaxed online bin-packing model. SIAM J. Comput. 30(5), 1532–1551 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garey, M.R., Graham, R.L., Ullman, J.D.: Worst-case analysis of memory allocation algorithms. In: Fischer, P.C., Zeiger, H.P., Ullman, J.D., Rosenberg, A.L. (eds.) STOC, pp 143–150. ACM (1972)Google Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing problems - a survey. In: Ausiello, G., Lucertini, M. (eds.) Analysis and Design of Algorithms in Combinatorial Optimization, pp 147–172. Springer, New York (1981)Google Scholar
  22. 22.
    Grove, E.F.: Online binpacking with lookahead. In: Proc. 6th Symp. on Discrete Algorithms (SODA), pp. 430–436 (1995)Google Scholar
  23. 23.
    Gupta, S., Kamali, S., López-Ortiz, A.: On advice complexity of the k-server problem under sparse metrics. Theory Comput. Syst. 59(3), 476–499 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Heydrich, S., van Stee, R.: Beating the harmonic lower bound for online bin packing. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pp. 41:1–41:14 (2016)Google Scholar
  25. 25.
    Johnson, D.S.: Near-Optimal Bin Packing Algorithms. Ph.D. thesis, MIT, Cambridge MA (1973)Google Scholar
  26. 26.
    Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J. Comput. 3, 256–278 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M., J.W.: Randomization can be as helpful as a glimpse of the future in online computation. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pp. 39:1–39:14 (2016)Google Scholar
  28. 28.
    Komm, D., Královič, R.: Advice complexity and barely random algorithms. RAIRO Inform Theor. Appl. 45(2), 249–267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Renault, M.P., Rosėn, A.: On online algorithms with advice for the k-server problem. Theory Comput. Syst. 56(1), 3–21 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Renault, M.P., Rosėn, A., van Stee, R.: Online algorithms with advice for bin packing and scheduling problems. Theor. Comput. Sci. 600, 155–170 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Sequence A000041Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Spyros Angelopoulos
    • 1
  • Christoph Dürr
    • 1
  • Shahin Kamali
    • 2
  • Marc P. Renault
    • 3
  • Adi Rosén
    • 4
    • 5
  1. 1.Laboratoire d’informatique de Paris-6, LIP6Sorbonne UniversitéParisFrance
  2. 2.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  3. 3.Department of Computer ScienceUniversity of WisconsinMadisonUSA
  4. 4.CNRSParisFrance
  5. 5.Université Paris DiderotParisFrance

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