ICPC 2012 - I. A Safe Bet
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/2012/I-a-safe-bet. Edit
competitive_programming/icpc/2012/I-a-safe-bet/solution.tex to update the written solution and
competitive_programming/icpc/2012/I-a-safe-bet/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
A Safe Bet
Problem ID: safe
Safe Ltd. is a company that manufactures high-quality safes. Its latest invention is an optical closure
mechanism that uses a laser beam passing through a rectangular grid with several mirrors.
Laser- @
@ Laser- @
@
@
@ @
@
@ @ -
@ @
@ - Detector @ Detector
@ @
Beam detected, safe open Beam not detected, alarm raised
When the laser is activated, a beam enters the top row of the grid horizontally from the left. The beam is
reflected by every mirror that it hits. Each mirror has a 45 degree diagonal orientation, either
or . If
the beam exits the bottom row of the grid horizontally to the right, it is detected and the safe opens (see
the left side of the figure above). Otherwise the safe remains closed and an alarm is raised.
Each safe has a missing mirror, which prevents the laser beam from traveling successfully through the
grid (see the right side of the figure above). The safe has a mechanism that enables the user to drop a
single mirror into any empty grid cell. A legitimate user knows the correct position and orientation of
the missing mirror (
in row 4 column 3 above) and can thus open the safe. Without this knowledge
the user has to guess correctly, which can be difficult for safes with large grids.
Your job is to determine if particular safes are actually secure. A secure safe does not open right away
without inserting a mirror, and there is at least one valid location and orientation for the missing mirror.
There may indeed be multiple such locations and orientations.
Input
Each test case describes a single safe and starts with a line containing four integer numbers r, c, m, and
n (1 ≤ r, c ≤ 1 000 000 and 0 ≤ m, n ≤ 200 000). The mechanism’s grid has r rows and c columns.
Each of the next m lines contains two integer numbers ri and ci (1 ≤ ri ≤ r and 1 ≤ ci ≤ c) specifying
that there is a
mirror in row ri column ci . The following n lines specify the positions of the mirrors
in the same way. The m + n positions of the mirrors are pairwise distinct.
Output
For each test case, display its case number followed by:
• 0 if the safe opens without inserting a mirror.
• k r c if the safe does not open without inserting a mirror, there are exactly k positions where
inserting a mirror opens the safe, and (r, c) is the lexicographically smallest such row, column
position. A position where both a
and a mirror open the safe counts just once.
• impossible if the safe cannot be opened with or without inserting a mirror.
Sample Input Output for Sample Input
5 6 1 4 Case 1: 2 4 3
2 3 Case 2: 0
1 2 Case 3: impossible
2 5
4 2
5 5
100 100 0 2
1 77
100 77
100 100 0 0
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 2012\\I. A Safe Bet}
\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;
}