Background

Translated from ROIR 2021 Day 1 T4 Антенна。

Problem Description

There are $n$ ropes. The $i$-th rope has length $s_i$ cm and has $m_i$ nodes. The $j$-th node is located at $p_{i,j}$ cm from the left endpoint of the rope.

Try to construct a plan to connect the ropes from left to right. Let the order of ropes in this plan be $q$. Clearly, $q$ is a permutation of $1 \sim n$, and it must satisfy the following requirement: after connecting the right endpoint of rope $q_i$ to the left endpoint of rope $q_{i+1}$ $(1\le i<n)$, the distances between adjacent nodes are equal.

Obviously, there may be no valid plan. In this case, output No.

Input Format

The first line contains an integer $n$.

The next $2\times n$ lines are:

• Line $2\times i(1\le i\le n)$ contains two integers $m_i$ and $s_i$.
• Line $2\times i+1(1\le i\le n)$ contains $m_i$ integers $p_{i,j}$.

Output Format

If a plan can be constructed, output Yes, and then output one more line containing $n$ integers $q_i$.

If there is no solution, output No.

3
1 7
3
1 8
6
2 8
1 6

Yes
2 1 3

1
1 7
5

Yes
1

1
3 10
2 5 9

No

3
1 5
3
1 3
3
1 6
3

No

4
1 5
0
1 0
0
1 3
3
1 0
0

Yes
3 2 4 1

Hint

[Sample Explanation 1]:

[Constraints]:

For all subtasks, $1\le n\le 10^5$, $1\le m_i\le 10^5$, $0\le s_i\le 10^9$, $0\le p_{i,1}<p_{i,2}<\cdots<p_{i,m_i}\le s_i$, $\sum m_i\le 10^5$.

Translated by ChatGPT 5