文章基本信息

标题：Randomized Algorithms for Finding a Majority Element
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作者：Pawel Gawrychowski ; Jukka Suomela ; Przemyslaw Uznanski 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2016
卷号：53
页码：9:1-9:14
DOI：10.4230/LIPIcs.SWAT.2016.9
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Given n colored balls, we want to detect if more than n/2 of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to minimize the total number of such operations. A well-known exercise is to show how to find such a ball with only 2n comparisons while using only a logarithmic number of bits for bookkeeping. The resulting algorithm is called the Boyer-Moore majority vote algorithm. It is known that any deterministic method needs 3n/2-2 comparisons in the worst case, and this is tight. However, it is not clear what is the required number of comparisons if we allow randomization. We construct a randomized algorithm which always correctly finds a ball of the majority color (or detects that there is none) using, with high probability, only 7n/6+o(n) comparisons. We also prove that the expected number of comparisons used by any such randomized method is at least 1.019n.
关键词：majority; randomized algorithms; lower bounds