For each of the following alternative greedy strategies, either prove that the resulting algorithm always constructs an optimal schedule, or describe a small input example for which the algorithm does not produce an optimal schedule. Assume that all algorithms break ties arbitrarily (that is, in a manner that is completely out of your control). a. Choose the course x that ends last, discard classes that conflict with x, and recurse.
b. Choose the course x that starts first, discard all classes that conflict with x, and recurse.
c. Choose the course x that starts last, discard all classes that conflict with x, and recurse.
d. Choose the course x with shortest duration, discard all classes that conflict with x, and recurse.
e. Choose a course x that conflicts with the fewest other courses, discard all classes that conflict with x, and recurse.