Skip to main content
SpringerLink
Log in

Unbiased Matrix Rounding

  • Conference paper
Algorithm Theory – SWAT 2006 (SWAT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4059))

Included in the following conference series:

Abstract

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics.

We improve earlier solutions of different authors in two ways. For rounding matrices of size m ×n, we reduce the runtime from O( (mn)2 ) to O(mn log(mn)). Second, our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. This is particularly useful in the statistics application of controlled rounding.

The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased, that is, for all entries y ij of our rounding, we have E(y ij ) = x ij , where x ij is the corresponding entry of the input matrix.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asano, T.: Digital halftoning: Algorithm engineering challenges. IEICE Trans. on Inf. and Syst. E86-D, 159–178 (2003)

    Google Scholar 

  2. Bacharach, M.: Matrix rounding problems. Management Science (Series A) 12, 732–742 (1966)

    MathSciNet  Google Scholar 

  3. Baranyai, Z.: On the factorization of the complete uniform hypergraph. In: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. I, Colloq. Math. Soc. Jánōs Bolyai, vol. 10, pp. 91–108. North-Holland, Amsterdam (1975)

    Google Scholar 

  4. Beck, J., Spencer, J.: Well distributed 2-colorings of integers relative to long arithmetic progressions. Acta Arithm. 43, 287–298 (1984)

    MATH  MathSciNet  Google Scholar 

  5. Bentley, J.L.: Algorithm design techniques. Commun. ACM 27, 865–871 (1984)

    Article  Google Scholar 

  6. Brauner, N., Crama, Y.: The maximum deviation just-in-time scheduling problem. Discrete Appl. Math. 134, 25–50 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Causey, B.D., Cox, L.H., Ernst, L.R.: Applications of transportation theory to statistical problems. Journal of the American Statistical Association 80, 903–909 (1985)

    Article  MathSciNet  Google Scholar 

  8. Cox, L.H.: A constructive procedure for unbiased controlled rounding. Journal of the American Statistical Association 82, 520–524 (1987)

    Article  MATH  Google Scholar 

  9. Cox, L.H., Ernst, L.R.: Controlled rounding. Informes 20, 423–432 (1982)

    MATH  Google Scholar 

  10. Doerr, B.: Linear and hereditary discrepancy. Combinatorics, Probability and Computing 9, 349–354 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Doerr, B.: Lattice approximation and linear discrepancy of totally unimodular matrices. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 119–125 (2001)

    Google Scholar 

  12. Doerr, B.: Global roundings of sequences. Information Processing Letters 92, 113–116 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Doerr, B.: Generating randomized roundings with cardinality constraints and derandomizations. In: 23rd Annual Symposium on Theoretical Aspects of Computer Science (2006)

    Google Scholar 

  14. Doerr, B., Friedrich, T., Klein, C., Osbild, R.: Rounding of sequences and matrices, with applications. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 96–109. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. Fellegi, I.P.: Controlled random rounding. Survey Methodology 1, 123–133 (1975)

    Google Scholar 

  16. Ford Jr., L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  17. Knuth, D.E.: Two-way rounding. SIAM J. Discrete Math. 8, 281–290 (1995)

    Article  MathSciNet  Google Scholar 

  18. Monden, Y.: What makes the Toyota production system really tick? Industrial Eng. 13, 36–46 (1981)

    Google Scholar 

  19. Monden, Y.: Toyota Production System. Industrial Engineering and Management Press, Norcross (1983)

    Google Scholar 

  20. Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. J. Comput. Syst. Sci. 37, 130–143 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sadakane, K., Takki-Chebihi, N., Tokuyama, T.: Combinatorics and algorithms on low-discrepancy roundings of a real sequence. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 166–177. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  22. Sadakane, K., Takki-Chebihi, N., Tokuyama, T.: Discrepancy-based digital halftoning: Automatic evaluation and optimization. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 301–319. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  23. Spencer, J.: Ten lectures on the probabilistic method. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 64. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)

    MATH  Google Scholar 

  24. Steiner, G., Yeomans, S.: Level schedules for mixed-model, just-in-time processes. Management Science 39, 728–735 (1993)

    Article  MATH  Google Scholar 

  25. Willenborg, L., de Waal, T.: Elements of Statistical Disclosure Control. Lecture Notes in Statistics, vol. 155. Springer, Heidelberg (2001)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Benjamin Doerr, Tobias Friedrich, Christian Klein & Ralf Osbild

Authors
  1. Benjamin Doerr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Friedrich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ralf Osbild
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. MADALGO, Department of Computer Science, University of Aarhus, Denmark

    Lars Arge

  2. Institute of Mathematics and Computer Science, University of Latvia, Raiņa bulvāris 29, Rīga, LV-1459, Latvia

    Rusins Freivalds

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Doerr, B., Friedrich, T., Klein, C., Osbild, R. (2006). Unbiased Matrix Rounding. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_12

Download citation

  • DOI: https://doi.org/10.1007/11785293_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35753-7

  • Online ISBN: 978-3-540-35755-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation