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Small fixes

This commit is contained in:
Antti H S Laaksonen 2017-04-17 12:18:29 +03:00
parent 55b55d1fc1
commit 8028fddfa6
6 changed files with 10 additions and 10 deletions
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2
chapter01.tex
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@ -836,7 +836,7 @@ calculate logarithms to some fixed base.
\index{natural logarithm} \index{natural logarithm}
The \key{natural logarithm} $\ln(x)$ of a number $x$ The \key{natural logarithm} $\ln(x)$ of a number $x$
is a logarithm whose base is $e \approx 2{,}71828$. is a logarithm whose base is $e \approx 2.71828$.
Another property of logarithms is that Another property of logarithms is that
the number of digits of an integer $x$ in base $b$ is the number of digits of an integer $x$ in base $b$ is

10
chapter05.tex
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@ -512,7 +512,7 @@ the lower-right corner and count the number of such paths.
\item \item
running time: 483 seconds running time: 483 seconds
\item \item
recursive calls: 76 billions recursive calls: 76 billion
\end{itemize} \end{itemize}
\subsubsection{Optimization 1} \subsubsection{Optimization 1}
@ -566,7 +566,7 @@ and finally multiply the number of the solutions by two.
\item \item
running time: 244 seconds running time: 244 seconds
\item \item
recursive calls: 38 billions recursive calls: 38 billion
\end{itemize} \end{itemize}
\subsubsection{Optimization 2} \subsubsection{Optimization 2}
@ -595,7 +595,7 @@ immediately if we reach the lower-right square too early.
\item \item
running time: 119 seconds running time: 119 seconds
\item \item
recursive calls: 20 billions recursive calls: 20 billion
\end{itemize} \end{itemize}
\subsubsection{Optimization 3} \subsubsection{Optimization 3}
@ -627,7 +627,7 @@ It turns out that this optimization is very useful:
\item \item
running time: 1.8 seconds running time: 1.8 seconds
\item \item
recursive calls: 221 millions recursive calls: 221 million
\end{itemize} \end{itemize}
\subsubsection{Optimization 4} \subsubsection{Optimization 4}
@ -664,7 +664,7 @@ very efficient:
\item \item
running time: 0.6 seconds running time: 0.6 seconds
\item \item
recursive calls: 69 millions recursive calls: 69 million
\end{itemize} \end{itemize}
~\\ ~\\

2
chapter09.tex
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@ -245,7 +245,7 @@ to the position of $X$.
Next we will see how we can Next we will see how we can
process range minimum queries in $O(1)$ time process range minimum queries in $O(1)$ time
after an $O(n \log n)$ time preprocessing using \index{sparse table} after an $O(n \log n)$ time preprocessing using \index{sparse table}
a data structure called a \emph{sparse table}\footnote{The a data structure called a \key{sparse table}\footnote{The
sparse table structure was introduced in \cite{ben00}. sparse table structure was introduced in \cite{ben00}.
There are also more sophisticated techniques \cite{fis06} where There are also more sophisticated techniques \cite{fis06} where
the preprocessing time of the array is only $O(n)$, but such algorithms the preprocessing time of the array is only $O(n)$, but such algorithms

2
chapter13.tex
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@ -603,7 +603,7 @@ is named after R. W. Floyd and S. Warshall
who published it independently in 1962 \cite{flo62,war62}.} who published it independently in 1962 \cite{flo62,war62}.}
is an alternative way to approach the problem is an alternative way to approach the problem
of finding shortest paths. of finding shortest paths.
Unlike the other algorihms in this chapter, Unlike the other algorithms in this chapter,
it finds all shortest paths between the nodes it finds all shortest paths between the nodes
in a single run. in a single run.

2
chapter25.tex
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@ -130,7 +130,7 @@ Of course, this strategy requires that
the number of sticks is \emph{not} divisible by 4 the number of sticks is \emph{not} divisible by 4
when it is our move. when it is our move.
If it is, there is nothing we can do, If it is, there is nothing we can do,
but the opponent will win the game if and the opponent will win the game if
they play optimally. they play optimally.
\subsubsection{State graph} \subsubsection{State graph}

2
chapter27.tex
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@ -401,7 +401,7 @@ and the next range is
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
there will be three steps: there will be three steps:
the left endpoint moves one step to the left, the left endpoint moves one step to the right,
and the right endpoint moves two steps to the right. and the right endpoint moves two steps to the right.
After each step, the array \texttt{c} After each step, the array \texttt{c}