Conquer the World

Time limit8sMemory limit1024 MB

Summary

Armies sit on the nodes of a weighted tree; move them along edges to satisfy each node's demand while minimizing total transport cost.

Level: Hard8 of 10
Topics: Tree, DFS, Greedy
Solved: No attempts yet

Problem

Bwahahahahaha!!! Your nemesis, the dashingly handsome spy Waco Powers, has at last fallen to your secret volcano base's deathtraps (or so you assume, being a little too busy to witness it firsthand). At long last, you are all set to CONQUER THE WORLD!

Nothing will stand in your way! Well, nothing except a minor problem of logistics. Your evil armies have announced that they will not continue carving their relentless path of destruction across the puny nations of the world without being paid. And unfortunately you are running low on cash. A volcano lair has many wonderful qualities, but "reasonably affordable" is not one of them. You have had to pull funds from the travel budget to pay your ungrateful underlings. Now you are not sure how you will actually get your armies into position to CONQUER THE WORLD.

You have a map of the nations of the world and all your available transport routes between them. Each route connects two nations and has a fixed cost per army that uses it. The routes are laid out such that there is exactly one way to travel between any two nations. You know the current position of each of your armies and how many you will need to place permanently in each nation in order to subjugate it. How can you move the armies into place as cheaply as possible so you can CONQUER THE WORLD?

Input

The first line contains an integer $n$ ( $1 \le n \le 250000$ ), the number of nations. Then follow $n-1$ lines, each containing three integers $u$ , $v$ , and $c$ ( $1 \le u, v \le n$ , $1 \le c \le 10^6$ ), meaning that a bidirectional route connects nations $u$ and $v$ and costs $c$ per army that uses it.

Then follow $n$ more lines. The $i$ -th of them contains two non-negative integers $x_i$ and $y_i$ , meaning that nation $i$ currently holds $x_i$ armies and that the final arrangement must leave at least $y_i$ armies there. The total number of armies, the sum of the $x_i$ values, is at least the sum of the $y_i$ values and at most $10^6$ .

Output

Print the minimum cost of moving the armies so that nation $i$ ends up with at least $y_i$ armies for every $i$ .

Examples3

Example 1
Input
```
3
1 2 5
3 1 5
2 1
5 0
1 3
```
Expected output
```
15
```

Example 2

Input

Expected output

Example 3
Input
```
1
5 3
```
Expected output
```
0
```