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2024-03-22 16:55:08 -07:00
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Advanced/De Bruijn/parts/1 words.tex
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@ -17,9 +17,9 @@ For example, \texttt{11} is a subword of \texttt{011}, but \texttt{00} is not.
\definition{}
Recall \ref{lockproblem}. From now on, we'll call this the \textit{$n$-subword problem}: \par
Given an alphabet $A$ and a positive integer $n$, we want a \par
word over $A$ that contains all possible length-$n$ subwords. \par
The shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
Given an alphabet $A$ and a positive integer $n$,
we want a word over $A$ that contains all possible length-$n$ subwords. \par
That shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
@ -55,7 +55,7 @@ Find the following:
\vfill
\pagebreak
\problem{}
\problem{}<sbounds>
Let $w$ be a word over an alphabet of size $k$. \par
Prove the following:
\begin{itemize}
@ -148,7 +148,7 @@ We'll call this the \textit{Fibonacci word} of order $k$.
% C_k is called the "Champernowne word" of order k.
\problem{}<cword>
Let $C_k$ denote the word over the alphabet $\{\texttt{0}, \texttt{1}\}$ obtained by \par
concatenating the binary representations of the integers $0,~...,~2^k -1$.\par
concatenating the binary representations of the integers $0,~...,~2^k -1$. \par
For example, $C_1 = \texttt{0}$, $C_2 = \texttt{011011}$, and $C_3 = \texttt{011011100101110111}$.
\begin{itemize}
\item How many symbols does the word $C_k$ contain?