Problem 3. Balancing a Tree

USACO 2022 US Open Contest, Gold

Farmer John has conducted an extensive study of the evolution of different cow breeds. The result is a rooted tree with $N$ ($2\le N\le 10^5$) nodes labeled $1\ldots N$, each node corresponding to a cow breed. For each $i\in [2,N]$, the parent of node $i$ is node $p_i$ ($1\le p_i<i$), meaning that breed $i$ evolved from breed $p_i$. A node $j$ is called an ancestor of node $i$ if $j=p_i$ or $j$ is an ancestor of $p_i$.

Every node $i$ in the tree is associated with a breed having an integer number of spots $s_i$. The "imbalance" of the tree is defined to be the maximum of $|s_i-s_j|$ over all pairs of nodes $(i,j)$ such that $j$ is an ancestor of $i$.

Farmer John doesn't know the exact value of $s_i$ for each breed, but he knows lower and upper bounds on these values. Your job is to assign an integer value of $s_i \in [l_i,r_i]$ ($0\le l_i\le r_i\le 10^9$) to each node such that the imbalance of the tree is minimized.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $T$ ($1\le T\le 10$), the number of independent test cases to be solved, and an integer $B\in \{0,1\}$.

Each test case starts with a line containing $N$, followed by $N-1$ integers $p_2,p_3,\ldots,p_N$.

The next $N$ lines each contain two integers $l_i$ and $r_i$.

It is guaranteed that the sum of $N$ over all test cases does not exceed $10^5$.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output one or two lines, depending on the value of $B$.

The first line for each test case should contain the minimum imbalance.

If $B=1,$ then print an additional line with $N$ space-separated integers $s_1,s_2,\ldots, s_N$ containing an assignment of spots that achieves the above imbalance. Any valid assignment will be accepted.

SAMPLE INPUT:

3 0
3
1 1
0 100
1 1
6 7
5
1 2 3 4
6 6
1 6
1 6
1 6
5 5
3
1 1
0 10
0 1
9 10

SAMPLE OUTPUT:

3
1
4

For the first test case, the minimum imbalance is $3$. One way to achieve imbalance $3$ is to set $[s_1,s_2,s_3]=[4,1,7]$.

SAMPLE INPUT:

3 1
3
1 1
0 100
1 1
6 7
5
1 2 3 4
6 6
1 6
1 6
1 6
5 5
3
1 1
0 10
0 1
9 10

SAMPLE OUTPUT:

3
3 1 6
1
6 5 5 5 5
4
5 1 9

This input is the same as the first one aside from the value of $B$. Another way to achieve imbalance $3$ is to set $[s_1,s_2,s_3]=[3,1,6]$.

SCORING: Test cases 3-4 satisfy $l_i=r_i$ for all $i$.Test cases 5-6 satisfy $p_i=i-1$ for all $i$.Test cases 7-16 satisfy no additional constraints. Within each subtask, the first half of the test cases will satisfy $B=0$, and the rest will satisfy $B=1$.

Problem credits: Andrew Wang