Dynamic Programming Problems Master List 🎯

Welcome to our comprehensive guide on Dynamic Programming Problems! In this lesson, we'll dive deep into the world of dynamic programming, a powerful algorithmic technique for solving complex problems efficiently. Let's get started! 📝

What is Dynamic Programming?

Dynamic programming is an algorithmic technique used to solve complex problems by breaking them down into simpler sub-problems. The solutions to these sub-problems are stored and reused, making the algorithm more efficient. 💡 Pro Tip: It's a bottom-up approach where we solve smaller problems before solving larger ones.

Why Dynamic Programming?

Dynamic programming is a must-know technique for any programmer. It's used to solve various real-world problems, including scheduling, knapsack problems, and even optimization problems. By understanding dynamic programming, you'll be able to solve these problems effectively and efficiently.

Dynamic Programming Problems Master List

We've compiled a list of dynamic programming problems that are suitable for both beginners and intermediates. Each problem comes with a detailed explanation, sample code, and a quiz to test your understanding.

Fibonacci Series

The Fibonacci series is a sequence of numbers where each number is the sum of the two preceding ones. This problem is a great introduction to dynamic programming.

Sample Code

python

def fibonacci(n):
    fib = [0, 1]
    for i in range(2, n+1):
        fib.append(fib[i-1] + fib[i-2])
    return fib

Quiz

Quick Quiz

Question 1 of 1

What is the 10th number in the Fibonacci series?

Longest Common Subsequence

The Longest Common Subsequence (LCS) problem involves finding the longest sequence present in two strings. This problem is a classic example of dynamic programming.

Sample Code

python

def lcs(x, y):
    dp = [[0] * (len(y) + 1) for _ in range(len(x) + 1)]

    for i in range(len(x)):
        for j in range(len(y)):
            if x[i] == y[j]:
                dp[i+1][j+1] = dp[i][j] + 1
            else:
                dp[i+1][j+1] = max(dp[i+1][j], dp[i][j+1])

    return dp[-1][-1]

Quiz

Quick Quiz

Question 1 of 1

What is the Longest Common Subsequence of "BANANA" and "APPLE"?

Knapsack Problem

The Knapsack problem is about maximizing the total value of items in a knapsack, given a set of items with weights and values, and a knapsack capacity. This problem is a popular example of dynamic programming.

Sample Code

python

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

    for i in range(n+1):
        for w in range(capacity+1):
            if i == 0 or w == 0:
                dp[i][w] = 0
            elif weights[i-1] <= w:
                dp[i][w] = max(values[i-1] + dp[i-1][w-weights[i-1]], dp[i-1][w])
            else:
                dp[i][w] = dp[i-1][w]

    return dp[n][capacity]

Quiz

Quick Quiz

Question 1 of 1

You have a knapsack of capacity 10. You have two items: item 1 with weight 5 and value 7, and item 2 with weight 4 and value 8. Which item should you pick to maximize the total value in the knapsack?

Remember, practice makes perfect! Keep solving these problems to improve your dynamic programming skills. Happy coding! 💡 Pro Tip: Try to solve the problems without looking at the sample code first, then compare your solution with ours. This will help reinforce your understanding.

📝 Note: Dynamic programming problems can be solved using different types of arrays, including 1D arrays, 2D arrays, and even matrices.