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De Bruijn edits

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2026-03-04 14:14:38 -08:00
parent 2de7ee0c22
commit 53c3e1859b
2 changed files with 8 additions and 5 deletions
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src/Advanced/De Bruijn/parts/0 intro.tex
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@@ -21,7 +21,7 @@ Unlock this lock with only 5 keypresses.
\end{solution} \end{solution}
\vfill \vfill
Now, consider the same lock, now set with a three-digit binary code. Now consider the same lock, but configured with a three-digit binary code.
\problem{} \problem{}
How many codes are possible? How many codes are possible?
\vfill \vfill

11
src/Advanced/De Bruijn/parts/1 words.tex
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@@ -20,7 +20,11 @@ We say $v$ is a \textit{subword} of $w$ if $v$ is contained in $w$. \par
For example, \texttt{11} is a subword of \texttt{011}, but \texttt{00} is not. For example, \texttt{11} is a subword of \texttt{011}, but \texttt{00} is not.
\definition{} \definition{}
Recall \ref{lockproblem}. Let's generalize this to the \textit{$n$-subword problem}: \par Recall the lock problem from the previous page.
Let's generalize this to the \textit{$n$-subword problem}:
\vspace{1mm}
Given an alphabet $A$ and a positive integer $n$, Given an alphabet $A$ and a positive integer $n$,
we want a word over $A$ that contains all possible length-$n$ subwords. we want a word over $A$ that contains all possible length-$n$ subwords.
The shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}. The shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
@@ -67,7 +71,7 @@ Find the following:
\problem{}<sbounds> \problem{}<sbounds>
Let $w$ be a word over an alphabet of size $k$. \par Let $w$ be a word over an alphabet of size $k$. \par
Prove the following: Show that all of the following are true:
\begin{itemize} \begin{itemize}
\item $\mathcal{S}_n(w) \leq k^n$ \item $\mathcal{S}_n(w) \leq k^n$
\item $\mathcal{S}_n(w) \geq \mathcal{S}_{n-1}(w) - 1$ \item $\mathcal{S}_n(w) \geq \mathcal{S}_{n-1}(w) - 1$
@@ -103,7 +107,7 @@ Prove the following:
\definition{} \definition{}
Let $v$ and $w$ be words over the same alphabet. \par Let $v$ and $w$ be words over the same alphabet. \par
The word $vw$ is the word formed by writing $v$ after $w$. \par The word $vw$ is the word formed by writing $w$ after $v$. \par
For example, if $v = \texttt{1001}$ and $w = \texttt{10}$, $vw$ is $\texttt{100110}$. For example, if $v = \texttt{1001}$ and $w = \texttt{10}$, $vw$ is $\texttt{100110}$.
\problem{} \problem{}
@@ -116,7 +120,6 @@ We'll call this the \textit{Fibonacci word} of order $k$.
\item What are $F_3$, $F_4$, and $F_5$? \item What are $F_3$, $F_4$, and $F_5$?
\item Compute $\mathcal{S}_0$ through $\mathcal{S}_5$ for $F_5$. \item Compute $\mathcal{S}_0$ through $\mathcal{S}_5$ for $F_5$.
\item Show that the length of $F_k$ is the $(k + 2)^\text{th}$ Fibonacci number. \par \item Show that the length of $F_k$ is the $(k + 2)^\text{th}$ Fibonacci number. \par
\hint{Induction.}
\end{itemize} \end{itemize}
\begin{solution} \begin{solution}