How does dynamic programming work for the knapsack problem?

Can you explain how dynamic programming is used to solve the knapsack problem?

Sure! Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. In the knapsack problem, we aim to maximize the total value of items in a knapsack without exceeding its weight capacity.

What are the key steps involved in the dynamic programming approach for this problem?

The key steps include defining the problem, creating a table to store results of subproblems, and filling this table based on whether to include each item or not.

Can you give me an example of how the table is filled?

Absolutely! For each item, you check if its weight is less than or equal to the current capacity. If it is, you decide whether to include it based on the maximum value achievable with or without the item.

What is the time complexity of this dynamic programming solution?

The time complexity is O(n * W), where n is the number of items and W is the maximum weight capacity of the knapsack.