9. (15 points) Dynamic Programming You are given a rod of length n. A rod of length j will be sold for P[j] dollars (for 1 j k). Cutting the rod is free. Let R(n) be the maximum earnings by cutting up a rod of length n and selling the pieces. a) (3 points) State a self-reduction from R(n) to its sub-problems. Consider at least a base case of n = 0 and make sure your self-reduction covers all n. b) (2 points) What data structure should be used to store the solutions to sub-problems and what order should the sub-problems be solved? c) (4 points) Write a dynamic programming algorithm based off your self-reduction to calculate the maximum earnings. You do not need to show how to recover the solution.