Problem Statement
The Fibonacci sequence is defined as follows: F(0) = 0, F(1) = 1, and for n > 1, F(n) = F(n-1) + F(n-2). The task is to compute the nth Fibonacci number efficiently.
Algorithmic Approach / Logic
To solve this problem using dynamic programming, we can use memoization. Memoization stores previously computed values of the Fibonacci sequence to avoid redundant calculations. This approach significantly reduces the time complexity compared to a naive recursive solution.
Code Implementation (Python)
def fibonacci(n):
# Base cases
if n == 0:
return 0
elif n == 1:
return 1
else:
memo = [None] * (n + 1)
memo[0] = 0
memo[1] = 1
for i in range(2, n + 1):
memo[i] = memo[i - 1] + memo[i - 2]
return memo[n]Complexity Analysis
The time complexity of the dynamic programming solution is O(n), where n is the position in the Fibonacci sequence, as each number is computed once. The space complexity is also O(n) due to the use of a memoization array.