Pow(x, n)
Problem Description
Implement pow(x, n), which calculates x raised to the power n (i.e., xn).
Example 1:
Input: x = 2.00000, n = 10
Output: 1024.00000
Example 2:
Input: x = 2.10000, n = 3
Output: 9.26100
Example 3:
Input: x = 2.00000, n = -2
Output: 0.25000
Explanation: 2-2 = 1/22 = 1/4 = 0.25
Constraints:
- -100.0 < x < 100.0
- -231 <= n <= 231-1
- n is an integer.
- Either x is not zero or n > 0.
- -104 <= xn <= 104
Solve it on LeetCode
Approaches
1. Iterative Multiplication
Intuition:
The simplest way to calculate x^n is to multiply x by itself n times. While this approach is straightforward, it's very inefficient for large values of n due to its linear complexity. Also, handling negative powers involves taking the reciprocal of positive power results.
Code:
- Time Complexity: O(n) - The loop runs
ntimes. - Space Complexity: O(1) - No additional space is used.
2. Recursive Approach with Divide and Conquer
Intuition:
The key observation is that x^n can be broken down recursively:
- If
nis even,x^n = x^(n/2) * x^(n/2) - If
nis odd,x^n = x * x^(n//2) * x^(n//2)
This approach divides the problem size by half at each step, leading to a more efficient solution.
Code:
- Time Complexity: O(log n) - Recursive function halves
nat each call. - Space Complexity: O(log n) - Call stack space for recursion.
3. Iterative Binary Exponentiation (Optimal)
Intuition:
The iterative method of binary exponentiation is similar to the recursive method, but it avoids the overhead of recursion. We use bit manipulation to quickly determine if the current power n is odd and needs additional multiplication by x. This results in efficient exponentiation with reduced time complexity.
Code:
- Time Complexity: O(log n) - The powering operation uses logarithmic number of multiplications.
- Space Complexity: O(1) - This approach does not use additional space.