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Average Case Analysis of Bounded Space Bin Packing Algorithms

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Abstract

We consider the one-dimensional bin packing problem and analyze the average case performance of bounded space algorithms. The analysis covers a wide variety of bin packing algorithms including Next-K Fit, K-Bounded Best Fit and Harmonic algorithms, as well as of others. We assume discrete item sizes, an assumption which is true in most real-world applications of bin packing. We present a Markov chains method which is general enough to calculate results for any i.i.d. discrete item size distribution. The paper presents many results heretofore unknown or conjectured from simulation. Some of the results are surprising as they differ considerably from results for continuous distributions.

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References

  1. Applegate, D.L., Buriol, L.S., Dillard, B.L., Johnson, D.S., Shor, P.W.: The cutting-stock approach to bin packing: theory and experiments. In: Proceedings of the Fifth Workshop on Algorithm Engineering and Experimentation, pp. 1–15 (2003)

  2. Coffman, E.G. Jr., Courcoubetis, C.A., Garey, M.R., Johnson, D.S., McGeogh, L.A., Shor, P.W., Weber, R.R., Yannakakis, M.: Fundamental discrepancies between average-case analyses under discrete and continuous distributions: A bin packing case study. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 230–240. ACM, New York (1991)

    Google Scholar 

  3. Coffman, E.G. Jr., Courcoubetis, C.A., Garey, M.R., Johnson, D.S., Shor, P.W., Weber, R.R., Yannakakis, M.: Bin packing with discrete item sizes, part I: Perfect packing theorems and the average case behavior of optimal packings. SIAM J. Discrete Math. 13, 384–402 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Coffman, E.G. Jr., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: A survey. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-Hard Problems, pp. 46–93. PWS, Boston (1996)

    Google Scholar 

  5. Coffman, E.G. Jr., Halfin, S., Jean-Marie, A., Robert, P.: Stochastic analysis of a slotted FIFO communication channel. IEEE Trans. Inf. Theory 39(5), 1555–1566 (1993)

    Article  MATH  Google Scholar 

  6. Coffman, E.G. Jr., Johnson, D.S., Shor, P.W., Weber, R.R.: Markov chains, computer proofs and average case analysis of best fit bin packing. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp. 412–421. ACM, New York (1993)

    Google Scholar 

  7. Coffman, E.G. Jr., Johnson, D.S., Shor, P.W., Weber, R.R.: Bin packing with discrete item sizes, part II: Tight bounds on first fit. Random Struct. Algorithms 10, 69–101 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Coffman, E.G. Jr., Lueker, G.S.: Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, New York (1991)

    Google Scholar 

  9. Coffman, E.G. Jr., So, K., Hofri, M.: A stochastic model of bin packing. Inf. Control 44, 105–115 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Courcoubetis, C.A., Weber, R.R.: Necessary and sufficient conditions for stability of a bin packing system. J. Appl. Probab. 23, 989–999 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Csirik, J., Frenk, J.B.G., Frieze, A.M., Galambos, G., Rinnoy Kan, A.H.G.: A probabilistic analysis of the next-fit decreasing bin packing heuristic. Oper. Res. Lett. 5(5), 233–236 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  12. Csirik, J., Johnson, D.S.: Bounded space on line bin packing: Best is better than first. Algorithmica 31(2), 115–138 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Csirik, J., Johnson, D.S., Kenyon, C., Orlin, J.B., Shor, P.W., Weber, R.R.: On the sum of-squares algorithm for bin packing. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 208–217 (2000)

  14. Csirik, J., Johnson, D.S., Kenyon, C., Shor, P.W., Weber, R.R.: A self-organizing bin packing heuristic. In: Proceedings of Workshop on Algorithm Engineering and Experimentation. Lecture Notes in Computer Science pp. 246–265. Springer, New York (1999)

    Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  16. Hofri, M.: A probabilistic analysis of the Next-Fit bin packing algorithm. J. Algorithms 5, 547–556 (1984)

    Article  MathSciNet  Google Scholar 

  17. Hofri, M., Kamhi, S.: A stochastic analysis of the NFD bin packing algorithm. J. Algorithms 7, 489–509 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. Johnson, D.S.: Near-optimal Bin Packing Algorithms. Doctoral dissertation, Mathematics, Massachusetts Institute of Technology, Cambridge, MA (1973)

  19. Johnson, D.S.: Fast algorithms for bin packing. J. Comput. System Sci. 8, 272–314 (1974)

    MATH  MathSciNet  Google Scholar 

  20. Karmarkar, N.: Probabilistic analysis of some bin packing algorithms. In: Proceedings 23rd Annual Symposium on Foundations of Computer Science, pp. 107–111 (1982)

  21. Lee, C.C., Lee, D.T.: A simple on-line packing algorithm. J. ACM 32, 562–572 (1985)

    Article  MATH  Google Scholar 

  22. Lee, C.C., Lee, D.T.: Robust on-line bin-packing algorithms. Technical report, Department of Electrical Engineering and CS. Northwestern University, Evanston, IL (1987)

  23. Mao, W.: Best-k-Fit bin packing. Computing 50, 265–270 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Naaman (Menakerman), N., Rom, R.: Analysis of bounded space bin packing algorithms. Technical report 340. Technion EE publication CCIT. ftp://ftp.technion.ac.il/pub/supported/ee/Network/NaRo-bounded05.pdf (March 2001)

  25. Ramanan, P.: Average case analysis of the smart next fit algorithm. Inf. Process. Lett. 31, 221–225 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rhee, T.: Probabilistic analysis of the next-fit decreasing algorithm for bin packing. Oper. Res. Lett. 6(4), 189–191 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rhee, T., Talagrand, M.: Martingale inequalities and NP-complete problems. Math. Oper. Res. 12, 177–181 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rhee, T., Talagrand, M.: Optimal bin packing with items of random sizes. SIAM J. Comput. 18, 139–151 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zhang, G.: Tight worst-case performance bound for AFB K . Technical report #015, Institute of Applied Mathematics, Beijing, China (1994)

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Authors and Affiliations

  1. Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Nir Naaman & Raphael Rom

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  1. Nir Naaman
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  2. Raphael Rom
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Correspondence to Nir Naaman.

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Naaman, N., Rom, R. Average Case Analysis of Bounded Space Bin Packing Algorithms. Algorithmica 50, 72–97 (2008). https://doi.org/10.1007/s00453-007-9073-y

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