DESIGN ANALYSIS OF ALGORITHMS

The knapsack problem is that given a set of positive integers { a1,……,an } , and knapsack of size s , find a subset A of { a1,……, an }such that the sum of elements in A is the largest , but at most s. Part 1. Use the dynamic programming method to design an algorithm for the knapsack problem. Prove the correctness of your algorithm. Show the computational time of your algorithm carefully. Part2. Use C++ to implement the function below int knapsack ( int *a, //the input integers int n, // the number of input integers int s, // knapsack size int *subset, //subset elements int &size_of_subset // the number of items in the subset ) Test your program for the following knapsack problem: Input list: 5, 23, 27, 37, 48, 51, 63, 67, 71, 75, 79, 83, 89, 91, 101, 112, 121, 132, 137, 141, 143, 147, 153, 159, 171, 181, 190, 191. With knapsack size 595. Print out a subset with the sum of its elements so that the sum has the closest distance to 595. Also print out your source code.