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 $1\ldots N$, 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 does not 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 $s_i\in[l_i,r_i]$ ($0\le l_i\le r_i\le 10^9$) to each node so that the imbalance of the tree is minimized.

The first line contains $T$ ($1\le T\le 10$), the number of independent test cases, and an integer $B\in\{0,1\}$. Each test case starts with a line containing $N$, followed by a line with $N-1$ 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 $10^5$.

For each test case, output one or two lines, depending on the value of $B$. The first line should contain the minimum imbalance. If $B=1$, print one more line with $N$ space-separated integers $s_1,s_2,\ldots,s_N$ that achieve that imbalance. Any valid assignment is accepted.