J. Pollution Solution
2013 World Finals St. Petersburg HOSTED BY ITMO Problem J Pollution Solution Time Limit: 1 second As an employee of Aqueous Contaminate Management, you must monitor the pollution that gets dumped (sometimes accidentally, sometimes purposefully) into river...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
As an employee of Aqueous Contaminate Management, you must monitor the pollution that gets dumped (sometimes accidentally, sometimes purposefully) into rivers, lakes and oceans. One of your jobs is to measure the impact of the pollution on various ecosystems in the water such as coral reefs, spawning grounds, and so on.
Figure J.1: Illustration of Sample Input 1.The model you use in your analysis is illustrated in Figure J.1. The shoreline (the horizontal line in the figure) lies on the x-axis with the source of the pollution located at the origin (0,0). The spread of the pollution into the water is represented by the semicircle, and the polygon represents the ecosystem of concern. You must determine the area of the ecosystem that is contaminated, represented by the dark blue region in the figure.
Input
The input consists of a single test case. A test case starts with a line containing two integers n and r, where 3 ≤ n ≤ 100 is the number of vertices in the polygon and 1 ≤ r ≤ 1 000 is the radius of the pollution field. This is followed by n lines, each containing two integers xi , yi , giving the coordinates of the polygon vertices in counter-clockwise order, where −1 500 ≤ xi ≤ 1 500 and 0 ≤ yi ≤ 1 500. The polygon does not self-intersect or touch itself. No vertex lies on the circle boundary.
Output
Display the area of the polygon that falls within the semicircle centered at the origin with radius r. Give the result with an absolute error of at most 10−3 .
ICPC 2013
2013 World Finals
St. Petersburg
HOSTED BY ITMOSample Tests
6 10
-8 2
8 2
8 14
0 14
0 6
-8 14101.576437872Editorial
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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