What are some dynamic examples in programming?

What are some dynamic examples in programming?

Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. Examples include the Fibonacci sequence, knapsack problem, and shortest path algorithms.

Can you explain the Fibonacci sequence using dynamic programming?

Sure! The Fibonacci sequence can be computed using dynamic programming by storing previously computed values to avoid redundant calculations. This can be done using either a top-down approach with memoization or a bottom-up approach.

What is the knapsack problem?

The knapsack problem is a classic optimization problem where you have to maximize the total value of items in a knapsack without exceeding its weight capacity. Dynamic programming helps find the optimal solution efficiently.

Can you give me a code example for the knapsack problem?

Certainly! Here's a simple Python implementation of the 0/1 knapsack problem using dynamic programming:

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], dp[i-1][w - weights[i-1]] + values[i-1])
            else:
                dp[i][w] = dp[i-1][w]

    return dp[n][capacity]