Dynamic Programming: A Comprehensive Guide with Examples for Fibonacci Numbers

The Fibonacci Number problem is to compute the nth number in the Fibonacci sequence. The sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones.

To solve this problem using dynamic programming, we can use an iterative approach to store previously computed Fibonacci numbers. This avoids the exponential time complexity associated with a naive recursive solution. The idea is to build up the solution incrementally by storing the results of subproblems in an array.

def fibonacci(n):
  if n <= 1:
    return n
  dp = [0] * (n + 1)
  dp[0], dp[1] = 0, 1
  for i in range(2, n + 1):
    dp[i] = dp[i - 1] + dp[i - 2]
  return dp[n]

In this implementation:

• Base case: If n is less than or equal to 1, return n.
• Create an array `dp` of size n + 1 and initialize the first two elements as 0 and 1.
• Iterate from 2 to n, filling in each element by summing the previous two elements in the array.
• Return the last element of the `dp` array, which contains the Fibonacci number at position n.

The time complexity of this dynamic programming solution is O(n), where n is the input value. This is because we perform a single pass through the array to compute each Fibonacci number, and the space complexity is also O(n) due to the storage of intermediate results in the `dp` array.