Advertisement

Journal of Scheduling

, Volume 20, Issue 5, pp 423–442 | Cite as

Reordering buffer management with advice

  • Anna Adamaszek
  • Marc P. Renault
  • Adi Rosén
  • Rob van Stee
Article

Abstract

In the reordering buffer management problem, a sequence of colored items arrives at a service station to be processed. Each color change between two consecutively processed items generates some cost. A reordering buffer of capacity k items can be used to preprocess the input sequence in order to decrease the number of color changes. The goal is to find a scheduling strategy that, using the reordering buffer, minimizes the number of color changes in the given sequence of items. We consider the problem in the setting of online computation with advice. In this model, the color of an item becomes known only at the time when the item enters the reordering buffer. Additionally, together with each item entering the buffer, we get a fixed number of advice bits, which can be seen as information about the future or as information about an optimal solution (or an approximation thereof) for the whole input sequence. We show that for any \(\varepsilon > 0\) there is a \((1+\varepsilon )\)-competitive algorithm for the problem which uses only a constant (depending on \(\varepsilon \)) number of advice bits per input item. This also immediately implies a \((1+\varepsilon )\)-approximation algorithm which has \(2^{O(n\log 1/\varepsilon )}\) running time (this should be compared to the trivial optimal algorithm which has a running time of \(k^{O(n)}\)). We complement the above result by presenting a lower bound of \(\varOmega (\log k)\) bits of advice per request for any 1-competitive algorithm.

Keywords

Reordering buffer management Online algorithms Online algorithms with advice Competitive analysis 

Notes

Acknowledgments

We would like to thank the reviewers for their thorough reading of the paper and their helpful comments, which helped us to improve the presentation of the paper. The first author is supported by the DFF-MOBILEX mobility grant from the Danish Council for Independent Research. The second and third authors were partially supported by ANR Project NeTOC.

References

  1. Adamaszek, A., Czumaj, A., Englert, M., & Räcke, H. (2011). Almost tight bounds for reordering buffer management. In L. Fortnow & S. P. Vadhan (Eds.), STOC (pp. 607–616). San Jose: ACM Press.Google Scholar
  2. Albers, S., & Hellwig, M. (2014). Online makespan minimization with parallel schedules. In: Ravi, R., Gørtz, I. L. (eds.) Algorithm theory—SWAT 2014—14th Scandinavian symposium and workshops, Copenhagen, July 2–4. Proceedings, Lecture notes in computer science (Vol. 8503, pp. 13–25). Springer. doi: 10.1007/978-3-319-08404-6_2.
  3. Asahiro, Y., Kawahara, K., & Miyano, E. (2012). Np-hardness of the sorting buffer problem on the uniform metric. Discrete Applied Mathematics, 160(10–11), 1453–1464.CrossRefGoogle Scholar
  4. Avigdor-Elgrabli, N., & Rabani, Y. (2010). An improved competitive algorithm for reordering buffer management. In M. Charikar (Ed.), SODA (pp. 13–21). Philadelphia: SIAM.Google Scholar
  5. Avigdor-Elgrabli, N., & Rabani, Y. (2013). A constant factor approximation algorithm for reordering buffer management. In S. Khanna (Ed.), SODA (pp. 973–984). Philadelphia: SIAM.Google Scholar
  6. Avigdor-Elgrabli, N., & Rabani, Y. (2013). An optimal randomized online algorithm for reordering buffer management. In 54th Annual IEEE Symposium on Foundations of Computer Science, (pp.1–10). Berkeley, CA. doi: 10.1109/FOCS.2013.9
  7. Blandford, D. K., & Blelloch, G. E. (2002). Index compression through document reordering. In: DCC (pp. 342–351). IEEE Computer Society.Google Scholar
  8. Böckenhauer, H., Hromkovic, J., Komm, D., Krug, S., Smula, J., & Sprock, A. (2014). The string guessing problem as a method to prove lower bounds on the advice complexity. Theoretical Computer Science, 554, 95–108. doi: 10.1016/j.tcs.2014.06.006.CrossRefGoogle Scholar
  9. Böckenhauer, H. J., Komm, D., Královic, R., & Královic, R. (2011). On the advice complexity of the k-server problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP (1), Lecture notes in computer science (Vol. 6755, pp 207–218). Springer. Also as technical report at ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/7xx/703.pdf.Google Scholar
  10. Böckenhauer, H. J., Komm, D., Královic, R., Královic, R., & Mömke, T. (2009). On the advice complexity of online problems. In: Dong, Y., Du, D.Z., Ibarra, O.H. (eds.) ISAAC, Lecture notes in computer science (Vol. 5878, pp. 331–340). Berlin: Springer.Google Scholar
  11. Chan, H. L., Megow, N., Sitters, R., & van Stee, R. (2012). A note on sorting buffers offline. Theoretical Computer Science, 423, 11–18.CrossRefGoogle Scholar
  12. Dobrev, S., Královič, R., & Pardubská, D. (2008). How much information about the future is needed? In: SOFSEM’08: Proceedings of the 34th conference on current trends in theory and practice of computer science (pp. 247–258). Berlin, Heidelberg: Springer.Google Scholar
  13. Dohrau, J. (2015). Online makespan scheduling with sublinear advice. In: Italiano, G. F., Margaria-Steffen, T., Pokorný, J., Quisquater, J., Wattenhofer, R. (eds.) SOFSEM 2015: Theory and practice of computer science—41st international conference on current trends in theory and practice of computer science, Pec pod Sněžkou, Czech Republic, January 24–29, 2015. Proceedings, Lecture Notes in Computer Science (Vol. 8939, pp. 177–188). Springer. doi: 10.1007/978-3-662-46078-8_15.
  14. Dorrigiv, R., He, M., & Zeh, N. (2012). On the advice complexity of buffer management. In: Chao, K. M., Sheng Hsu, T., Lee, D. T. (eds.) ISAAC, Lecture notes in computer science (Vol. 7676, pp. 136–145). New York: Springer.Google Scholar
  15. Emek, Y., Fraigniaud, P., Korman, A., & Rosén, A. (2011). Online computation with advice. Theoretical Computer Science, 412(24), 2642–2656.CrossRefGoogle Scholar
  16. Englert, M., Räcke, H., & Westermann, M. (2010). Reordering buffers for general metric spaces. Theory of Computing, 6(1), 27–46.CrossRefGoogle Scholar
  17. Englert, M., & Westermann, M. (2005). Reordering buffer management for non-uniform cost models. In: Caires, L., Italiano, G. F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP, Lecture notes in computer science (Vol. 3580, pp 627–638). Berlin: Springer.Google Scholar
  18. Gutenschwager, K., Spiekermann, S., & Voß, S. (2004). A sequential ordering problem in automotive paint shops. International Journal of Production Research, 42(9), 1865–1878. doi: 10.1080/00207540310001646821.CrossRefGoogle Scholar
  19. Hromkovic, J., Královic, R., & Královic, R. (2010). Information complexity of online problems. In: Hlinený, P., Kucera, A. (eds.) MFCS, Lecture notes in computer science (Vol. 6281, pp 24–36). Berlin: Springer.Google Scholar
  20. Komm, D., & Královic, R. (2011). Advice complexity and barely random algorithms. RAIRO-Theoretical Informatics and Applications, 45(2), 249–267.CrossRefGoogle Scholar
  21. Krokowski, J., Räcke, H., Sohler, C., & Westermann, M. (2004). Reducing state changes with a pipeline buffer. In: Girod, B., Magnor, M. A., Seidel, H. P. (eds.) VMV (p. 217). Aka GmbH.Google Scholar
  22. Lewis, H. R., & Papadimitriou, C. H. (1997). Elements of the theory of computation (2nd ed.). Upper Saddle River, NJ: Prentice Hall PTR.Google Scholar
  23. Räcke, H., Sohler, C., & Westermann, M. (2002). Online scheduling for sorting buffers. In: Möhring, R. H., Raman, R. (eds.) ESA, Lecture Notes in Computer Science (Vol. 2461, pp. 820–832). Berlin: Springer.Google Scholar
  24. Renault, M. P., & Rosén, A. (2012). On online algorithms with advice for the k-server problem. Theory of Computing Systems, 56, 1–19. doi: 10.1007/s00224-012-9434-z.Google Scholar
  25. Renault, M. P., Rosén, A., & van Stee, R. (2015). Online algorithms with advice for bin packing and scheduling problems. Theoretical Computer Science, 600, 155–170. doi: 10.1016/j.tcs.2015.07.050.CrossRefGoogle Scholar
  26. Standard for information technology portable operating system interface (posix(r)) base specifications, issue 7. IEEE Std 1003.1, 2013 Edition (incorporates IEEE Std 1003.1-2008, and IEEE Std 1003.1-2008/Cor 1-2013) pp. 1–3906 (2013). doi: 10.1109/IEEESTD.2013.6506091.

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Anna Adamaszek
    • 1
  • Marc P. Renault
    • 2
  • Adi Rosén
    • 2
  • Rob van Stee
    • 3
  1. 1.University of CopenhagenCopenhagenDenmark
  2. 2.CNRS and Université Paris DiderotParisFrance
  3. 3.University of LeicesterLeicesterUnited Kingdom

Personalised recommendations