Small fixes

chapter01.tex

@@ -836,7 +836,7 @@ calculate logarithms to some fixed base.
 \index{natural logarithm}
 
 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
 the number of digits of an integer $x$ in base $b$ is

chapter05.tex

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

chapter09.tex

@@ -245,7 +245,7 @@ to the position of $X$.
 Next we will see how we can
 process range minimum queries in $O(1)$ time
 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}.
 There are also more sophisticated techniques \cite{fis06} where
 the preprocessing time of the array is only $O(n)$, but such algorithms

chapter13.tex

@@ -603,7 +603,7 @@ is named after R. W. Floyd and S. Warshall
 who published it independently in 1962 \cite{flo62,war62}.}
 is an alternative way to approach the problem
 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
 in a single run.
 

chapter25.tex

@@ -130,7 +130,7 @@ Of course, this strategy requires that
 the number of sticks is \emph{not} divisible by 4
 when it is our move.
 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.
 
 \subsubsection{State graph}

chapter27.tex

@@ -401,7 +401,7 @@ and the next range is
 \end{tikzpicture}
 \end{center}
 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.
 
 After each step, the array \texttt{c}