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Average-Case Lower Bounds for the Plurality Problem

Laurent Alonso 1 Edward M. Reingold 2
1 ALICE - Geometry and Lighting
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Given a set of $n$ elements, each of which is colored one of $c \geq 2$ colors, we have to determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We derive lower bounds for the expected number of color comparisons when the $c^n$ colorings are equally probable. We prove a general lower bound of $\frac{c}{3} n - O( \sqrt n)$ for $c \geq 2$; we prove the stronger particular bounds of $\frac{7}{6} n - O(\sqrt n)$ for $c=3$, $\frac{54}{35} n - O(\sqrt n)$ for $c = 4$, $\frac{607}{315}n-O(\sqrt n)$ for $c=5$, $\frac{1592}{693}n - O(\sqrt n)$ for $c = 6$, $\frac{7985}{3003} n - O(\sqrt n)$ for $c = 7$, and $\frac{19402}{6435} n - O(\sqrt n)$ for $c = 8$.
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Submitted on : Sunday, September 2, 2007 - 8:43:38 AM
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Laurent Alonso, Edward M. Reingold. Average-Case Lower Bounds for the Plurality Problem. ACM Transactions on Algorithms, Association for Computing Machinery, 2008, 4 (3), pp.27:1,27:17. ⟨10.1145/1367064.1367067⟩. ⟨inria-00169183⟩



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