H. Room Service
Problem ID: room You are working for a company designing cute, funny robot vacuum cleaners. At a high level, the robots’ behavior is divided into three modes: 1. Exploration 2. Vacuuming 3. Rampant Killing Unfortunately, while consumer testing shows that th...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
You are working for a company designing cute, funny robot vacuum cleaners. At a high level, the robots’ behavior is divided into three modes:
- Exploration
- Vacuuming
- Rampant Killing
Unfortunately, while consumer testing shows that the last two modes are working perfectly, the explo- ration mode still has bugs. You’ve been put in charge of debugging. At the beginning of the exploration mode, the robot is placed into a convex polygonal room. It has sensors that should tell it where all the walls are. Your job is to write a program that verifies that these readings are correct. To do this, the robot needs to physically touch every wall in the room. Your problem is this: given the shape of a convex polygonal room with N walls and a starting point P inside it, determine the shortest route that touches each wall and then returns to P . Touching a corner counts as touching both incident walls.
Input
Each test case starts with a line containing the number of vertices N of the polygon (3 ≤ N ≤ 100) and the integer coordinates Px and Py of the robot’s starting point (−10 000 ≤ Px , Py ≤ 10 000). This is followed by N lines, each containing two integers x, y (−10 000 ≤ x, y ≤ 10 000) defining a vertex of the polygon. Vertices are given in counterclockwise order, all interior angles are less than 180 degrees, the polygon does not self-intersect, and the robot’s starting point is strictly inside the polygon.
Output
For each test case, display the case number and the length of the desired route, accurate to two decimal places.
Sample Input Output for Sample Input 4 0 0 Case 1: 5.66 -1 -1 Case 2: 36.73 1 -1 1 1 -1 1 3 10 1 0 0 30 0 0 20
16Editorial
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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