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Toward a Deterministic Polynomial Time Algorithm with Optimal Additive Query Complexity

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Mathematical Foundations of Computer Science 2010 (MFCS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6281))

Abstract

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 ≤ n c/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.

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References

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Alon, N., Beigel, R., Kasif, S., Rudich, S., Sudakov, B.: Learning a Hidden Matching. SIAM J. Comput. 33(2), 487–501 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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. Angluin, D., Chen, J.: Learning a Hidden Hypergraph. Journal of Machine Learning Research 7, 2215–2236 (2006)

    MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Bshouty, N.H., Mazzawi, H.: On Parity Check (0,1)-Matrix over ℤ p . TR09-067. In: ECCC (2009)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  23. Mazzawi, H.: Optimally Reconstructing Weighted Graphs Using Queries. In: Symposium on Discrete Algorithms, pp. 608–615 (2010)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Technion - Israel Institute of Technology,  

    Nader H. Bshouty & Hanna Mazzawi

Authors
  1. Nader H. Bshouty
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  2. Hanna Mazzawi
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Editors and Affiliations

  1. Falculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Petr Hliněný

  2. Faculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Antonín Kučera

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Bshouty, N.H., Mazzawi, H. (2010). Toward a Deterministic Polynomial Time Algorithm with Optimal Additive Query Complexity. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_21

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  • DOI: https://doi.org/10.1007/978-3-642-15155-2_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15154-5

  • Online ISBN: 978-3-642-15155-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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