Probabilistic strategies for the partition and plurality problems

Abstract

ABSTRACT:We consider a game played by two players, Paul and Carol. Carol fixes a coloring of n balls with three colors. At each step, Paul chooses a pair of balls and asks Carol whether the balls have the same color. Carol truthfully answers yes or no. In the Plurality problem, Paul wants to find a ball with the most common color. In the Partition problem, Paul wants to partition the balls according to their colors. Paul's goal is to ask Carol the fewest number of questions to reach his goal. We find optimal probabilistic strategies for the Partition problem and the Plurality problem in the considered setting.

Computing Majority with Triple Queries

“…In oblivious strategies the questioner has to ask all questions in advance before getting any answer from the oracle. Finally, bounds for randomized algorithms can be found in [10,13].…”

Average-case analysis of some plurality algorithms

“…The case c = 3 was also studied in Dvořák et al [2007], where 3n/2 − O( √ n log n) comparisons are proved necessary and 3n/2 + O(1) are proved sufficient when randomized algorithms are allowed. For c colors, Král' et al [2008] proved a lower bound of (cn) for randomized algorithms.…”

“…For c colors, Král' et al [2008] proved a lower bound of (cn) for randomized algorithms. Dvořák et al [2007] also studied the partition problem in which the three color classes of the elements must be completely determined; they proved that 2n − 3 comparisons are necessary and sufficient in the ordinary case and (5n − 8)/3 + o(1) in the randomized case; for the ordinary case of the partition problem with c colors and n ≥ c, they proved that (c − 1)n − c 2 comparisons are necessary and sufficient; Král' et al [2008] showed that (c − 1)(n − c)/4 are necessary for randomized algorithms. With an unknown number of colors, n−1 2 comparisons are necessary and sufficient to determine a color that occurs at least as often as any other color (the "non-strict" plurality problem) [Srivastava and Taylor 2005].…”

Finding a non-minority ball with majority answers

“…There are several variants of the majority problem [1]. The plurality problem, where we have to find a plurality ball (or show that none exists) was considered, among others, in [1,8,10]. Another possible direction is to use sets of size greater than two as queries [6,5].…”

“…Katona and studied by Johnson and Mészáros [13]. They have shown that if all elements are different, then they can be almost completely sorted 8 using O(n log n) queries in the adaptive model and O(n q−t+1 ) queries in the non-adaptive model and both results are sharp. However, their algorithms fail if not all elements are different.…”