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From Discrepancy to Majority

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Abstract

We show how to select an item with the majority color from n two-colored items, given access to the items only through an oracle that returns the discrepancy of subsets of k items (the absolute value of the difference between the numbers of items with each color). We use \(n/\lfloor \tfrac{k}{2}\rfloor +O(k)\) queries, improving a previous method by De Marco and Kranakis that used \(n-k+k^2/2\) queries. We also prove a lower bound of \(n/(k-1)-O (n^{1/3})\) on the number of queries needed, both for discrepancy queries and to queries that return the partition of items into monochromatic subsets. This improves a lower bound of \(\lfloor n/k\rfloor \) by De Marco and Kranakis.

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Notes

  1. There is a bug in their method for odd k, in Case 1 of Theorem 4.1, when \(i=\lfloor k/2\rfloor \).

References

  1. Aigner, M.: Variants of the majority problem. Discret. Appl. Math. 137(1), 3–25 (2004). doi:10.1016/S0166-218X(03)00186-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Aigner, M.: Two colors and more. Entropy, Search, Complexity. pp. 9–26. Springer, Bolyai Soc. Math. Stud. 16 (2007). doi:10.1007/978-3-540-32777-6_1

  3. Aigner, M., De Marco, G., Montangero, M.: The plurality problem with three colors and more. Theor. Comput. Sci. 337(1–3), 319–330 (2005). doi:10.1016/j.tcs.2004.12.035

  4. Alonso, L., Reingold, E.M., Schott, R.: Determining the majority. Inf. Process. Lett. 47(5), 253–255 (1993). doi:10.1016/0020-0190(93)90135-V

    Article  MathSciNet  MATH  Google Scholar 

  5. Alonso, L., Reingold, E.M., Schott, R.: The average-case complexity of determining the majority. SIAM J. Comput. 26(1), 1–14 (1997). doi:10.1137/S0097539794275914

    Article  MathSciNet  MATH  Google Scholar 

  6. Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge Tracts in Mathematics 89. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  7. Chung, F., Graham, R., Mao, J., Yao, A.: Oblivious and adaptive strategies for the majority and plurality problems. Algorithmica 48(2), 147–157 (2007). doi:10.1007/s00453-007-0060-0

    Article  MathSciNet  MATH  Google Scholar 

  8. De Marco, G., Kranakis, E.: Searching for majority with \(k\)-tuple queries. Discret. Math. Algorithms Appl. 7(2), 1550009 (2015). doi:10.1142/S1793830915500093

    Article  MathSciNet  MATH  Google Scholar 

  9. De Marco, G., Kranakis, E., Wiener, G.: Computing majority with triple queries. Theor. Comput. Sci. 461, 17–26 (2012). doi:10.1016/j.tcs.2012.03.039

    Article  MathSciNet  MATH  Google Scholar 

  10. Du, D.-Z., Hwang, F.K.: Combinatorial Group Testing and its Applications, 2nd edn. Ser. Appl. Math. 12. World Scientific (2000)

  11. Dvořák, Z., Jelínek, V., Král, D., Kynčl, J., Saks, M.: Probabilistic strategies for the partition and plurality problems. Random Struct. Algorithms 30(1–2), 63–77 (2007). doi:10.1002/rsa.20148

  12. Eppstein, D., Goodrich, M.T., Hirschberg, D.S.: Improved combinatorial group testing algorithms for real-world problem sizes. SIAM J. Comput. 36(5), 1360–1375 (2007). doi:10.1137/050631847

    Article  MathSciNet  MATH  Google Scholar 

  13. Eppstein, D., Goodrich, M. T., Hirschberg, D. S.: Combinatorial pair testing: distinguishing workers from slackers. In: Proceedings of 13th International Symposium Algorithms and Data Structures (WADS 2013). Lecture Notes in Computer Science, vol. 8037, pp. 316–327. Springer (2013). doi:10.1007/978-3-642-40104-6_28

  14. Gerbner, D., Keszegh, B., Pálvölgyi, D., Patkós, B., Vizer, M., Wiener, G.: Finding a non-minority ball with majority answers. Discret. Appl. Math. 219, 18–31 (2017). doi:10.1016/j.dam.2016.11.012

  15. Gerbner, D., Vizer, M.: Majority problems of large query size. Electronic preprint. arxiv:1610.09114, (2016)

  16. Kráľ, D., Sgall, J., Tichý, T.: Randomized strategies for the plurality problem. Discret. Appl. Math. 156(17), 3305–3311 (2008). doi:10.1016/j.dam.2008.05.014

  17. Saks, M.E., Werman, M.: On computing majority by comparisons. Combinatorica 11(4), 383–387 (1991). doi:10.1007/BF01275672

  18. Valiant, L.G.: Short monotone formulae for the majority function. J. Algorithms 5(3), 363–366 (1984). doi:10.1016/0196-6774(84)90016-6

    Article  MathSciNet  MATH  Google Scholar 

  19. Wiener, G.: Search for a majority element. J. Stat. Plan. Inference 100(2), 313–318 (2002). doi:10.1016/S0378-3758(01)00142-2

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of California, Irvine, CA, USA

    David Eppstein & Daniel S. Hirschberg

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  1. David Eppstein
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  2. Daniel S. Hirschberg
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Correspondence to David Eppstein.

Additional information

David Eppstein was supported in part by NSF Grants CCF-1228639, CCF-1618301, and CCF-1616248. A preliminary version of this paper appeared at the 12th Latin American Theoretical Informatics Symposium (LATIN 2016), Lecture Notes in Computer Science 9644 (2016), pp. 390–402.

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Eppstein, D., Hirschberg, D.S. From Discrepancy to Majority. Algorithmica 80, 1278–1297 (2018). https://doi.org/10.1007/s00453-017-0303-7

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