L. Swap Space
You administer a large cluster of computers with hard drives that use various file system types to store data. You recently decided to unify the file systems to the same type. That is quite a challenge since all the drives are currently in use, all of them...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
You administer a large cluster of computers with hard drives that use various file system types to store data. You recently decided to unify the file systems to the same type. That is quite a challenge since all the drives are currently in use, all of them are filled with important data to the limits of their capacities, and you cannot afford to lose any of the data. Moreover, reformatting a drive to use a new file system may significantly change the drive’s capacity. To make the reformat possible, you will have to buy an extra hard drive. Obviously, you want to save money by minimizing the size of such extra storage. You can reformat the drives in any order. Prior to reformatting a drive, you must move all data from that drive to one or more other drives, splitting the data if necessary. After a drive is reformatted, you can immediately start using it to store data from other drives. It is not necessary to put all the data on the same drives they originally started on – in fact, this might even be impossible if some of the drives have smaller capacity with the new file system. It is also allowed for some data to end up on the extra drive. As an example, suppose you have four drives A, B, C, and D with drive capacities 6, 1, 3, and 3 GB. Under the new file system, the capacities become 6, 7, 5, and 5 GB, respectively. If you buy only 1 GB of extra space, you can move the data from drive B there and then reformat drive B. Now you have 7 GB free on drive B, so you can move the 6 GB from drive A there and reformat drive A. Finally, you move the six total gigabytes from drives C and D to drive A, and reformat C and D.
Input
The input begins with a line containing one integer n (1 ≤ n ≤ 106 ), which is the number of drives in your cluster. Following this are n lines, each describing a drive as two integers a and b, where a is the capacity with the old file system and b is the capacity with the new file system. All capacities are given in gigabytes and satisfy 1 ≤ a, b ≤ 109 . (One thousand petabytes should be enough for everyone, right?)
Output
Display the total extra capacity in gigabytes you must buy to reformat the drives.
Sample Tests
4
6 6
1 7
3 5
3 514
2 2
3 3
5 1
5 105Editorial
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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