[LeetCode] Valid Perfect Square 检验完全平方数

时间:2022-12-24 01:11:44

Given a positive integer num, write a function which returns True if num is a perfect square else False.

Note: Do not use any built-in library function such as sqrt.

Example 1:

Input: 16
Returns: True

Example 2:

Input: 14
Returns: False

Credits:
Special thanks to @elmirap for adding this problem and creating all test cases.

这道题给了我们一个数,让我们判断其是否为完全平方数,那么显而易见的是,肯定不能使用 brute force,这样太不高效了,那么最小是能以指数的速度来缩小范围,那么我最先想出的方法是这样的,比如一个数字 49,我们先对其除以2,得到 24,发现 24 的平方大于 49,那么再对 24 除以2,得到 12,发现 12 的平方还是大于 49,再对 12 除以2,得到6,发现6的平方小于 49,于是遍历6到 12 中的所有数,看有没有平方等于 49 的,有就返回 true,没有就返回 false,参见代码如下:

解法一:

class Solution {
public:
bool isPerfectSquare(int num) {
if (num == ) return true;
long x = num / , t = x * x;
while (t > num) {
x /= ;
t = x * x;
}
for (int i = x; i <= * x; ++i) {
if (i * i == num) return true;
}
return false;
}
};

下面这种方法也比较高效,从1搜索到 sqrt(num),看有没有平方正好等于 num 的数:

解法二:

class Solution {
public:
bool isPerfectSquare(int num) {
for (int i = ; i <= num / i; ++i) {
if (i * i == num) return true;
}
return false;
}
};

我们也可以使用二分查找法来做,要查找的数为 mid*mid,参见代码如下:

解法三:

class Solution {
public:
bool isPerfectSquare(int num) {
long left = , right = num;
while (left <= right) {
long mid = left + (right - left) / , t = mid * mid;
if (t == num) return true;
if (t < num) left = mid + ;
else right = mid - ;
}
return false;
}
};

下面这种方法就是纯数学解法了,利用到了这样一条性质,完全平方数是一系列奇数之和,例如:

1 = 1
4 = 1 + 3
9 = 1 + 3 + 5
16 = 1 + 3 + 5 + 7
25 = 1 + 3 + 5 + 7 + 9
36 = 1 + 3 + 5 + 7 + 9 + 11
....
1+3+...+(2n-1) = (2n-1 + 1)n/2 = n*n

这里就不做证明了,我也不会证明,知道了这条性质,就可以利用其来解题了,时间复杂度为 O(sqrt(n))。

解法四:

class Solution {
public:
bool isPerfectSquare(int num) {
int i = ;
while (num > ) {
num -= i;
i += ;
}
return num == ;
}
};

下面这种方法是第一种方法的类似方法,更加精简了,时间复杂度为 O(lgn):

解法五:

class Solution {
public:
bool isPerfectSquare(int num) {
long x = num;
while (x * x > num) {
x = (x + num / x) / ;
}
return x * x == num;
}
};

这道题其实还有 O(1) 的解法,这你敢信?简直太丧心病狂了,详情请参见论坛上的这个帖子

Github 同步地址:

https://github.com/grandyang/leetcode/issues/367

类似题目:

Sqrt(x)

参考资料:

https://leetcode.com/problems/valid-perfect-square/

https://leetcode.com/problems/valid-perfect-square/discuss/83872/O(1)-time-c%2B%2B-solution-inspired-by-Q_rsqrt

https://leetcode.com/problems/valid-perfect-square/discuss/83874/A-square-number-is-1%2B3%2B5%2B7%2B...-JAVA-code

https://leetcode.com/problems/valid-perfect-square/discuss/83902/Java-Three-Solutions-135..-SequenceBinary-SearchNewton

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