Question

Conquer the World

Time limit: 8 seconds

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 of input contains an integer n (1 ≤ n ≤ 250000), the number of nations. This is followed by n − 1 lines, each containing three integers u, v, and c (1 ≤ u,v ≤ n, 1 ≤ c ≤ 10⁶), indicating that there is a bidirectional route connecting nations u and v, which costs c per army to use.

Finally, another n lines follow, the i^th of which contains two non-negative integers x_i and y_i, indicating that there are currently x_i armies in nation i, and you need at least y_i armies to end up in that nation in the final configuration. The total number of armies (the sum of the x_i values) is at least the sum of the y_i values, and no more than 10⁶.

Output

Display the minimum cost to move your armies such that there are at least y_i armies in nation i for all i.

Sample Input 1                                                      Sample Output 1

3 1 2 5 3 1 5 2 1 5 0 1 3

15

Sample Input 2 Sample Output 2

6 1 2 2 1 3 5 1 4 1 2 5 5 2 6 1 0 0 1 0 2 1 2 1 0 1 0 1

9

Q:What is Actually asked in the problem briefly define the problem and give its solution?

Answer 1

The problem statement decribes a situation which can be compared to the Travelling Salesman problem.

Here, the question states that there are 'n' country, let's refer to them as 'n' locations on a map. It goes on to state that these locations are in return connected by 'n-1' bidirectional paths, let suppose these paths are bidirectional roads. Now, between any two given locations (say 'u' and 'v'), there's a toll or a fee 'c' associated with anyone who wishes to use these connecting roads.

Once these inputs are acquired, we then have to provide number of armies we have present at each of these 'n' locations, along with number of armies that should be present in order to conquer that location. And in order to match the number of armies currently at a given location with the number of armies that should be there (in order to conquer it), it's understood that we'd have to move armies one-by-one to the respective locations to satisfy the required configuration. While doing so, we also have to keep in mind the cost 'c' for moving each army in either direction on any of the aforementioned roads.

Hence, the question requires the minimum cost which would be associated with achieving such a task.

Since it hasn't been specified which language the answer is supposed to be written in or if an algorithm is required, I wouldn't try attempting that part of your question in its entirety. Instead, let me outline an algorithm that would need to be built for it:

FOR EACH Country :

IF number of armies needed to conquer Country are less than required configuration :

    Find shortest path/ least cost to get required number of armies to Country

ELSE :

    Continue