The Partitioning Problem: Given a set ofnpositive integers, partition the integers into two subsets of equal or almost equal sum. The goal is to partition the input numbers into two groups such that the difference between the sums of the elements in the two group is minimum or as small as possible. a)  Since no polynomial solutions are known for the above problem, implement an exponential algorithm for finding the optimal solution for any instance of the problem and test it for small input samples. Identify the maximum input size for which an optimal solution is possible to obtain using your algorithm in reasonable time (for example 3 hours). Hint: Every possible partition (solution) can be represented by a binary array of sizen

Solve by Algorithm

I need the algorithm, proof of correctness and runtime analysis for the problem. No code necessary ONLY algorithm. And runtime should be O(n log n) as stated in the question.

Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies.   Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…