How can I solve the knapsack problem using dynamic programming?

How can I solve the knapsack problem using dynamic programming?

The knapsack problem involves selecting items with given weights and values to maximize value without exceeding capacity. Dynamic programming is a great approach to solve it efficiently.

Can you explain the dynamic programming approach step by step?

Sure! First, define a 2D array where rows represent items and columns represent weights. Then, fill this array based on whether to include each item or not.

What is the time complexity of this approach?

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

Can you provide a code example for this?

Absolutely! Here's a simple Python implementation:

def knapsack(weights, values, capacity):
    n = len(values)
    dp = [[0 for _ in range(capacity + 1)] for _ in range(n + 1)]

    for i in range(1, n + 1):
        for w in range(1, capacity + 1):
            if weights[i-1] <= w:
                dp[i][w] = max(dp[i-1][w], values[i-1] + dp[i-1][w - weights[i-1]])
            else:
                dp[i][w] = dp[i-1][w]
    return dp[n][capacity]