F. Deer-Proof Fence
Input: fence.in Uncle Magnus has planted some young saplings on his farm as part of his reforestation project. Unfortunately, deer like to eat tender sapling shoots and leaves, making it necessary to build protective fences around them. Since deer and other...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Uncle Magnus has planted some young saplings on his farm as part of his reforestation project. Unfortunately, deer like to eat tender sapling shoots and leaves, making it necessary to build protective fences around them. Since deer and other sapling nibblers can reach partway over the fence, every fence must lie at least a minimum distance (a margin) from each sapling.
Deer-proof fencing is quite expensive, so Uncle Magnus wants to minimize the total length of fencing used. Your job is to write a program that computes the minimum length of fencing that is required to enclose and protect the saplings. Fences may include both straight and curved segments. You may design a single fence that encloses all saplings or multiple fences that enclose separate groups of saplings.
Figure 6 shows two example configurations, each consisting of three saplings with different margin requirements. In the top configuration, which corresponds to the first sample input, the minimum-length solution consists of two separate fences. In the bottom configuration, which corresponds to the second sample input, the minimum-length solution consists of a single fence.
Figure 6: Deerproof fences.Input
The input consists of multiple test cases. The first line of each test case contains integers N (0 < N ≤ 9), which is the number of saplings, and M (0 < M ≤ 200), which is the margin required around each sapling. This line is followed by N additional lines. Each of these N lines contains two integers x and y that describe the Cartesian coordinates of a sapling (|x| ≤ 100 and |y| ≤ 100). No two saplings are in the same location. For simplicity the saplings can all be considered as points and the thickness of deer-proof fences can be considered zero.
The last test case is followed by a line containing two zeros.
Output
For each test case, print the case number (starting with 1) followed by the minimum total length of fencing required to protect the saplings with the given margin. Print the length with two digits to the right of the decimal point. Follow the format of the sample output.
Sample Tests
3 2
0 0
2 0
10 0
3 4
0 0
2 0
10 0
0 0Case 1: length = 29.13
Case 2: length = 45.13Editorial
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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