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ICPC 2018
ICPC 2018

G. Panda Preserve

Last month, Sichuan province secured funding to establish the Great Panda National Park, a natural preserve for a population of more than 1 800 giant pandas. The park will be surrounded by a polygonal fence. In order for researchers to track the pandas, wir...

Updated May 21, 2026
Track ICPC
Year 2018
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Problem statement

Problem Statement

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Time limit 10 seconds

Last month, Sichuan province secured funding to establish the Great Panda National Park, a natural preserve for a population of more than 1 800 giant pandas. The park will be surrounded by a polygonal fence. In order for researchers to track the pandas, wireless receivers will be placed at each vertex of the enclosing polygon and each animal will be outfitted with a wireless transmitter. Each wireless receiver will cover a circular area centered at the location of the receiver, and all receivers will have the same range. Naturally, receivers with smaller range are cheaper, so your goal is to determine the smallest possible range that suffices to cover the entire park. As an example, Figure G.1 shows the park described by the first sample input. Notice that a wireless range of 35 does not suffice (a), while the optimal range of 50 covers the entire park (b).

(a) An insufficient range for covering the park.         (b) The minimal range for covering the park.
Figure G.1: Illustration of Sample Input 1.

Input

The first line of the input contains an integer n (3 ≤ n ≤ 2 000) specifying the number of vertices of the polygon bounding the park. This is followed by n lines, each containing two integers x and y (|x|, |y| ≤ 104 ) that give the coordinates (x, y) of the vertices of the polygon in counter-clockwise order. The polygon is simple; that is, its vertices are distinct and no two edges of the polygon intersect or touch, except that consecutive edges touch at their common vertex.

Output

Display the minimum wireless range that suffices to cover the park, with an absolute or relative error of at most 10−6 .

Sample Tests

Sample 1
Sample Input 
5
0 0
170 0
140 30
60 30
0 70
Sample Output 
50
Sample 2
Sample Input 
5
0 0
170 0
140 30
60 30
0 100
Sample Output 
51.538820320
Sample 3
Sample Input 
5
0   0
1   2
1   5
0   2
0   1
Sample Output 
1.581138830
Editorial

Editorial

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

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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/2018/G-panda-preserve/solution.texC++ implementationcompetitive_programming/icpc/2018/G-panda-preserve/solution.cppStatement textcompetitive_programming/icpc/2018/G-panda-preserve/statement.txtStatement PDFcompetitive_programming/icpc/2018/G-panda-preserve/statement.pdfMetadatacompetitive_programming/icpc/2018/G-panda-preserve/meta.jsonYear packetcompetitive_programming/icpc/2018/contest_problems.pdf