Small fixes

chapter01.tex

@@ -35,9 +35,9 @@ straightforward and concise.
 Programs should be written quickly,
 because there is not much time available.
 Unlike in traditional software engineering,
-the programs are short (usually at most some
+the programs are short (usually at most a few
-hundreds of lines) and it is not needed to 
+hundred lines of code), and they do not need to 
-maintain them after the contest.
+be maintained after the contest.
 
 \section{Programming languages}
 
@@ -56,7 +56,7 @@ Many people think that C++ is the best choice
 for a competitive programmer,
 and C++ is nearly always available in
 contest systems.
-The benefits in using C++ are that
+The benefits of using C++ are that
 it is a very efficient language and
 its standard library contains a 
 large collection
@@ -154,7 +154,7 @@ This kind of code always works,
 assuming that there is at least one space
 or newline between each element in the input.
 For example, the above code can read
-both the following inputs:
+both of the following inputs:
 \begin{lstlisting}
 123 456 monkey
 \end{lstlisting}

chapter02.tex

@@ -14,7 +14,7 @@ The \key{time complexity} of an algorithm
 estimates how much time the algorithm will use
 for some input.
 The idea is to represent the efficiency
-as an function whose parameter is the size of the input.
+as a function whose parameter is the size of the input.
 By calculating the time complexity,
 we can find out whether the algorithm is fast enough
 without implementing it.

chapter05.tex

@@ -248,7 +248,7 @@ a solution can be constructed.
 
 As an example, consider the problem of
 calculating the number
-of ways $n$ queens can be placed to
+of ways $n$ queens can be placed on
 an $n \times n$ chessboard so that
 no two queens attack each other.
 For example, when $n=4$,
@@ -276,10 +276,10 @@ there are two possible solutions:
 The problem can be solved using backtracking
 by placing queens to the board row by row.
 More precisely, exactly one queen will
-be placed to each row so that no queen attacks
+be placed on each row so that no queen attacks
 any of the queens placed before.
 A solution has been found when all
-$n$ queens have been placed to the board.
+$n$ queens have been placed on the board.
 
 For example, when $n=4$,
 some partial solutions generated by
@@ -358,7 +358,7 @@ void search(int y) {
 \end{lstlisting}
 \end{samepage}
 The search begins by calling \texttt{search(0)}.
-The size of the board is $n$,
+The size of the board is $n \times n$,
 and the code calculates the number of solutions
 to \texttt{count}.
 
@@ -366,7 +366,7 @@ The code assumes that the rows and columns
 of the board are numbered from 0 to $n-1$.
 When the function \texttt{search} is
 called with parameter $y$,
-it places a queen to row $y$
+it places a queen on row $y$
 and then calls itself with parameter $y+1$.
 Then, if $y=n$, a solution has been found
 and the variable \texttt{count} is increased by one.
@@ -446,11 +446,11 @@ the $4 \times 4$ board are numbered as follows:
 \end{center}
 
 Let $q(n)$ denote the number of ways
-to place $n$ queens to an $n \times n$ chessboard.
+to place $n$ queens on an $n \times n$ chessboard.
 The above backtracking
 algorithm tells us that, for example, $q(8)=92$.
 When $n$ increases, the search quickly becomes slow,
-because the number of the solutions increases
+because the number of solutions increases
 exponentially.
 For example, calculating $q(16)=14772512$
 using the above algorithm already takes about a minute

chapter10.tex

@@ -333,8 +333,8 @@ the sets $x=\{1,3,4,8\}$ and $y=\{3,6,8,9\}$,
 and then constructs the set $z = x \cup y = \{1,3,4,6,8,9\}$:
 
 \begin{lstlisting}
-int x = (1<<1)+(1<<3)+(1<<4)+(1<<8);
+int x = (1<<1)|(1<<3)|(1<<4)|(1<<8);
-int y = (1<<3)+(1<<6)+(1<<8)+(1<<9);
+int y = (1<<3)|(1<<6)|(1<<8)|(1<<9);
 int z = x|y;
 cout << __builtin_popcount(z) << "\n"; // 6
 \end{lstlisting}
@@ -804,7 +804,7 @@ that only elements $0 \ldots k$
 may be removed from $S$.
 For example,
 \[\texttt{partial}(\{0,2\},1)=\texttt{value}[\{2\}]+\texttt{value}[\{0,2\}],\]
-because we only may remove elements $0 \ldots 1$.
+because we may only remove elements $0 \ldots 1$.
 We can calculate values of \texttt{sum} using
 values of \texttt{partial}, because
 \[\texttt{sum}(S) = \texttt{partial}(S,n-1).\]

chapter12.tex

@@ -350,7 +350,7 @@ while (!q.empty()) {
 Using the graph traversal algorithms,
 we can check many properties of graphs.
 Usually, both depth-first search and
-bredth-first search may be used,
+breadth-first search may be used,
 but in practice, depth-first search
 is a better choice, because it is
 easier to implement.

chapter15.tex

@@ -3,7 +3,7 @@
 \index{spanning tree}
 
 A \key{spanning tree} of a graph consists of
-the nodes of the graph and some of the
+all nodes of the graph and some of the
 edges of the graph so that there is a path
 between any two nodes.
 Like trees in general, spanning trees are

preface.tex

@@ -5,8 +5,8 @@
 The purpose of this book is to give you
 a thorough introduction to competitive programming.
 It is assumed that you already
-know the basics of programming, but previous
+know the basics of programming, but no previous
-background on competitive programming is not needed.
+background in competitive programming is needed.
 
 The book is especially intended for
 students who want to learn algorithms and