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The binary search method utilizes the principle that for a number n to be a perfect square, there exists an integer x such that x * x = n. By searching between 1 and n, binary search will narrow down the possibilities by checking the middle value, squaring it, and adjusting the search range based on whether the squared value is less than, greater than, or equal to n. This ensures a time complexity of O(log n).
Time Complexity: O(log n)
Space Complexity: O(1)
1public class Solution {
2 public bool IsPerfectSquare(int num) {
3 long left = 1, right = num;
4 while (left <= right) {
5 long mid = left + (right - left) / 2;
6 long square = mid * mid;
7 if (square == num) return true;
8 if (square < num) left = mid + 1;
9 else right = mid - 1;
10 }
11 return false;
12 }
13}The C# solution holds strong to the binary search principles typing long for larger ranges, verifying roots square towards the number.
Newton's method is an iterative numerical approach to approximate solutions to functions. For finding a square root, the function is designed around n = x^2. By iteratively refining the guess x using x = (x + n/x) / 2, the method converges to an integer square root if one exists. This method can be faster than binary search with careful tuning.
Time Complexity: O(log n)
Space Complexity: O(1)
1JavaScript leverages Math function support for approximating root determination iteratively through Newton's Method adjustments in numerical calculation stability.