Sending a Sequence Over the Network
Contest: Codeforces Round 826 (Div. 3)
Difficulty rating: 1600
Tags: dp

The sequence $$$a$$$ is sent over the network as follows:

1. sequence $$$a$$$ is split into segments (each element of the sequence belongs to exactly one segment, each segment is a group of consecutive elements of sequence);
2. for each segment, its length is written next to it, either to the left of it or to the right of it;
3. the resulting sequence $$$b$$$ is sent over the network.

For example, we needed to send the sequence $$$a = [1, 2, 3, 1, 2, 3]$$$. Suppose it was split into segments as follows: $$$[\color{red}{1}] + [\color{blue}{2, 3, 1}] + [\color{green}{2, 3}]$$$. Then we could have the following sequences:

- $$$b = [1, \color{red}{1}, 3, \color{blue}{2, 3, 1}, \color{green}{2, 3}, 2]$$$,
- $$$b = [\color{red}{1}, 1, 3, \color{blue}{2, 3, 1}, 2, \color{green}{2, 3}]$$$,
- $$$b = [\color{red}{1}, 1, \color{blue}{2, 3, 1}, 3, 2, \color{green}{2, 3}]$$$,
- $$$b = [\color{red}{1}, 1,\color{blue}{2, 3, 1}, 3, \color{green}{2, 3}, 2]$$$.

If a different segmentation had been used, the sent sequence might have been different.

The sequence $$$b$$$ is given. Could the sequence $$$b$$$ be sent over the network? In other words, is there such a sequence $$$a$$$ that converting $$$a$$$ to send it over the network could result in a sequence $$$b$$$?

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

Each test case consists of two lines.

The first line of the test case contains an integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the size of the sequence $$$b$$$.

The second line of test case contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 10^9$$$) — the sequence $$$b$$$ itself.

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

Output format:
For each test case print on a separate line:

- YES if sequence $$$b$$$ could be sent over the network, that is, if sequence $$$b$$$ could be obtained from some sequence $$$a$$$ to send $$$a$$$ over the network.
- NO otherwise.

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

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first case, the sequence $$$b$$$ could be obtained from the sequence $$$a = [1, 2, 3, 1, 2, 3]$$$ with the following partition: $$$[\color{red}{1}] + [\color{blue}{2, 3, 1}] + [\color{green}{2, 3}]$$$. The sequence $$$b$$$: $$$[\color{red}{1}, 1, \color{blue}{2, 3, 1}, 3, 2, \color{green}{2, 3}]$$$.

In the second case, the sequence $$$b$$$ could be obtained from the sequence $$$a = [12, 7, 5]$$$ with the following partition: $$$[\color{red}{12}] + [\color{green}{7, 5}]$$$. The sequence $$$b$$$: $$$[\color{red}{12}, 1, 2, \color{green}{7, 5}]$$$.

In the third case, the sequence $$$b$$$ could be obtained from the sequence $$$a = [7, 8, 9, 10, 3]$$$ with the following partition: $$$[\color{red}{7, 8, 9, 10, 3}]$$$. The sequence $$$b$$$: $$$[5, \color{red}{7, 8, 9, 10, 3}]$$$.

In the fourth case, there is no sequence $$$a$$$ such that changing $$$a$$$ for transmission over the network could produce a sequence $$$b$$$.

Example input:
7
9
1 1 2 3 1 3 2 2 3
5
12 1 2 7 5
6
5 7 8 9 10 3
4
4 8 6 2
2
3 1
10
4 6 2 1 9 4 9 3 4 2
1
1

Example output:
YES
YES
YES
NO
YES
YES
NO

Editorial:
Let's introduce the dynamics. $$$dp[i] = true$$$ if on the prefix $$$i$$$ the answer is Yes.
Then in this sequence $$$b$$$ the numbers corresponding to the sizes of the segments from the partition $$$a$$$ into subsegments will be called interesting.
A number at position $$$i$$$ in the sequence $$$b$$$, if it is interesting, is either to the right or to the left of the segment.
• If it is to the left of the segment, it can only be interesting if $$$dp[i-1] = true$$$. Then $$$dp[i + b[i]] = true$$$.
• If it is on the right side of the segment, then if $$$dp[i - b[i] - 1] = true$$$, then $$$dp[i] = true$$$.
The answer for the whole sequence is Yes if $$$dp[n] = true$$$.