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ICPC 2013
ICPC 2013

G. Map Tiles

2013 World Finals St. Petersburg HOSTED BY ITMO Problem G Map Tiles Time Limit: 20 seconds Publishing maps is not an easy task. First you need some appropriate transformation to display the earth’s spherical shape in a two-dimensional plane. Then another...

Updated May 21, 2026
Track ICPC
Year 2013
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

Statement text Statement PDF
Time limit 20 seconds

Publishing maps is not an easy task. First you need some appropriate transformation to display the earth’s spherical shape in a two-dimensional plane. Then another issue arises – most high-quality maps are too large to be printed on a single page of paper. To cope with that, map publishers often split maps into several rectangular tiles, and print each tile on one page. In this problem, you will examine this “tiling” process. The International Cartographic Publishing Company (ICPC) needs to cut their printing costs by mini- mizing the number of tiles used for their maps. Even with a fixed tile size (determined by the page size) and map scale, you can still optimize the situation by adjusting the tile grid. The left side of Figure G.1 shows 14 map tiles covering a region. The right side shows how you can cover the same region with only 10 tiles, without changing the tile sizes or orientation.

Figure G.1: Two possible ways of tiling Texas.

Your task is to help the ICPC find the minimum number of tiles needed to cover a given region. For simplicity, the region will be given as a closed polygon that does not intersect itself. Note that the tiles must be part of a rectangular grid aligned with the x-axis and y-axis. That is, they touch each other only with their whole sides and cannot be rotated. Also note that although all input coordinates are integers, tiles may be located at non-integer coordinates. The polygon may touch the edges of marginal lines (as in Sample Input 2). However, to avoid floating- point issues, you may assume the optimal answer will not change even if the polygon is allowed to go outside the map tiles by a distance of 10−6 .

Input

The input consists of a single test case. The first line of a test case contains three integers: n, xs , and ys . The number of polygon vertices is n (3 ≤ n ≤ 50), and xs and ys (1 ≤ xs , ys ≤ 100) are the dimensions of each tile. Each of the next n lines contains two integers x and y (0 ≤ x ≤ 10xs , 0 ≤ y ≤ 10ys ), specifying the vertices of the polygon representing the region (in either clockwise or counter-clockwise order).

                                                             ICPC 2013
2013 World Finals
                                                        St. Petersburg
                                                        HOSTED BY   ITMO

Output

Display the minimal number of tiles necessary to cover the whole interior of the polygon.

Sample Tests

Sample 1
Sample Input 
 12 9 9
 1 8
 1 16
 6 16
 9 29
 19 31
 23 24
 30 23
 29 18
 20 12
 22 8
 14 0
 14 8
Sample Output 
10
Sample 2
Sample Input 
 4 5 7
 10 10
 15 10
 15 17
 10 17
Sample Output 
1
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/2013/G-map-tiles/solution.texC++ implementationcompetitive_programming/icpc/2013/G-map-tiles/solution.cppStatement textcompetitive_programming/icpc/2013/G-map-tiles/statement.txtStatement PDFcompetitive_programming/icpc/2013/G-map-tiles/statement.pdfMetadatacompetitive_programming/icpc/2013/G-map-tiles/meta.jsonYear packetcompetitive_programming/icpc/2013/contest_problems.pdf