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Oblivious and Adaptive Strategies for the Majority and Plurality Problems

  • Fan Chung
  • Ron Graham
  • Jia Mao
  • Andrew Yao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3595)

Abstract

In the well-studied Majority problem, we are given a set of n balls colored with two or more colors, and the goal is to use the minimum number of color comparisons to find a ball of the majority color (i.e., a color that occurs for more than ⌈ n/2 ⌉ times). The Plurality problem has exactly the same setting while the goal is to find a ball of the dominant color (i.e., a color that occurs most often). Previous literature regarding this topic dealt mainly with adaptive strategies, whereas in this paper we focus more on the oblivious (i.e., non-adaptive) strategies. Given that our strategies are oblivious, we establish a linear upper bound for the Majority problem with arbitrarily many different colors. We then show that the Plurality problem is significantly more difficult by establishing quadratic lower and upper bounds. In the end, we also discuss some generalized upper bounds for adaptive strategies in the k-color Plurality problem.

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References

  1. 1.
    Aigner, M.: Variants of the Majority Problem. Applied Discrete Mathematics 137, 3–25 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aigner, M., De Marco, G., Montangero, M.: The Plurality Problem with Three Colors. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 513–521. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Aigner, M., De Marco, G., Montangero, M.: The Plurality Problem with Three Colors and More. Theoretical Computer Science (2005) (to appear)Google Scholar
  4. 4.
    Alon, N.: Eigenvalues and Expanders. Combinatorica 6, 86–96 (1986)Google Scholar
  5. 5.
    Alonso, L., Reingold, E., Schott, R.: Average-case Complexity of Determining the Majority. SIAM J. Computing 26, 1–14 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blecher, P.M.: On a Logical Problem. Discrete Mathematics 43, 107–110 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bollobás, B.: Random graphs, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chung, F.R.K.: Spectral Graph Theory. CBMS Lecture Notes. AMS Publications, Providence (1997)Google Scholar
  9. 9.
    Chung, F.R.K., Graham, R.L., Mao, J., Yao, A.C.: Finding Favorites. In: Electronic Colloquium on Computational Complexity (ECCC), p–78 (2003)Google Scholar
  10. 10.
    Fischer, M.J., Salzberg, S.L.: Finding a Majority among n Votes. J. Algorithms 3, 375–379 (1982)Google Scholar
  11. 11.
    Knuth, D.E.: personal communicationGoogle Scholar
  12. 12.
    Knuth, D.E.: The Art of Computer Programming, Volume 3. Sorting and Searching. Addison-Wesley Publishing Co., Reading (1973)zbMATHGoogle Scholar
  13. 13.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan Graphs. Combinatorica 8, 261–277 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moore, J.: Proposed problem 81-5. J. Algorithms 2, 208–210 (1981)CrossRefGoogle Scholar
  15. 15.
    Saks, M., Werman, M.: On Computing Majority by Comparisons. Combinatorica 11(4), 383–387 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Taylor, A., Zwicker, W.: personal communicationGoogle Scholar
  17. 17.
    Wiener, G.: Search for a Majority Element. J. Statistical Planning and Inference 100, 313–318 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Fan Chung
    • 1
  • Ron Graham
    • 1
  • Jia Mao
    • 1
  • Andrew Yao
    • 2
  1. 1.Department of Computer Science and EngineeringUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of Computer ScienceTsinghua UniversityChina

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