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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 \(i=\lfloor k/2\rfloor \).

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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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