The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible [Wikipedia].

Please analyze the problem and find the recursive solution showing the optimal substructure.

Please design your own dynamic programming algorithm and analyze its running time.

Please use C++ or Java to implement your algorithm.

Please test your program with different input and record the results.