Your task is to find a grouping of the jugs into pairs of red and blue jugs that hold the same amount of water. To do so, you may perform the following operation: pick a pair of jugs in which one is red and one is blue, fill the red jug with water, and then pour the water into the blue jug. This operation will tell you whether the red or the blue jug can hold more water, or that they have the same volume. Assume that such a comparison takes one time unit. Your goal is to find an algorithm that makes a minimum number of comparisons to determine the grouping. Remember that you may not directly compare two red jugs or two blue jugs. Required:
a. Prove a lower bound of Ω(n lg n) for the number of comparisons that an algorithm solving this problem must make.
b. Give a randomized algorithm whose expected number of comparisons is O(n lg n), and prove that this bound is correct. What is the worst-case number of comparisons for your algorithm?