Division by Two and Permutation

3000ms

262144K

Description:

You are given an array $$$a$$$ consisting of $$$n$$$ positive integers. You can perform operations on it.

In one operation you can replace any element of the array $$$a_i$$$ with $$$\lfloor \frac{a_i}{2} \rfloor$$$, that is, by an integer part of dividing $$$a_i$$$ by $$$2$$$ (rounding down).

See if you can apply the operation some number of times (possible $$$0$$$) to make the array $$$a$$$ become a permutation of numbers from $$$1$$$ to $$$n$$$ —that is, so that it contains all numbers from $$$1$$$ to $$$n$$$, each exactly once.

For example, if $$$a = [1, 8, 25, 2]$$$, $$$n = 4$$$, then the answer is yes. You could do the following:

Replace $$$8$$$ with $$$\lfloor \frac{8}{2} \rfloor = 4$$$, then $$$a = [1, 4, 25, 2]$$$.
Replace $$$25$$$ with $$$\lfloor \frac{25}{2} \rfloor = 12$$$, then $$$a = [1, 4, 12, 2]$$$.
Replace $$$12$$$ with $$$\lfloor \frac{12}{2} \rfloor = 6$$$, then $$$a = [1, 4, 6, 2]$$$.
Replace $$$6$$$ with $$$\lfloor \frac{6}{2} \rfloor = 3$$$, then $$$a = [1, 4, 3, 2]$$$.

Input:

The first line of input data contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) —the number of test cases.

Each test case contains exactly two lines. The first one contains an integer $$$n$$$ ($$$1 \le n \le 50$$$), the second one contains integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

Output:

For each test case, output on a separate line:

YES if you can make the array $$$a$$$ become a permutation of numbers from $$$1$$$ to $$$n$$$,
NO otherwise.

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

Sample Input:

6
4
1 8 25 2
2
1 1
9
9 8 3 4 2 7 1 5 6
3
8 2 1
4
24 7 16 7
5
22 6 22 4 22

Sample Output:

YES
NO
YES
NO
NO
YES

Note:

The first test case is explained in the text of the problem statement.

In the second test case, it is not possible to get a permutation.

Informação

Provedor Codeforces

Código CF1624C