Toward a Deterministic Polynomial Time Algorithm with Optimal Additive Query Complexity

  • Nader H. Bshouty
  • Hanna Mazzawi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)


In this paper, we study two combinatorial search problems: The coin weighing problem with a spring scale (also known as the vector reconstructing problem using additive queries) and the problem of reconstructing weighted graphs using additive queries. Suppose we are given n identical looking coins. Suppose that m out of the n coins are counterfeit and the rest are authentic. Assume that we are allowed to weigh subsets of coins with a spring scale. It is known that the optimal number of weighing for identifying the counterfeit coins and their weights is at least
$$\Omega\left(\frac{m\log n}{\log m}\right).$$
We give a deterministic polynomial time adaptive algorithm for identifying the counterfeit coins and their weights using
$$ O\left(\frac{m\log n}{\log m}+ m\log \log m\right) $$
weighings, assuming that the weight of the counterfeit coins are greater than the weight of the authentic coin. This algorithm is optimal when m ≤ nc/loglogn, where c is any constant. Also our weighing complexity is within loglogm times the optimal complexity for all m.

To obtain this result, our algorithm makes use of search matrices, the divide and conquer approach and the guess and check approach.

When combining these methods with the technique introduced in [Optimally Reconstructing Weighted Graphs Using Queries. SODA, 2010], we get a similar positive result for the problem of reconstructing a hidden weighted graph using additive queries.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aigner, M.: Combinatorial Search. John Wiley and Sons, Chichester (1988)MATHGoogle Scholar
  2. 2.
    Alon, N., Asodi, V.: Learning a Hidden Subgraph. SIAM J. Discrete Math 18(4), 697–712 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Beigel, R., Kasif, S., Rudich, S., Sudakov, B.: Learning a Hidden Matching. SIAM J. Comput. 33(2), 487–501 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Angluin, D., Chen, J.: Learning a Hidden Graph Using O(logn) Queries per Edge. In: Conference on Learning Theory, pp. 210–223 (2004)Google Scholar
  5. 5.
    Angluin, D., Chen, J.: Learning a Hidden Hypergraph. Journal of Machine Learning Research 7, 2215–2236 (2006)MathSciNetGoogle Scholar
  6. 6.
    Bouvel, M., Grebinski, V., Kucherov, G.: Combinatorial Search on Graphs Motivated by Bioinformatics Applications: A Brief Survey. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 16–27. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Bshouty, N.H.: Optimal Algorithms for the Coin Weighing Problem with a Spring Scale. In: Conference on Learning Theory (2009)Google Scholar
  8. 8.
    Bshouty, N.H., Mazzawi, H.: Reconstructing Weighted Graphs with Minimal Query Complexity. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds.) ALT 2009. LNCS (LNAI), vol. 5809, pp. 97–109. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Bshouty, N.H., Mazzawi, H.: On Parity Check (0,1)-Matrix over ℤp. TR09-067. In: ECCC (2009)Google Scholar
  10. 10.
    Cantor, D.: Determining a set from the cardinalities of its intersections with other sets. Canadian Journal of Mathematics 16, 94–97 (1962)Google Scholar
  11. 11.
    Cantor, D., Mills, W.: Determining a Subset from Certain Combinatorial Properties. Canad. J. Math. 18, 42–48 (1966)MATHMathSciNetGoogle Scholar
  12. 12.
    Choi, S., Han Kim, J.: Optimal Query Complexity Bounds for Finding Graphs. In: STOC, pp. 749–758 (2008)Google Scholar
  13. 13.
    Du, D., Hwang, F.K.: Combinatorial group testing and its application. Series on applied mathematics, vol. 3. World Science (1993)Google Scholar
  14. 14.
    Erdös, Rényi, A.: On two problems of information theory. Publ. Math. Inst. Hung. Acad. Sci. 8, 241–254 (1963)Google Scholar
  15. 15.
    Grebinski, V., Kucherov, G.: Optimal Reconstruction of Graphs Under the Additive Model. Algorithmica 28(1), 104–124 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Grebiniski, V., Kucherov, G.: Reconstructing a hamiltonian cycle by querying the graph: Application to DNA physical mapping. Discrete Applied Mathematics 88, 147–165 (1998)CrossRefMathSciNetGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Lindström, B.: On a combinatorial problem in number theory. Canad. Math. Bull. 8, 477–490 (1965)MATHMathSciNetGoogle Scholar
  19. 19.
    Lindström, B.: On a combinatorial detection problem II. Studia Scientiarum Mathematicarum Hungarica 1, 353–361 (1966)MATHMathSciNetGoogle Scholar
  20. 20.
    Lindström, B.: On Möbius functions and a problem in combinatorial number theory. Canad. Math. Bull. 14(4), 513–516 (1971)MATHMathSciNetGoogle Scholar
  21. 21.
    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
  22. 22.
    Li, M., Vitányi, P.M.B.: Combinatorics and Kolmogorov Complexity. In: Structure in Complexity Theory Conference, pp. 154–163 (1991)Google Scholar
  23. 23.
    Mazzawi, H.: Optimally Reconstructing Weighted Graphs Using Queries. In: Symposium on Discrete Algorithms, pp. 608–615 (2010)Google Scholar
  24. 24.
    Moser, L.: The second moment method in combinatorial analysis. In: Combinatorial Structure and their applications, pp. 283–384. Gordon and Breach, New York (1970)Google Scholar
  25. 25.
    Pippenger, N.: An Informtation Theoretic Method in Combinatorial Theory. J. Comb. Theory, Ser. A 23(1), 99–104 (1977)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Pippenger, N.: Bounds on the performance of protocols for a multiple-access broadcast channel. IEEE Transactions on Information Theory 27(2), 145–151 (1981)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Reyzin, L., Srivastava, N.: Learning and Verifying Graphs using Queries with a Focus on Edge Counting. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds.) ALT 2007. LNCS (LNAI), vol. 4754, pp. 277–289. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  28. 28.
    Ruszinkó, M., Vanroose, P.: How an Erdős-Rényi-type search approach gives an explicit code construction of rate 1 for random access with multiplicity feedback. IEEE Transactions on Information Theory 43(1), 368–372 (1997)MATHCrossRefGoogle Scholar
  29. 29.
    Soderberg, S., Shapiro, H.S.: A combinatory detection problem. American Mathematical Monthly 70, 1066–1070 (1963)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nader H. Bshouty
    • 1
  • Hanna Mazzawi
    • 1
  1. 1.Technion - Israel Institute of Technology 

Personalised recommendations