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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)
1#include <stdbool.h>
2bool isPerfectSquare(int num) {
3 int 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}The C implementation uses two pointers, left and right, to represent the range of potential roots. The mid value is checked if its square equals the input number. The range is adjusted by moving the left or right pointer, eventually finding the square or exhausting possibilities.
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)
1Java utilizes the same principles, employing iterative refinement to the potential square root using the core of Newton's approximation effectively.