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# [Solution] Path Prefixes Codeforces Solution

G. Path Prefixes
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given a rooted tree. It contains $n$ vertices, which are numbered from $1$ to $n$. The root is the vertex $1$.

Each edge has two positive integer values. Thus, two positive integers ${a}_{j}$ and ${b}_{j}$ are given for each edge.

Output $n-1$ numbers ${r}_{2},{r}_{3},\dots ,{r}_{n}$, where ${r}_{i}$ is defined as follows.

Consider the path from the root (vertex $1$) to $i$ ($2\le i\le n$). Let the sum of the costs of ${a}_{j}$ along this path be ${A}_{i}$. Then ${r}_{i}$ is equal to the length of the maximum prefix of this path such that the sum of ${b}_{j}$ along this prefix does not exceed ${A}_{i}$. Example for $n=9$. The blue color shows the costs of ${a}_{j}$, and the red color shows the costs of ${b}_{j}$.

Consider an example. In this case:

• ${r}_{2}=0$, since the path to $2$ has an amount of ${a}_{j}$ equal to $5$, only the prefix of this path of length $0$ has a smaller or equal amount of ${b}_{j}$;
• ${r}_{3}=3$, since the path to $3$ has an amount
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•  of ${a}_{j}$ equal to $5+9+5=19$, the prefix of length $3$ of this path has a sum of ${b}_{j}$ equal to $6+10+1=17$ ( the number is $17\le 19$);
• ${r}_{4}=1$, since the path to $4$ has an amount of ${a}_{j}$ equal to $5+9=14$, the prefix of length $1$ of this path has an amount of ${b}_{j}$ equal to $6$ (this is the longest suitable prefix, since the prefix of length $2$ already has an amount of ${b}_{j}$ equal to $6+10=16$, which is more than $14$);
• ${r}_{5}=2$, since the path to $5$ has an amount of ${a}_{j}$ equal to $5+9+2=16$, the prefix of length $2$ of this path has a sum of ${b}_{j}$ equal to $6+10=16$ (this is the longest suitable prefix, since the prefix of length $3$ already has an amount of ${b}_{j}$ equal to $6+10+1=17$, what is more than $16$);
• ${r}_{6}=1$, since the path up to $6$ has an amount of ${a}_{j}$ equal to $2$, the prefix of length $1$ of this path has an amount of ${b}_{j}$ equal to $1$;
• ${r}_{7}=1$, since the path to $7$ has an amount of ${a}_{j}$ equal to $5+3=8$, the prefix of length $1$ of this path has an amount of ${b}_{j}$ equal to $6$ (this is the longest suitable prefix, since the prefix of length $2$ already has an amount of ${b}_{j}$ equal to $6+3=9$, which is more than $8$);
• ${r}_{8}=2$, since the path up to $8$ has an amount of ${a}_{j}$ equal to $2+4=6$, the prefix of length $2$ of this path has an amount of ${b}_{j}$ equal to $1+3=4$;
• ${r}_{9}=3$, since the path to $9$ has an amount of ${a}_{j}$ equal to $2+4+1=7$, the prefix of length $3$ of this path has a sum of ${b}_{j}$ equal to $1+3+3=7$.
Input

The first line contains an integer $t$ ($1\le t\le {10}^{4}$) — the number of test cases in the test.

The descriptions of test cases follow.

Each description begins with a line that contains an integer $n$ ($2\le n\le 2\cdot {10}^{5}$) — the number of vertices in the tree.

This is followed by $n-1$ string, each of which contains three numbers ${p}_{j},{a}_{j},{b}_{j}$ ($1\le {p}_{j}\le n$$1\le {a}_{j},{b}_{j}\le {10}^{9}$) — the ancestor of the vertex $j$, the first and second values an edge that leads from ${p}_{j}$ to $j$. The value of $j$ runs through all values from $2$ to $n$ inclusive. It is guaranteed that each set of input data has a correct hanged tree with a root at the vertex $1$.

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

Output

For each test case, output $n-1$ integer in one line: ${r}_{2},{r}_{3},\dots ,{r}_{n}$.