Problem Statement
The task is to calculate x raised to the power of n (i.e., x^n) using recursion.
Definition
- The power function calculates
x raised to the power n where x is the base and n is the exponent.
- Formula:
x^n = x imes x imes ... imes x (n times)
Example Inputs
Explanation: 2^3 = 2 imes 2 imes 2 = 8.
Approach - Use Recursion
Approach
Recursion is a technique in which a function calls itself to break a problem into smaller sub-problems. For the power function, the idea is to calculate x^n by multiplying x by x^{n-1} recursively, until the exponent becomes 0 (the base case).
Steps
- If
n = 0, return 1 (base case: any number raised to the power 0 is 1).
- If
n > 0, recursively calculate x^n = x imes x^{n-1}.
- Return the result.
Time & Space Complexity
- Time Complexity: O(n), where
n is the exponent. We make n recursive calls until the exponent becomes 0.
- Space Complexity: O(n), due to the recursion stack. Each recursive call adds a layer to the stack.
Code Snippet
Dry Run
Let’s dry run the code with the input x = 2 and n = 3.
- power(2, 3): 2 * power(2, 2)
- power(2, 2): 2 * power(2, 1)
- power(2, 1): 2 * power(2, 0)
- power(2, 0): 1 (base case)
- Return from power(2, 1): 2 * 1 = 2
- Return from power(2, 2): 2 * 2 = 4
- Return from power(2, 3): 2 * 4 = 8
Output:
Complexity Analysis
Time and Space Complexity
- Time Complexity: O(n) because we make
n recursive calls.
- Space Complexity: O(n) due to the recursion stack. The depth of the recursion will be
n before reaching the base case.
Conclusion
Using recursion to calculate x^n is a simple and intuitive approach. The recursive method breaks the problem into smaller subproblems by reducing the exponent at each step. However, for larger values of n, the time complexity can be inefficient. An optimized solution could involve using the method of exponentiation by squaring, which reduces the time complexity to O(log n).