ICPC 2010 - C. Tracking Bio-bots
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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competitive_programming/icpc/2010/C-tracking-bio-bots/solution.tex to update the written solution and
competitive_programming/icpc/2010/C-tracking-bio-bots/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem C
Tracking Bio-bots
Problem ID: biobots
The researchers at International Bio-bot Makers (IBM) have invented a new kind of Bio-bot, a robot with
behavior mimicking biological organisms. The development of the new robot is at a primitive stage; they now
resemble simple four-wheeled rovers. And like most modern robots, Bio-bots are not very mobile. Their weak
motors and limited turning capability put considerable limitations on their movement, even in simple, relatively
obstacle-free environments.
Currently the Bio-bots operate in a room which can be described as an m×n grid. A Bio-bot occupies a full
square on this grid. The exit is in the northeast corner, and the room slopes down towards it, which means the
Bio-bots are only capable of moving north or east at any time. Some squares in the room are also occupied by
walls, which completely block the robot. Figure 1, which corresponds to the sample input, shows an example of
such a room.
Figure 1
Clearly, a Bio-bot located on square A is capable of leaving the room, while one at square B is trapped inside it,
no matter what it does. Locations like B are called “stuck squares.” (Walls do not count as stuck squares.) Given
the description of a room, your job is to count the total number of stuck squares in the room.
Input
Input consists of multiple test cases, each describing one room. Each test case begins with a line containing
three integers m, n, and w (1 ≤ m, n ≤ 106, 0 ≤ w ≤ 1000). These indicate that the room contains m rows, n
columns, and w horizontal walls.
Each of the next w lines contains four integers x1, y1, x2, y2, the coordinates of the squares delimiting one wall.
All walls are aligned from west to east, so 0 ≤ x1 ≤ x2 < n and 0 ≤ y1 = y2 < m. Walls do not overlap each other.
The southwest corner of the room has coordinates (0,0) and the northeast corner has coordinates (n -1, m -1).
The last test case is followed by a line containing three zeros.
Output
For each test case, display one line of output containing the test case number followed by the number of stuck
squares in the given room. Follow the format shown in the sample output.
Sample Input Output for the Sample Input
8 8 3 Case 1: 8
1 6 3 6
2 4 2 4
4 2 7 2
0 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 2010\\C. Tracking Bio-bots}
\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;
}