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492. Construct the Rectangle

https://leetcode.com/problems/construct-the-rectangle/description/

Description

For a web developer, it is very important to know how to design a web page's size. So, given a specific rectangular web page’s area, your job by now is to design a rectangular web page, whose length L and width W satisfy the following requirements:

  1. The area of the rectangular web page you designed must equal to the given target area.

  2. The width W should not be larger than the length L, which means L >= W.

  3. The difference between length L and width W should be as small as possible.
    You need to output the length L and the width W of the web page you designed in sequence.

Example:

Input: 4
Output: [2, 2]
Explanation: The target area is 4, and all the possible ways to construct it are [1,4], [2,2], [4,1].
But according to requirement 2, [1,4] is illegal; according to requirement 3, [4,1] is not optimal compared to [2,2]. So the length L is 2, and the width W is 2.

Note:

Solution

版本1:

class Solution:
    def constructRectangle(self, area):
        """
        :type area: int
        :rtype: List[int]
        """
        import math
        L = math.ceil(area**.5)
        while area%L:
            L += 1
        return L,area//L

该算法在 LeetCode 上测试的结果为 1596 ms。

非常耗时,因为在以 k=areak=\sqrt{area} 为分界点,前后能够整除 areaarea 的因子个数是相同的,设为 nn 个,而前面的因子的密度为 n/kn/k,而后面的因子的密度约为 n/(k2k)n/(k^2-k),而 k<<k2kk << k^2-k,当 areaarea 较大时,搜寻 kk 之前的部分效率远比搜寻 kk 之后的部分要高。

版本2:

class Solution:
    def constructRectangle(self, area):
        """
        :type area: int
        :rtype: List[int]
        """
        W = int(area**.5)
        while area%W:
            W -= 1
        return [area//W,W]

该算法在 LeetCode 上测试的结果为 60 ms,与上一版本相比快了很多。

版本3:

class Solution:
    def constructRectangle(self, area):
        """
        :type area: int
        :rtype: List[int]
        """
        import math
        W = math.floor(area**.5)
        while area%W:
            W -= 1
        return [area//W,W]

该算法在 LeetCode 上测试的结果为 40 ms,可见使用 math 的 floor() 运算比 int() 要高效很多。