ICPC 2018 - I. Triangles
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
Source-first archive entry
This page is built from the copied files in competitive_programming/icpc/2018/I-triangles. Edit
competitive_programming/icpc/2018/I-triangles/solution.tex to update the written solution and
competitive_programming/icpc/2018/I-triangles/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem I
Triangles
Time limit: 6 seconds
For your trip to Beijing, you have brought plenty of puzzle books, many of them containing challenges
like the following: how many triangles can be found in Figure I.1?
Figure I.1: Illustration of Sample Input 2.
While these puzzles keep your interest for a while, you quickly get bored with them and instead start
thinking about how you might solve them algorithmically. Who knows, maybe a problem like that will
actually be used in this year’s contest. Well, guess what? Today is your lucky day!
Input
The first line of input contains two integers r and c (1 ≤ r ≤ 3 000, 1 ≤ c ≤ 6 000), specifying the
picture size, where r is the number of rows of vertices and c is the number of columns. Following this are
2r − 1 lines, each of them having at most 2c − 1 characters. Odd lines contain grid vertices (represented
as lowercase x characters) and zero or more horizontal edges, while even lines contain zero or more
diagonal edges. Specifically, picture lines with numbers 4k + 1 have vertices in positions 1, 5, 9, 13, . . .
while lines with numbers 4k + 3 have vertices in positions 3, 7, 11, 15, . . . . All possible vertices are
represented in the input (for example, see how Figure I.1 is represented in Sample Input 2). Horizontal
edges connecting neighboring vertices are represented by three dashes. Diagonal edges are represented
by a single forward slash (‘/’) or backslash (‘\’) character. The edge characters will be placed exactly
between the corresponding vertices. All other characters will be space characters. Note that if any input
line could contain trailing whitespace, that whitespace may be omitted.
Output
Display the number of triangles (of any size) formed by grid edges in the input picture.
Sample Input 1 Sample Output 1
3 3 1
x---x
\ /
x
/ \
x x
Sample Input 2 Sample Output 2
4 10 12
x x---x---x x
\ / / \
x x---x x x
/ \ / \ \
x x---x---x---x
/ / \ \ / \
x---x---x---x---x
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
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
Exact copied C++ implementation from solution.cpp.
#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;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2018\\I. Triangles}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#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;
}