[Leetcode series] [algorithm] [simple] effective perfect square (Newton iteration)

topic:

Title link:  https://leetcode-cn.com/problems/valid-perfect-square/

 

Problem-solving ideas:

Method 1: Dichotomy

Find from the half of num, then find the middle number every half

 

Method 2: Newton iteration

As long as it is a continuous function, and find one of the points that meet the requirements, in most cases, you can use Newton iteration

For this problem, the function expression can be understood as:

Where x is the value we want to request and num is the value entered by the program

The iterative idea of ​​Newton's method is as follows:

1. Pick a value to start iteration
2. Calculating the tangent slope at
3. Find the slope , and the straight line passing through the point , the x coordinate of the intersection point with the x axis , that is, find the solution to the equation:
   1. Represents the function of the left and right image translation units, high school (middle school?) Knowledge, plus the left and right cut, so that the purpose of the function when the image , the linear function
   2. Represents the slope of the tangent equation
   3. For the required value this time, the converted expression is:
   4. For this problem ,, bring the above formula to get:
4. Use to continue iterating until you find a solution that makes the original formula

Borrowing the wiki animation, the whole process is as follows:

 

Code:

method one:

class Solution:
    def isPerfectSquare(self, num: int) -> bool:
        if num < 2:
            return True
        
        left, right = 0, num // 2
        while left <= right:
            mid = (left + right) // 2
            if mid == 0:
                left = mid + 1
                continue
                
            tmp = num // mid
            mod = num % mid
            if tmp == mid and mod == 0:
                return True
            elif tmp > mid:
                left = mid + 1
            else:
                right = mid - 1
                
        return False

 

Method Two:

class Solution:
    def isPerfectSquare(self, num: int) -> bool:
        if num < 2:
            return True
        
        x = num // 2
        while x * x > num:
            x = (x + num // x) // 2
            
        return x * x == num