H. Painter
Input file: painter.in You probably never heard of the painter Peer. He is not well known, much to his regret. Peer was one of the inventors of monochromy, which means that each of his paintings has a single color, but in different shades. He also believed...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
You probably never heard of the painter Peer. He is not well known, much to his regret. Peer was one of the inventors of monochromy, which means that each of his paintings has a single color, but in different shades. He also believed in the use of simple geometric forms.
During his triangle period, Peer drew triangles on a rectangular canvas, making sure their borders did not intersect. He would then choose a color, and fill the regions. Peer would paint the outermost region (the canvas itself) with the lightest shade of the color chosen. Then step by step, he would fill more inner regions with a darker shade of the same color. The image below is one of his “Forms in Green” paintings.
In a way the process was quite mechanical. The only thing Peer considered difficult was to decide, after drawing the triangles, how many different shades he would need. You must write a program to do that calculation for him. Your program will have a collection of triangles as its input. It should calculate the number of different shades needed to paint the regions according to the given rule.
Your program must also detect the rare times that Peer makes a mistake and draws triangles that intersect. Two triangles are considered intersecting if the edges of one triangle have at least one point in common with the edges of the other. In that case, the collection of triangles is invalid.
Input
The input file contains multiple test cases. The first line of each test case contains a single non-negative integer n (n ≤ 100000), which is the number of triangles in the test case. The following n lines of the test case contain the descriptions of triangles in the format x1 y1 x2 y2 x3 y3, where xi, yi are integers (-100000 < xi, yi < 100000) that are the
coordinates of the vertices of the triangles. The three points are guaranteed not to be collinear.
The last test case is followed by -1 on a line by itself.
Output
For each test case, print the case number (beginning with 1) and the number of shades needed to fill the regions if the test case is valid. Print the word ERROR if the test case is invalid (two or more triangles in the test case intersect).
Sample Tests
8
8 3 8 4 7 4
14 13 -1 9 9 0
1 8 7 7 4 10
5 10 11 8 13 12
9 10 11 10 11 9
2 7 9 1 10 6
5 5 5 6 8 6
9 2 9 5 6 4
2
0 0 1 0 0 1
2 0 1 1 1 -1
-1Case 1: 5 shades
Case 2: ERROREditorial
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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