Famous Computer Science Algorithms - Full Course

Core recursion structure

Recursive algorithms require defining base cases where answers are known (e.g., Fibonacci terms 1 and 2 equal 1) and recursive cases where the function calls itself to break problems into smaller subproblems.

Divide and conquer strategy

The Fibonacci sequence demonstrates how f(n) = f(n-1) + f(n-2) breaks complex calculations into smaller subproblems that build upon base cases to solve the larger problem.

Basic implementation

A naive recursive Fibonacci function checks if n ≤ 2 to return 1 (base case), otherwise returns fib(n-1) + fib(n-2), causing the function to call itself repeatedly until reaching the base cases.

Exponential time complexity

The naive recursive approach runs in O(2^n) time, becoming infeasible for inputs as small as n=70 due to redundant calculations of identical Fibonacci terms across different branches.

Memoization caching

Storing computed results in a cache object reduces time complexity to O(n) by checking if the value exists before computing, eliminating redundant recursive calls and enabling instant execution for large inputs.

Space trade-offs

While memoization improves time performance to O(n), it requires O(n) space complexity to store the cache of previously calculated values in memory.

Iterative conversion

Replacing recursion with a for loop that iterates from 2 to n eliminates the function call stack overhead while maintaining O(n) time complexity.

Constant space optimization

Tracking only the previous two Fibonacci terms (last and secondLast) reduces space complexity from O(n) to O(1), requiring memory for just two variables regardless of input size.

Production-ready solution

The iterative approach achieves optimal O(n) time with O(1) space, making it the preferred implementation for production environments handling large datasets.