Randomized strategies for the plurality problem

AbstractWe consider a game played by two players, Paul and Carol. At the beginning of the game, Carol fixes a coloring of n balls. At each turn, Paul chooses a pair of the balls and asks Carol whether the balls have the same color. Carol truthfully answers his question. Paul’s goal is to determine the most frequent (plurality) color in the coloring by asking as few questions as possible. The game is studied in the probabilistic setting when Paul is allowed to choose his next question randomly.We give asymptotically tight bounds both for the case of two colors and many colors. For the balls colored by k colors, we prove a lower bound Ω(kn) on the expected number of questions; this is asymptotically optimal. For the balls colored by two colors, we provide a strategy for Paul to determine the plurality color with the expected number of 2n/3+O(nlogn) questions; this almost matches the lower bound 2n/3−O(n)

Majority in the Three-Way Comparison Model

In this thesis, we study comparison based problems in a new comparison model called three-way, where a comparison can result in { >, =, < }. We consider a set of n balls with fixed ordered coloring. Particularly, we are interested in finding a ball of the majority color, the color that occurs more than half, when there are 2 colors, partition problem, where the goal is to determine groups of balls with the same color when there are 2 and 3 colors, respectively. We study these problems using both deterministic and randomized approaches