Skip to main content
SpringerLink
Log in

On the Coin Weighing Problem with the Presence of Noise

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2012, RANDOM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7408))

Abstract

In this paper we consider the following coin weighing problem: Given n coins for which some of them are counterfeit with the same weight. The problem is: given the weights of the counterfeit coin and the authentic coin, detect the counterfeit coins a with minimal number of weighings. This problem has many applications in computational learning theory, compressed sensing and multiple access adder channels.

An old optimal non-adaptive polynomial time algorithm of Lindstrom can detect the counterfeit coins with O(n/logn) weighings. An information theoretic proof shows that Lindstrom’s algorithm is optimal. In this paper we study non-adaptive algorithms for this problem when some of the answers of the weighings received are incorrect or unknown.

We show that no coin weighing algorithm exists that can detect the counterfeit coins when the number of incorrect weighings is more than 1/4 fraction of the number of weighings. We also give the tight bound Θ(n/logn) for the number of weighings when the number of incorrect answers is less than 1/4 fraction of the number of weighings.

We then give a non-adaptive polynomial time algorithm that detects the counterfeit coins with k = O(n/loglogn) weighings even if some constant fraction of the answers of the weighings received are incorrect. This improves Bshouty and Mazzawi’s algorithm [7] that uses O(n) weighings. This is the first sublinear algorithm for this problem.

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. Aigner, M.: Combinatorial Search. John Wiley and Sons (1988)

    Google Scholar 

  2. Alon, N., Asodi, V.: Learning a Hidden Subgraph. SIAM J. Discrete Math. 18(4), 697–712 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biglieri, E., Györfi, L.: Multiple Access Channels Theory and Practice Volume 10 NATO Security through Science Series - D: Information and Communication Security (April 2007)

    Google Scholar 

  4. Bruneau, L., Germinet, F.: On the singularity of random matrices with independent entries. Proc. Amer. Math. Soc. 137, 787–792 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bshouty, N.H.: Optimal Algorithms for the Coin Weighing Problem with a Spring Scale. In: Conference on Learning Theory (2009)

    Google Scholar 

  6. Bshouty, N.H., Mazzawi, H.: Toward a Deterministic Polynomial Time Algorithm with Optimal Additive Query Complexity. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 221–232. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Bshouty, N.H., Mazzawi, H.: Algorithms for the Coin Weighing Problems with the Presence of Noise. ECCC, TR11-124

    Google Scholar 

  8. Cantor, D.: Determining a set from the cardinalities of its intersections with other sets. Canadian Journal of Athematics 16, 94–97 (1962)

    Article  Google Scholar 

  9. Cheng, J., Kamoi, K., Watanabe, Y.: User Identification by Signature Code for Noisy Multiple-Access Adder Channel. In: ISIT (2006)

    Google Scholar 

  10. Cantor, D., Mills, W.: Determining a Subset from Certain Combinatorial Properties. Canad. J. Math. 18, 42–48 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang, S.C., Weldon, E.J.: Coding for T-user multiple access channels. IEEE Transactions on Information Theory 25(6), 684–691 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, J., Watanabe, Y.: A Multiuser k-Ary Code for the Noisy Multiple-Access Adder Channel. IEEE Transactions on Information Theory 47, 6 (2001)

    Article  MathSciNet  Google Scholar 

  13. Cheng, J., Watanabe, Y.: Affine Code for T-User Noisy Multiple Access Adder Channel. IEICE Trans. Fundamentals E83-A(3) (2000)

    Google Scholar 

  14. Choi, S., Han Kim, J.: Optimal Query Complexity Bounds for Finding Graphs. In: STOC, pp. 749–758 (2008)

    Google Scholar 

  15. Cheng, J., Kamoi, K., Watanabe, Y.: User Identification by Signature Code for Noisy Multiple-Access Adder Channel. In: IEEE International Symposium on Information Theory, pp. 1974–1977 (2006)

    Google Scholar 

  16. Du, D., Hwang, F.K.: Combinatorial group testing and its application. Series on applied mathematics, vol. 3. World Science (1993)

    Google Scholar 

  17. Danev, D., Laczay, B., Ruszinkó, M.: Multiple Access Adder Channel. Multiple Access Channels - Theory and Practice, pp. 26–53. IOS Press (2007)

    Google Scholar 

  18. Erdös, Rényi, A.: On two problems of information theory. Publ. Math. Inst. Hung. Acad. Sci. 8, 241–254 (1963)

    Google Scholar 

  19. Grebinski, V., Kucherov, G.: Optimal Reconstruction of Graphs Under the Additive Model. Algorithmica 28(1), 104–124 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grebiniski, V., Kucherov, G.: Reconstructing a hamiltonian cycle by querying the graph: Application to DNA physical mapping. Discrete Applied Mathematics 88, 147–165 (1998)

    Article  MathSciNet  Google Scholar 

  21. Grebinski, V.: On the Power of Additive Combinatorial Search Model. In: Hsu, W.-L., Kao, M.-Y. (eds.) COCOON 1998. LNCS, vol. 1449, pp. 194–203. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  22. Indyk, P., Ruzic, M.: Near-Optimal Sparse Recovery in the L1 Norm. In: FOCS 2008, pp. 199–207 (2008)

    Google Scholar 

  23. Khachatrian, G.K., Martirossian, S.S.: Codes for T-user Noiseless Adder Channel. Problems of Control and Information Theory 16, 187–192 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Komlós, J.: On the determinant of matrices. Studia. Sci. Math. Hungar. 2, 7–21 (1967)

    MathSciNet  MATH  Google Scholar 

  25. Laczay, B.: Coding for the Multiple Access Adder Channel (2003)

    Google Scholar 

  26. Lindström, B.: On a combinatorial problem in number theory. Canad. Math. Bull. 8, 477–490 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lindström, B.: On a combinatorial detection problem II. Studia Scientiarum Mathematicarum Hungarica 1, 353–361 (1966)

    MathSciNet  MATH  Google Scholar 

  28. Lindström, B.: On Möbius functions and a problem in combinatorial number theory. Canad. Math. Bull. 14(4), 513–516 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lindström, B.: Determining subsets by unramified experiments. In: Srivastava, J.N. (ed.) A Survey of Statistical Designs and Linear Models, pp. 407–418. North Holland, Amsterdam (1975)

    Google Scholar 

  30. Li, M., Vitányi, P.M.B.: Combinatorics and Kolmogorov Complexity. In: Structure in Complexity Theory Conference, pp. 154–163 (1991)

    Google Scholar 

  31. Moser, L.: The second moment method in combinatorial analysis. In: Combinatorial Structure and their Applications, pp. 283–384. Gordon and Breach (1970)

    Google Scholar 

  32. Pippenger, N.: An Informtation Theoretic Method in Combinatorial Theory. J. Comb. Theory, Ser. A 23(1), 99–104 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pippenger, N.: Bounds on the performance of protocols for a multiple-access broadcast channel. IEEE Transactions on Information Theory 27(2), 145–151 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  34. Roth, R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  35. Soderberg, S., Shapiro, H.S.: A combinatory detection problem. American Mathematical Monthly 70, 1066–1070 (1963)

    Article  MathSciNet  Google Scholar 

  36. Wilson, J.H.: Error-Correcting Codes for a T-User Binary Adder Channel. IEEE Transactions of Information Theory 34(4) (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technion, Israel

    Nader H. Bshouty

Authors
  1. Nader H. Bshouty
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Gates Building 7203, 15213, Pittsburgh, PA, USA

    Anupam Gupta

  2. Department of Computer Science, University Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, University of Geneva, Battelle A, 7 route de Drize, 1227, Carouge, Switzerland

    José Rolim

  4. Department of Computer Science, Foundation School of Engineering and Applied Science, Columbia University, Amsterdam Avenue 1214, 10027-7003, New York, NY, USA

    Rocco Servedio

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bshouty, N.H. (2012). On the Coin Weighing Problem with the Presence of Noise. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-32512-0_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32511-3

  • Online ISBN: 978-3-642-32512-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation