0/1 Knapsack Problem

Write a C++ program to solve the 0/1 knapsack problem using dynamic programming. Given a set of items, each with a weight and a value, determine the maximum value you can carry in a knapsack of a given capacity.

Example 1

Input: 
   weights = [1, 2, 3]
   values = [10, 15, 40]
   capacity = 5
Output: 55
Explanation:
   

   
Step 1: Create a dynamic programming table to store the maximum value for different subproblems.
   
Step 2: Fill the table by considering each item and each capacity value from 0 to the maximum capacity.
   
Step 3: For this example, the optimal solution is to take items with weights 2 and 3, giving a total value of 15 + 40 = 55.
   
Step 4: Return the maximum value possible.
   

Example 2

Input: 
   weights = [1, 1, 1]
   values = [10, 20, 30]
   capacity = 2
Output: 50
Explanation:
   

   
Step 1: Create a dynamic programming table to store the maximum value for different subproblems.
   
Step 2: Fill the table by considering each item and each capacity value from 0 to the maximum capacity.
   
Step 3: For this example, the optimal solution is to take the second and third items (values 20 and 30), giving a total value of 50.
   
Step 4: Return the maximum value possible.
   

Constraints

1 ≤ number of items ≤ 1000
1 ≤ capacity ≤ 1000
Time Complexity: O(n * capacity), where n is the number of items
Space Complexity: O(n * capacity)