Computational Aspects of Ordered Integer Partitions with Bounds

Abstract

This paper is dedicated to the counting problem of writing an integer number z as a sum of an ordered sequence of n integers from n given intervals, i.e., counting the number of configurations \((z_1,\ldots ,z_n)\) with \(z = z_1 + \cdots + z_n\) for \(z_i \in [x_i, y_i]\) with integers \(x_i\) and \(y_i\) and \(1 \le i \le n\). We show an algorithm computing this number in \( \mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n z^{\lg n}\right) \) average time, and a data structure computing this number in \(\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n\right) \) time, independently of z. The data structure is constructed in \( \mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n n^3\right) \) time. Its construction algorithm only depends on the intervals \([x_i,y_i]\) (\(1 \le i \le n\)). This construction algorithm can be parallelized with \(\pi = \mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n^3\right) \) processors, yielding \(\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n \frac{n^3}{\pi }\right) \) construction time with high probability.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. 1.

    In particular, we allow negative integer values.

References

  1. 1.

    Andrews, G.E.: The Theory of Partitions, 1st edn. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  2. 2.

    Baker Jr., H.C., Hewitt, C.: The incremental garbage collection of processes. SIGPLAN 12(8), 55–59 (1977)

    Article  Google Scholar 

  3. 3.

    Beihoffer, D., Hendry, J., Nijenhuis, A., Wagon, S.: Faster algorithms for Frobenius numbers. Electron. J. Combin. 12(27), 1–38 (2005)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Carter, M.: Foundations of Mathematical Economics, 1st edn. MIT Press, Cambridge (2001)

    Google Scholar 

  5. 5.

    Coello, C.A.C., Dhaenens, C., Jourdan, L.: Multi-objective combinatorial optimization: Problematic and context. In: Advances in Multi-Objective Nature Inspired Computing, Studies in Computational Intelligence, vol. 272, pp. 1–21. Springer (2010)

  6. 6.

    Conway, H., Guy, R.: The Book of Numbers, 1st edn. Copernicus, Torun (1995)

    Google Scholar 

  7. 7.

    Endres, M.: Semi-Skylines and Skyline Snippets—Theory and Applications, 1st edn. Books on Demand, Norderstedt (2011)

    Google Scholar 

  8. 8.

    Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley publications in statistics, 2nd edn. Chapman & Hall, London (2008)

    Google Scholar 

  9. 9.

    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 1st edn. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  10. 10.

    Glück, R., Köppl, D., Wirsching, G.: Computational aspects of ordered integer partition with upper bounds. In: Proc. SEA. LNCS, vol. 7933, pp. 79–90. Springer (2013)

  11. 11.

    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 3rd edn. Oxford University Press, Oxford (1954)

    Google Scholar 

  12. 12.

    Klamroth, K.: Discrete multiobjective optimization. In: Proc. EMO. LNCS, vol. 5467, p. 4. Springer (2009)

  13. 13.

    Knuth, D.E.: Johann Faulhaber and Sums of Powers. Math. Comput. 61(203), 277–294 (1993)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Knuth, D.E.: Seminumerical Algorithms, 3rd edn. Addison-Wesley, Boston (1997)

    Google Scholar 

  15. 15.

    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics-Non-relativistic Theory, 2nd edn. Pergamon Press, Oxford (1965)

    Google Scholar 

  16. 16.

    Matoušek, J., Nešetřil, J.: Invitation to Discrete Mathematics, 2nd edn. Oxford University Press, Oxford (2009)

    Google Scholar 

  17. 17.

    Park, G.: A generalization of multiple choice balls-into-bins: tight bounds. Algorithmica 77(4), 1159–1193 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Preisinger, T.: Graph-based algorithms for Pareto preference query evaluation. Ph.D. thesis, University of Augsburg, Germany (2009)

  19. 19.

    Raab, M., Steger, A.: “Balls into bins”—A simple and tight analysis. In: Proc. RANDOM. LNCS, vol. 1518, pp. 159–170. Springer (1998)

  20. 20.

    Wenzel, F., Köppl, D., Kießling, W.: Interactive toolbox for spatial-textual preference queries. In: Proc. SSTD. LNCS, vol. 8098, pp. 462–466. Springer (2013)

  21. 21.

    Wirsching, G.: Balls in constrained urns and Cantor-like sets. Zeitschrift für Analysis und ihre Anwendungen 17, 979–996 (1998)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work is partly funded by the JSPS KAKENHI Grant Number JP18F18120.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dominik Köppl.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Parts of this work have already been presented at the 12th International Symposium on Experimental Algorithms [10].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Glück, R., Köppl, D. Computational Aspects of Ordered Integer Partitions with Bounds. Algorithmica 82, 2955–2984 (2020). https://doi.org/10.1007/s00453-020-00713-7

Download citation