1968: [USACO 2022 US Open Contest Gold] Problem 3. Balancing a Tree

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 $1\dots N$, 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.

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

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 ${10}^5$.

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 $s_1,s_2,\dots ,s_N$ containing an assignment of spots that achieves the above imbalance. Any valid assignment will be accepted.

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

3
1
4