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ICPC 2014
ICPC 2014

G. Metal Processing Plant

Yulia works for a metal processing plant in Eka- terinburg. This plant processes ores mined in the Ural mountains, extracting precious metals such as chalcopyrite, platinum and gold from the ores. Every month the plant receives n shipments of un- processed...

Updated May 21, 2026
Track ICPC
Year 2014
Statement Text + PDF
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Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

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Time limit 4 seconds

Yulia works for a metal processing plant in Eka- terinburg. This plant processes ores mined in the Ural mountains, extracting precious metals such as chalcopyrite, platinum and gold from the ores. Every month the plant receives n shipments of un- processed ore. Yulia needs to partition these ship- ments into two groups based on their similarity. Then, each group is sent to one of two ore pro- cessing buildings of the plant. To perform this partitioning, Yulia first calculates a numeric distance d(i, j) for each pair of ship- ments 1 ≤ i ≤ n and 1 ≤ j ≤ n, where the Picture from Wikimedia Commons

smaller the distance, the more similar the ship- ments i and j are. For a subset S ⊆ {1, . . . , n} of shipments, she then defines the disparity D of S as the maximum distance between a pair of shipments in the subset, that is,

D(S) = max d(i, j).
          i,j∈S

Yulia then partitions the shipments into two subsets A and B in such a way that the sum of their dispar- ities D(A) + D(B) is minimized. Your task is to help her find this partitioning.

Input

The input consists of a single test case. The first line contains an integer n (1 ≤ n ≤ 200) indicating the number of shipments. The following n − 1 lines contain the distances d(i, j). The ith of these lines contains n − i integers and the j th integer of that line gives the value of d(i, i + j). The distances are symmetric, so d(j, i) = d(i, j), and the distance of a shipment to itself is 0. All distances are integers between 0 and 109 (inclusive).

Output

Display the minimum possible sum of disparities for partitioning the shipments into two groups.

Sample Tests

Sample 1
Sample Input 
 5
 4 5 0 2
 1 3 7
 2 0
 4
Sample Output 
4
Sample 2
Sample Input 
7
1 10 5 5 5 5
5 10 5 5 5
100 100 5 5
10 5 5
98 99
3
Sample Output 
15
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

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

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. 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.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#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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TeX write-upcompetitive_programming/icpc/2014/G-metal-processing-plant/solution.texC++ implementationcompetitive_programming/icpc/2014/G-metal-processing-plant/solution.cppStatement textcompetitive_programming/icpc/2014/G-metal-processing-plant/statement.txtStatement PDFcompetitive_programming/icpc/2014/G-metal-processing-plant/statement.pdfMetadatacompetitive_programming/icpc/2014/G-metal-processing-plant/meta.jsonYear packetcompetitive_programming/icpc/2014/contest_problems.pdf