Equate Multisets

题目描述

Multiset —is a set of numbers in which there can be equal elements, and the order of the numbers does not matter. Two multisets are equal when each value occurs the same number of times. For example, the multisets {2,2,4}\{2,2,4\}{2,2,4} and {2,4,2}\{2,4,2\}{2,4,2} are equal, but the multisets {1,2,2}\{1,2,2\}{1,2,2} and {1,1,2}\{1,1,2\}{1,1,2} — are not.

You are given two multisets aaa and bbb , each consisting of nnn integers.

In a single operation, any element of the bbb multiset can be doubled or halved (rounded down). In other words, you have one of the following operations available for an element xxx of the bbb multiset:

replace xxx with x⋅2x \cdot 2x⋅2 ,
or replace xxx with ⌊x2⌋\lfloor \frac{x}{2} \rfloor⌊2x⌋ (round down).

Note that you cannot change the elements of the aaa multiset.

See if you can make the multiset bbb become equal to the multiset aaa in an arbitrary number of operations (maybe 000 ).

For example, if n=4n = 4n=4 , a={4,24,5,2}a = \{4, 24, 5, 2\}a={4,24,5,2} , b={4,1,6,11}b = \{4, 1, 6, 11\}b={4,1,6,11} , then the answer is yes. We can proceed as follows:

Replace 111 with 1⋅2=21 \cdot 2 = 21⋅2=2 . We get b={4,2,6,11}b = \{4, 2, 6, 11\}b={4,2,6,11} .
Replace 111111 with ⌊112⌋=5\lfloor \frac{11}{2} \rfloor = 5⌊211⌋=5 . We get b={4,2,6,5}b = \{4, 2, 6, 5\}b={4,2,6,5} .
Replace 666 with 6⋅2=126 \cdot 2 = 126⋅2=12 . We get b={4,2,12,5}b = \{4, 2, 12, 5\}b={4,2,12,5} .
Replace 121212 with 12⋅2=2412 \cdot 2 = 2412⋅2=24 . We get b={4,2,24,5}b = \{4, 2, 24, 5\}b={4,2,24,5} .
Got equal multisets a={4,24,5,2}a = \{4, 24, 5, 2\}a={4,24,5,2} and b={4,2,24,5}b = \{4, 2, 24, 5\}b={4,2,24,5} .

输入格式

The first line of input data contains a single integer ttt ( 1≤t≤1041 \le t \le 10^41≤t≤104 ) —the number of test cases.

Each test case consists of three lines.

The first line of the test case contains an integer nnn ( 1≤n≤2⋅1051 \le n \le 2 \cdot 10^51≤n≤2⋅105 ) —the number of elements in the multisets aaa and bbb .

The second line gives nnn integers: a1,a2,…,ana_1, a_2, \dots, a_na1,a2,…,an ( 1≤a1≤a2≤⋯≤an≤1091 \le a_1 \le a_2 \le \dots \le a_n \le 10^91≤a1≤a2≤⋯≤an≤109 ) —the elements of the multiset aaa . Note that the elements may be equal.

The third line contains nnn integers: b1,b2,…,bnb_1, b_2, \dots, b_nb1,b2,…,bn ( 1≤b1≤b2≤⋯≤bn≤1091 \le b_1 \le b_2 \le \dots \le b_n \le 10^91≤b1≤b2≤⋯≤bn≤109 ) — elements of the multiset bbb . Note that the elements may be equal.

It is guaranteed that the sum of nnn values over all test cases does not exceed 2⋅1052 \cdot 10^52⋅105 .

输出格式

For each test case, print on a separate line:

YES if you can make the multiset bbb become equal to aaa ,
NO otherwise.

You can output YES and NO in any case (for example, strings yEs, yes, Yes and YES will be recognized as positive answer).

样例 #1

样例输入 #1

5
4
2 4 5 24
1 4 6 11
3
1 4 17
4 5 31
5
4 7 10 13 14
2 14 14 26 42
5
2 2 4 4 4
28 46 62 71 98
6
1 2 10 16 64 80
20 43 60 74 85 99

样例输出 #1

YES
NO
YES
YES
YES

提示

The first example is explained in the statement.

In the second example, it is impossible to get the value $ 31 $ from the numbers of the multiset $ b $ by available operations.

In the third example, we can proceed as follows:

Replace 222 with 2⋅2=42 \cdot 2 = 42⋅2=4 . We get b={4,14,14,26,42}b = \{4, 14, 14, 26, 42\}b={4,14,14,26,42} .
Replace 141414 with ⌊142⌋=7\lfloor \frac{14}{2} \rfloor = 7⌊214⌋=7 . We get b={4,7,14,26,42}b = \{4, 7, 14, 26, 42\}b={4,7,14,26,42} .
Replace 262626 with ⌊262⌋=13\lfloor \frac{26}{2} \rfloor = 13⌊226⌋=13 . We get b={4,7,14,13,42}b = \{4, 7, 14, 13, 42\}b={4,7,14,13,42} .
Replace 424242 with ⌊422⌋=21\lfloor \frac{42}{2} \rfloor = 21⌊242⌋=21 . We get b={4,7,14,13,21}b = \{4, 7, 14, 13, 21\}b={4,7,14,13,21} .
Replace 212121 with ⌊212⌋=10\lfloor \frac{21}{2} \rfloor = 10⌊221⌋=10 . We get b={4,7,14,13,10}b = \{4, 7, 14, 13, 10\}b={4,7,14,13,10} .
Got equal multisets a={4,7,10,13,14}a = \{4, 7, 10, 13, 14\}a={4,7,10,13,14} and b={4,7,14,13,10}b = \{4, 7, 14, 13, 10\}b={4,7,14,13,10} .

中文翻译

给你两个长度为 nnn 的数组 aaa 和数组 bbb ，定义x的一次操作（二选一）：：：

把 xxx 替换成 x⋅2x \cdot 2x⋅2 ,
把 xxx 替换成 ⌊x2⌋\lfloor \frac{x}{2} \rfloor⌊2x⌋ (下取整).

aaa 数组不能做操作， bbb 数组不能做操作。

问经过若干操作，能否讲 bbb 数组变成 aaa 数组？（数的顺序可以不一样）。

题解

先把 aaa 数组一直除以 222 (能除得尽就除），用一个 mapmapmap 统计个数，然后贪心地处理每个 bib_ibi ,能用就用。

#include<bits/stdc++.h>
using namespace std;
int n,a[200010],b[200010];
map<int,int> mp;
void solve(){mp.clear();cin>>n;for(int i=1;i<=n;i++){cin>>a[i];while(a[i]%2==0){a[i]/=2;}mp[a[i]]++;}for(int i=1;i<=n;i++)cin>>b[i];for(int i=1;i<=n;i++){int tmp=b[i];bool flag=0;while(tmp){if(!mp[tmp]){tmp/=2;continue;}mp[tmp]--;flag=1;break;}if(!flag){puts("NO");return ;}}for(auto it:mp){if(it.second>0){puts("NO");}}puts("YES");return ;
}
int main(){int TTT;cin>>TTT;while(TTT--){solve();}return 0;
}

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