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The recursive approach involves directly applying the Fibonacci formula: F(n) = F(n-1) + F(n-2). This approach naturally follows the definition of the Fibonacci sequence, solving subproblems by calling itself. However, this naive implementation leads to overlapping subproblems and higher time complexity.
This recursive solution is easy to implement but inefficient for larger n due to exponential time complexity.
Time Complexity: O(2^n) because it grows exponentially, with overlapping subproblems.
Space Complexity: O(n) due to the recursion stack required by function calls.
1function fibonacci(n) {
2 if (n <= 1) return n;
3 return fibonacci(n - 1) + fibonacci(n - 2);
4}
5
6let n = 4;
7console.log(fibonacci(n));This JavaScript function defines 'fibonacci' recursively computing the nth Fibonacci number. It checks for the base case and performs recursive calls analogous to other language solutions.
The dynamic programming approach involves using an array to store previously computed Fibonacci numbers, which prevents redundant calculations found in the recursive approach. This method significantly improves efficiency by transforming the exponential recursive solution into a linear solution, using merely linear space.
Time Complexity: O(n) due to linear iteration.
Space Complexity: O(n) for the array storing Fibonacci numbers.
1
In this C solution, an array 'fib' is used to store Fibonacci numbers up to n. The base cases are initialized, and the array is filled iteratively, computing each Fibonacci number from the two preceding numbers, achieving a linear time complexity.