Codeforces Round #775 (Div. 2,based on Moscow Open Olympiad in Informatics) - D. Integral Array - 题解

Posted 2022-03-10 Tisfy

目录

• • Codeforces Round #775 (Div. 2, based on Moscow Open Olympiad in Informatics)-D. Integral Array
  • • Problem Description
    • Input
    • Output
    • Sample Input
    • Sample Onput
    • Note
• 题目大意
• 解题思路
• AC代码

Codeforces Round #775 (Div. 2, based on Moscow Open Olympiad in Informatics)-D. Integral Array

传送门
Time Limit: 2 seconds
Memory Limit: 512 megabytes

Problem Description

You are given an array $a$ of $n$ positive integers numbered from $1$ to $n$. Let’s call an array integral if for any two, not necessarily different, numbers $x$ and $y$ from this array, $x\ge y$, the number $\lfloor \frac{x}{y}\rfloor$ ($x$ divided by $y$ with rounding down) is also in this array.

You are guaranteed that all numbers in $a$ do not exceed $c$. Your task is to check whether this array is integral.

Input

The input consists of multiple test cases. The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. Description of the test cases follows.

The first line of each test case contains two integers $n$ and $c$ ($1\le n\le 10^6$, $1\le c\le 10^6$) — the size of $a$ and the limit for the numbers in the array.

The second line of each test case contains $n$ integers $a_1$, $a_2$, …, $a_n$ ($1\le a_i\le c$) — the array $a$.

Let $N$ be the sum of $n$ over all test cases and $C$ be the sum of $c$ over all test cases. It is guaranteed that $N\le 10^6$ and $C\le 10^6$.

Output

For each test case print Yes if the array is integral and No otherwise.

Sample Input

4
3 5
1 2 5
4 10
1 3 3 7
1 2
2
1 1
1

Sample Onput

Yes
No
No
Yes

Note

In the first test case it is easy to see that the array is integral:

Thus, the condition is met and the array is integral.

In the second test case it is enough to see that

$\lfloor \frac{7}{3}\rfloor =\lfloor 2\frac{1}{3}\rfloor =2$, this number is not in $a$, that’s why it is not integral.

In the third test case $\lfloor \frac{2}{2}\rfloor =1$, but there is only $2$ in the array, that’s why it is not integral.

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题目大意

给你一个数组$a$，如果$\forall x,y\in a$且$x\ge y$，都有$\lfloor \frac{x}{y}\rfloor$，那么就输出Yes；否则No。

解题思路

假设 $x,y\in a$并且$r\notin a$，如果$y\cdot r\le x<y\cdot (r+1)$，那么$\lfloor \frac{x}{y}\rfloor$ 就等于 $r$($\notin a$)，因此输出No

所以，对于所有的“不属于$a$的$r$、属于$a$的$y$”，如果存在“$y\cdot r\le x<y\cdot (r+1)$”的$x$，那么就输出No。

也就是说，枚举所有不属于$a$的$r$和所有属于$a$的$y$，对于这个“$r$和$y$”，看是否有属于$[(y\cdot r),(y\cdot (r+1)))$的元素$x$在$a$中，如果有就输出No；如果枚举完也没有不符合条件的$x$，就输出Yes。

具体方法：
第一层循环从$1$到$c$枚举$r$，第二层循环从$1$开始枚举$y$,直到$y\cdot r>c$时结束。对于不存在于$a$中的$r$和存在于$a$中的$y$，查看是否有存在于不可行范围内的$a$。

复杂度分析：
循环复杂度为$O(c+\frac{c}{2}+\frac{c}{3}+\cdots +1)=O(c\log c)$
对于给定的$r$和$y$，可以用前缀和以$O(1)$的复杂度求出一定范围内是否有$x\in a$
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