Subsequence Addition (Hard Version)

2000ms

262144K

Description:

The only difference between the two versions is that in this version, the constraints are higher.

Initially, array $$$a$$$ contains just the number $$$1$$$. You can perform several operations in order to change the array. In an operation, you can select some subsequence$$$^{\dagger}$$$ of $$$a$$$ and add into $$$a$$$ an element equal to the sum of all elements of the subsequence.

You are given a final array $$$c$$$. Check if $$$c$$$ can be obtained from the initial array $$$a$$$ by performing some number (possibly 0) of operations on the initial array.

$$$^{\dagger}$$$ A sequence $$$b$$$ is a subsequence of a sequence $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by the deletion of several (possibly zero, but not all) elements. In other words, select $$$k$$$ ($$$1 \leq k \leq |a|$$$) distinct indices $$$i_1, i_2, \dots, i_k$$$ and insert anywhere into $$$a$$$ a new element with the value equal to $$$a_{i_1} + a_{i_2} + \dots + a_{i_k}$$$.

Input:

The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) — the number of elements the final array $$$c$$$ should have.

The second line of each test case contains $$$n$$$ space-separated integers $$$c_i$$$ ($$$1 \leq c_i \leq 2 \cdot 10^5$$$) — the elements of the final array $$$c$$$ that should be obtained from the initial array $$$a$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, output "YES" (without quotes) if such a sequence of operations exists, and "NO" (without quotes) otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes" and "YES" will be recognized as a positive answer).

Sample Input:

Sample Output:

YES
NO
YES
NO
YES
YES

Note:

For the first test case, the initial array $$$a$$$ is already equal to $$$[1]$$$, so the answer is "YES".

For the second test case, performing any amount of operations will change $$$a$$$ to an array of size at least two which doesn't only have the element $$$2$$$, thus obtaining the array $$$[2]$$$ is impossible and the answer is "NO".

For the third test case, we can perform the following operations in order to obtain the final given array $$$c$$$:

Initially, $$$a = [1]$$$.
By choosing the subsequence $$$[1]$$$, and inserting $$$1$$$ in the array, $$$a$$$ changes to $$$[1, 1]$$$.
By choosing the subsequence $$$[1, 1]$$$, and inserting $$$1+1=2$$$ in the middle of the array, $$$a$$$ changes to $$$[1, 2, 1]$$$.
By choosing the subsequence $$$[1, 2]$$$, and inserting $$$1+2=3$$$ after the first $$$1$$$ of the array, $$$a$$$ changes to $$$[1, 3, 2, 1]$$$.
By choosing the subsequence $$$[1, 3, 1]$$$ and inserting $$$1+3+1=5$$$ at the beginning of the array, $$$a$$$ changes to $$$[5, 1, 3, 2, 1]$$$ (which is the array we needed to obtain).

Informação

Provedor Codeforces

Código CF1807G2