Welcome to our lesson on Dynamic Programming! This powerful technique is invaluable for tackling complex problems in optimization, combinatorics, and beyond. Dynamic Programming allows you to approach problems by breaking them down into subproblems and storing results of these smaller calculations for future reference.

This method prevents redundant calculations, significantly enhancing the efficiency of your solutions!

One of the classic examples to illustrate Dynamic Programming is calculating the Fibonacci series. When you compute the N-th Fibonacci number recursively, smaller subproblems overlap, and without optimization, this leads to exponential time complexity.

By using Dynamic Programming, specifically memoization, we can save the results of these overlapping subproblems in a table, allowing us to retrieve them directly instead of recalculating each time. This reduces time complexity dramatically.

Here's an example:

Ruby
Copy to clipboard
1def fibonacci(n, memo = {})
2  # Return the cached result if it exists
3  return memo[n] if memo.key?(n)
4  
5  # Base cases: fibonacci(0) = 0, fibonacci(1) = 1
6  return n if n <= 1
7  
8  # Compute and cache the Fibonacci number
9  memo[n] = fibonacci(n - 1, memo) + fibonacci(n - 2, memo)
10  
11  # Return the computed Fibonacci number
12  memo[n]
13end
14
15# Test case
16puts fibonacci(10)  # Output: 55

1. Memoization Check:

   Ruby
   Copy to clipboard
   1return memo[n] if memo.key?(n)

   Before performing any calculations, the function checks if the n-th Fibonacci number has already been computed and stored in the memo hash. If it exists, the cached value is returned immediately, avoiding redundant computations.

2. Base Cases:

   Ruby
   Copy to clipboard
   1return n if n <= 1

   The Fibonacci sequence is defined such that fibonacci(0) = 0 and fibonacci(1) = 1. These base cases terminate the recursion.

3. Recursive Computation and Caching:

   Ruby
   Copy to clipboard
   1memo[n] = fibonacci(n - 1, memo) + fibonacci(n - 2, memo)

   For n > 1, the function recursively computes fibonacci(n - 1) and fibonacci(n - 2), sums them up, and stores the result in the memo hash for future reference.

4. Returning the Result:

   Ruby
   Copy to clipboard
   1memo[n]

   After computing, the function returns the n-th Fibonacci number, now stored in memo[n].

• Time Complexity:
  The memoized version reduces the time complexity from O(2ⁿ) in the naive recursive approach to O(n). Each Fibonacci number up to n is computed exactly once and then retrieved from the memo hash when needed.

• Space Complexity:
  The space complexity is O(n) due to the storage required for the memo hash, which stores the Fibonacci numbers from 0 to n, and the recursion stack depth, which also grows linearly with n.

By leveraging memoization, this Dynamic Programming approach efficiently computes large Fibonacci numbers that would be computationally expensive with a straightforward recursive method.

Grasping Dynamic Programming requires a shift in thinking—it's about recognizing interrelated subproblems and making use of previously computed solutions. Take a moment to understand the concept.

Next, we’ll dive into hands-on exercises designed to solidify your Dynamic Programming skills. These exercises aim to equip you with the tools to simplify complex challenges into manageable, efficient solutions. Let’s get started!