C. Conquer The World
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 C ONQUER T HE W ORLD! No...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
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 C ONQUER T HE W ORLD! 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 C ONQUER T HE W ORLD. 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 C ONQUER T HE W ORLD?
Input
The first line of input contains an integer n (1 ≤ n ≤ 250 000), 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 ith of which contains two non-negative integers xi and yi , indicating that there are currently xi armies in nation i, and you need at least yi armies to end up in that nation in the final configuration. The total number of armies (the sum of the xi values) is at least the sum of the yi values, and no more than 106 .
Output
Display the minimum cost to move your armies such that there are at least yi armies in nation i for all i.
Sample Tests
3
1 2 5
3 1 5
2 1
5 0
1 3156
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 19Editorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
Key Observations
Write the structural observations that make the problem tractable.
State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
If the constraints matter, explain exactly which part of the solution they enable.
Algorithm
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
Mention corner cases explicitly if they affect the implementation.
Correctness Proof
We prove that the algorithm returns the correct answer.
Lemma 1.
State the first key claim.
Proof.
Provide a concise proof.
Lemma 2.
State the next claim if needed.
Proof.
Provide a concise proof.
Theorem.
The algorithm outputs the correct answer for every valid input.
Proof.
Combine the lemmas and finish the argument.
Complexity Analysis
State the running time and memory usage in terms of the input size.
Implementation Notes
Mention any non-obvious implementation detail that is easy to get wrong.
Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
Code
C++ solution used for this page.
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}
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