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. 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, improving a lower bound of \(\lfloor n/k\rfloor \) by De Marco and Kranakis.
David Eppstein was supported in part by NSF grant CCF-1228639.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
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
Du, D.Z., Hwang, F.K.: Combinatorial Group Testing and its Applications. Ser. Appl. Math., vol. 12, 2nd edn. World Scientific, New York (2000)
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)
Valiant, L.G.: Short monotone formulae for the majority function. J. Algor. 5(3), 363–366 (1984)
Eppstein, D., Goodrich, M.T., Hirschberg, D.S.: Combinatorial pair testing: distinguishing workers from slackers. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 316–327. Springer, Heidelberg (2013)
De Marco, G., Kranakis, E.: Searching for majority with \(k\)-tuple queries. Discrete Math. Algor. Appl. 7(2), 1550009 (2015)
Alonso, L., Reingold, E.M., Schott, R.: Determining the majority. Inform. Process. Lett. 47(5), 253–255 (1993)
Alonso, L., Reingold, E.M., Schott, R.: The average-case complexity of determining the majority. SIAM J. Comput. 26(1), 1–14 (1997)
Saks, M.E., Werman, M.: On computing majority by comparisons. Combinatorica 11(4), 383–387 (1991)
Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press, Cambridge (2008)
Gerbner, D., Keszegh, B., Pálvölgyi, D., Patkós, B., Vizer, M., Wiener, G.: Finding a majority ball with majority answers. In: Proceedings of the 8th European Conference on Combinatorics, Graph Theory, and Applications (EuroComb 2015). Elect. Notes Discrete Math., vol. 49, pp. 345–351. Elsevier (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eppstein, D., Hirschberg, D.S. (2016). From Discrepancy to Majority. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_29
Download citation
DOI: https://doi.org/10.1007/978-3-662-49529-2_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49528-5
Online ISBN: 978-3-662-49529-2
eBook Packages: Computer ScienceComputer Science (R0)