For each of the following a e problems, design (or give) an algorithm. Based on your obtained algorithm, give (i) a natural size metric for its inputs; (ii) its basic operation; (iii) whether the basic operation count can be different for inputs of the same size; and (iv) What is the efficiency class of this algorithm:

a. Computing n!

b. Euclids algorithm

c. Cutting a stick: A sticks n inches long needs to be cut into n 1-inch pieces. Outline an algorithm that performs this task with the minimum number of cuts if several pieces of the stick can be cut at the same time. Also give a formula for the minimum number of cuts.

d. Merge sort

e. Lighter or heavier? You have n > 2 identicallooking coins and a two-pan balance scale with no weights. One of the coins is a fake, but you do not know whether it is lighter or heavier than the genuine coins, which all weigh the same. Design a (1) algorithm to determine whether the fake coin is lighter or heavier than the others.