ICPC 2012 - D. Fibonacci Words
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/2012/D-fibonacci-words/solution.tex to update the written solution and
competitive_programming/icpc/2012/D-fibonacci-words/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem D
Fibonacci Words
Problem ID: fibonacci
The Fibonacci word sequence of bit strings is defined as:
if n = 0
0
F (n) = 1 if n = 1
F (n − 1) + F (n − 2) if n ≥ 2
Here + denotes concatenation of strings. The first few elements are:
n F (n)
0 0
1 1
2 10
3 101
4 10110
5 10110101
6 1011010110110
7 101101011011010110101
8 1011010110110101101011011010110110
9 1011010110110101101011011010110110101101011011010110101
Given a bit pattern p and a number n, how often does p occur in F (n)?
Input
The first line of each test case contains the integer n (0 ≤ n ≤ 100). The second line contains the bit
pattern p. The pattern p is nonempty and has a length of at most 100 000 characters.
Output
For each test case, display its case number followed by the number of occurrences of the bit pattern p in
F (n). Occurrences may overlap. The number of occurrences will be less than 263 .
Sample Input Output for Sample Input
6 Case 1: 5
10 Case 2: 8
7 Case 3: 4
10 Case 4: 4
6 Case 5: 7540113804746346428
01
6
101
96
10110101101101
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8
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\\D. Fibonacci Words}
\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;
}