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Tree Broadcast - Repovive Starter Round 7 | Repovive

21G. Tree Broadcast

3500 pts2s256 MB

You are given a network of $n$ servers, numbered from $1$ to $n$ . The network has $n-1$ links and forms a tree.

Transmitting an entire file through link $i$ takes exactly $w_i$ seconds.

Initially, exactly one server has the complete file. File transmissions follow these rules:

A server can transmit the file only after it has received the complete file.
During a transmission, both endpoints of the link are busy.
A server can participate in at most one transmission at any moment.
Transmissions using disjoint pairs of servers can happen simultaneously.
A transmission cannot be paused.

For each $s$ with $1 \le s \le n$ , independently suppose that server $s$ is the only server that has the file initially. Find the minimum time needed for every server to receive the complete file.

Constraints

$1 \le t \le 10^4$
$1 \le n \le 2 \times 10^5$
The sum of $n$ over all test cases is at most $2 \times 10^5$
$1 \le u_i, v_i \le n$
$u_i \ne v_i$
$1 \le w_i \le 10^9$
The given links form a tree

Input

t

\left. \begin{array}{l} n \\ u_1 \quad v_1 \quad w_1 \\ u_2 \quad v_2 \quad w_2 \\ \vdots \\ u_{n-1} \quad v_{n-1} \quad w_{n-1} \end{array} \right\} \times t

Output

For each test case, output the minimum required time for every possible initially informed server:

\left. \begin{array}{l} ans_1 \quad ans_2 \quad \cdots \quad ans_n \end{array} \right\} \times t

Samples

Input

Case 1

Case 2

2
1 2 7

Case 3

3
1 2 2
2 3 5

Case 4

Case 5

Explanation

Case 1

The only server already has the file, so no transmission is needed.

Case 2

The file must cross the only link, which takes $7$ seconds, regardless of the starting server.

Case 3

Starting from an endpoint, the two transmissions take $2+5=7$ seconds. Starting from server $2$ , both links must be used one after the other, so the same total is required.

Case 4

Starting from server $1$ , first send to server $2$ and then to server $3$ . The two transmissions of length $10$ can then overlap, and all servers are informed after $12$ seconds. Similar schedules work from servers $2$ and $3$ . From server $4$ or $5$ , the file must travel along the full weighted path of length $22$ to reach the other endpoint.

Case 5

Starting from server $1$ , send to server $2$ first. The transmissions from $2$ to $4$ and from $1$ to $3$ can then happen simultaneously, finishing at time $101$ . The same is possible from server $2$ . From either endpoint, the file must traverse the full path, whose total weight is $201$ .

Output

Case 1

Case 2

7 7

Case 3

7 7 7

Case 4

12 12 12 22 22

Case 5

101 101 201 201

You are given a network of $n$ servers, numbered from $1$ to $n$ . The network has $n-1$ links and forms a tree.

Transmitting an entire file through link $i$ takes exactly $w_i$ seconds.

Initially, exactly one server has the complete file. File transmissions follow these rules:

A server can transmit the file only after it has received the complete file.
During a transmission, both endpoints of the link are busy.
A server can participate in at most one transmission at any moment.
Transmissions using disjoint pairs of servers can happen simultaneously.
A transmission cannot be paused.

For each $s$ with $1 \le s \le n$ , independently suppose that server $s$ is the only server that has the file initially. Find the minimum time needed for every server to receive the complete file.

$1 \le t \le 10^4$
$1 \le n \le 2 \times 10^5$
The sum of $n$ over all test cases is at most $2 \times 10^5$
$1 \le u_i, v_i \le n$
$u_i \ne v_i$
$1 \le w_i \le 10^9$
The given links form a tree

t

\left. \begin{array}{l} n \\ u_1 \quad v_1 \quad w_1 \\ u_2 \quad v_2 \quad w_2 \\ \vdots \\ u_{n-1} \quad v_{n-1} \quad w_{n-1} \end{array} \right\} \times t

For each test case, output the minimum required time for every possible initially informed server:

\left. \begin{array}{l} ans_1 \quad ans_2 \quad \cdots \quad ans_n \end{array} \right\} \times t

The file must cross the only link, which takes $7$ seconds, regardless of the starting server.

Starting from an endpoint, the two transmissions take $2+5=7$ seconds. Starting from server $2$ , both links must be used one after the other, so the same total is required.

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21G. Tree Broadcast

Constraints

Input

Output

Samples

21G. Tree Broadcast

Constraints

Input

Output

Samples