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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)
1class Solution {
2public:
3 bool isPerfectSquare(int num) {
4 long left = 1, right = num;
5 while (left <= right) {
6 long mid = left + (right - left) / 2;
7 long square = mid * mid;
8 if (square == num) return true;
9 if (square < num) left = mid + 1;
10 else right = mid - 1;
11 }
12 return false;
13 }
14};This C++ implementation mirrors the logic of binary search, looping with mid-point square checks against the number until confirmed a perfect square or intervals overlap to false.
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)
1Starts with an initial guess and iteratively applies Newton's formula to converge the guess to the correct number, utilizing standard while and arithmetic operations.