Average-case analysis of some plurality algorithms

Abstract

Given a set of n elements, each of which is colored one of c colors, we must determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We focus on the expected number of color comparisons when the c n colorings are equally probable. We analyze an obvious algorithm, showing that its expected performance is c 2 +c−2 2c n− O(c 2 ), with variance (c 2 n). We present and analyze an algorithm for the case c = 3 colors whose average complexity on the 3 n equally probable inputs is 7083 5425 n + O( √ n) = 1.3056 · · · n + O( √ n), substantially better than the expected complexity 5 3 n + O(1) = 1.6666 · · · n + O(1) of the obvious algorithm. We describe a similar algorithm for c = 4 colors whose average complexity on the 4 n equally probable inputs is 761311 402850 n + O(log n) = 1.8898 · · · n + O(log n), substantially better than the expected complexity 9 4 n + O(1) = 2.25n + O(1) of the obvious algorithm. ACM Reference Format:Alonso, L. and Reingold, E. M. 2009. Average-case analysis of some plurality algorithms.

A Double Binomial Sum: Putnam 35–A4 Revisited

“…Obvious ones, such as, ballot problems, where the expected margin of victory is an object of interest [3,4], and less obvious ones, such as, the excess of Hadamard matrices [5]. The latter arises in connection with the probabilistic method, as pioneered by Paul Erdös [2].…”

Determining plurality

“…For the average case of the plurality problem, assuming that all c n colorings are equally likely, Alonso and Reingold [2006] gives a general algorithm that uses an average of c 2 +c−2 2c n − O(c 2 ) color comparisons, an algorithm for c = 3 that uses an average of 7083 5425 n + O( √ n) = 1.3056 · · · n + O( √ n) color comparisons, and an algorithm for c = 4 that uses an average of 761311 402850 n + O( log n) = 1.8898 · · · n + O( log n) color comparisons. Alonso and Reingold [2008] establishes lower bounds on the average case of the plurality problem: a general lower bound of c 3 n − O( √ n) for c ≥ 3, as well as stronger particular bounds of 7…”

A plurality problem with three colors and query size three

“…Since then many research was carried out about the Plurality problem, see e.g. [2,3,4,5,7,10,11,16]. In [7] it is written, that "[The Plurality problem] seems to be a more difficult variant".…”