1. Three colors, n balls and two optimal algorithms (to say nothing of the asymptotically optimal one)

2. The setting A ‘colorblind’ algorithm obtains a set of n balls colored with three colors Queries of type “is the color of ball i equal to the color of ball j?” The aim: minimize the number of queries (assuming worst possible input)

3. Two problems The Partition Problem Partition the balls into three groups, according to their colors The Plurality Problem Find a ball whose color is the most common (or say that there is a draw)

4. Two types of algorithms Deterministic algorithm Each step is determined by the outcome of the previous queries Randomized algorithm Allowed to ‘toss a coin’ We consider the expected number of queries (over all possible outcomes of the coin tosses)

5. Results PluralityPartition lowerupper Det. Rand. Aigner, De Marco, Montangero: The plurality problem with three colors

6. Partition, deterministic case The optimal algorithm The first ball represents color A The first ball of different color represents color B Compare the remaining balls with these two

7. Partition, deterministic case The lower bound: the ‘evil’ oracle strategy The oracle always answers NO, unless such answer would contradict previous answers An optimal algorithm has all its queries answered by NO In the end, every two color classes A and B must have induced at least |A|+|B|-1 queries

8. Future plans If possible, improve the deterministic bounds for the plurality problem Write a paper Try to generalize to k colors

9. Acknowledgement We appreciate the valuable comments of János Komlós We acknowledge the hospitality of DIMACS and the support of DIMATIA