Small fixes

luku01.tex

@@ -759,7 +759,7 @@ up to an integer. For example,
 
 The functions $\min(x_1,x_2,\ldots,x_n)$
 and $\max(x_1,x_2,\ldots,x_n)$
-return the smallest and largest of values
+give the smallest and largest of values
 $x_1,x_2,\ldots,x_n$.
 For example,
 \[ \min(1,2,3)=1 \hspace{10px} \textrm{and} \hspace{10px} \max(1,2,3)=3.\]
@@ -793,7 +793,7 @@ There is also a closed-form formula
 for calculating Fibonacci numbers:
 \[f(n)=\frac{(1 + \sqrt{5})^n - (1-\sqrt{5})^n}{2^n \sqrt{5}}.\]
 
-\subsubsection{Logarithm}
+\subsubsection{Logarithms}
 
 \index{logarithm}
 

luku04.tex

@@ -20,7 +20,7 @@ Later in the book we will learn more sophisticated
 data structures that are not available
 in the standard library.
 
-\section{Dynamic array}
+\section{Dynamic arrays}
 
 \index{dynamic array}
 \index{vector}
@@ -135,7 +135,7 @@ string c = b.substr(3,4);
 cout << c << "\n"; // tiva
 \end{lstlisting}
 
-\section{Set structure}
+\section{Set structures}
 
 \index{set}
 
@@ -243,7 +243,7 @@ s.erase(s.find(5));
 cout << s.count(5) << "\n"; // 2
 \end{lstlisting}
 
-\section{Map structure}
+\section{Map structures}
 
 \index{map}
 
@@ -469,7 +469,7 @@ element that corresponds to $a$ or the previous element.
 
 \section{Other structures}
 
-\subsubsection{Bitset}
+\subsubsection{Bitsets}
 
 \index{bitset}
 
@@ -524,7 +524,7 @@ cout << (a|b) << "\n"; // 1011111110
 cout << (a^b) << "\n"; // 1001101110
 \end{lstlisting}
 
-\subsubsection{Deque}
+\subsubsection{Deques}
 
 \index{deque}
 
@@ -552,7 +552,7 @@ For this reason, a deque is slower than a vector.
 Still, the time complexity of adding and removing
 elements is $O(1)$ on average at both ends.
 
-\subsubsection{Stack}
+\subsubsection{Stacks}
 
 \index{stack}
 
@@ -574,7 +574,7 @@ cout << s.top(); // 5
 s.pop();
 cout << s.top(); // 2
 \end{lstlisting}
-\subsubsection{Queue}
+\subsubsection{Queues}
 
 \index{queue}
 
@@ -596,7 +596,7 @@ s.pop();
 cout << s.front(); // 2
 \end{lstlisting}
 
-\subsubsection{Priority queue}
+\subsubsection{Priority queues}
 
 \index{priority queue}
 \index{heap}

luku07.tex

@@ -119,7 +119,7 @@ we should form the sum $x-3$ or $x-4$.
 
 Thus, the recursive formula is
 \[f(x) = \min(f(x-1),f(x-3),f(x-4))+1\]
-where the function $\min$ returns the smallest
+where the function $\min$ gives the smallest
 of its parameters.
 In the general case, for the coin set
 $\{c_1,c_2,\ldots,c_k\}$,
@@ -250,7 +250,7 @@ that are implemented using loops.
 Still, the underlying idea is the same as
 in the recursive function.
 
-\subsubsection{Constructing the solution}
+\subsubsection{Constructing a solution}
 
 Sometimes we are asked both to find the value
 of an optimal solution and also to give
@@ -319,7 +319,7 @@ we maximize or minimize something in the recursion,
 but now we will calculate sums of numbers of solutions.
 
 To solve the problem, we can define a function $f(x)$
-that returns the number of ways to construct
+that gives the number of ways to construct
 the sum $x$ using the coins.
 For example, $f(5)=6$ when the coins are $\{1,3,4\}$.
 The value of $f(x)$ can be calculated recursively
@@ -505,9 +505,9 @@ go through all previous values
 $f(1),f(2),\ldots,f(k-1)$ and select the best solution.
 The time complexity of such an algorithm is $O(n^2)$.
 Surprisingly, it is also possible to solve the
-problem in $O(n \log n)$ time, but this is more difficult.
+problem in $O(n \log n)$ time. Can you see how?
 
-\section{Path in a grid}
+\section{Paths in a grid}
 
 Our next problem is to find a path
 in an $n \times n$ grid

luku08.tex

@@ -245,13 +245,13 @@ Since both the left and right pointer
 move $O(n)$ steps during the algorithm,
 the time complexity is $O(n)$.
 
-\subsubsection{2SUM-problem}
+\subsubsection{2SUM problem}
 
-\index{2SUM-problem}
+\index{2SUM problem}
 
 Another problem that can be solved using
 the two pointers method is the following problem,
-also known as the \key{2SUM-problem}:
+also known as the \key{2SUM problem}:
 We are given an array of $n$ numbers and
 a target sum $x$, and our task is to find two numbers
 in the array such that their sum is $x$,
@@ -414,13 +414,13 @@ number such that the sum is $x$.
 This can be done by performing $n$ binary searches,
 and each search takes $O(\log n)$ time.
 
-\index{3SUM-problem}
+\index{3SUM problem}
 A more difficult problem is 
-the \key{3SUM-problem} that asks to
+the \key{3SUM problem} that asks to
 find \emph{three} numbers in the array
 such that their sum is $x$.
 This problem can be solved in $O(n^2)$ time.
-Can you see how it is possible?
+Can you see how?
 
 \section{Nearest smaller elements}
 

luku09.tex

@@ -432,7 +432,7 @@ the union of the ranges $[2,5]$ and $[4,7]$:
 Since $\textrm{rmq}(2,5)=3$ and $\textrm{rmq}(4,7)=1$,
 we can conclude that $\textrm{rmq}(2,7)=1$.
 
-\section{Binary indexed tree}
+\section{Binary indexed trees}
 
 \index{binary indexed tree}
 \index{Fenwick tree}
@@ -567,7 +567,7 @@ corresponds to a range in the array:
 \end{tikzpicture}
 \end{center}
 
-\subsubsection{Sum query}
+\subsubsection{Sum queries}
 
 The values in the binary indexed tree
 can be used to efficiently calculate
@@ -629,7 +629,7 @@ we can use the same trick that we used with sum arrays:
 \[ \textrm{sum}(a,b) = \textrm{sum}(1,b) - \textrm{sum}(1,a-1).\]
 Also in this case, only $O(\log n)$ values are needed.
 
-\subsubsection{Array update}
+\subsubsection{Array updates}
 
 When a value in the array changes,
 several values in the binary indexed tree should be updated.
@@ -731,7 +731,7 @@ values in the binary indexed tree, and each move
 to the next position
 takes $O(1)$ time using bit operations.
 
-\section{Segment tree}
+\section{Segment trees}
 
 \index{segment tree}
 
@@ -841,7 +841,7 @@ node is the sum of the corresponding array elements,
 and it can be calculated as the sum of
 the values of its left and right child node.
 
-\subsubsection{Range query}
+\subsubsection{Range queries}
 
 The sum of elements in a given range
 can be calculated as a sum of values in the segment tree.
@@ -922,7 +922,7 @@ of the tree are needed.
 Hence, the total number of nodes
 is only $O(\log n)$.
 
-\subsubsection{Array update}
+\subsubsection{Array updates}
 
 When an element in the array changes,
 we should update all nodes in the tree
@@ -1200,7 +1200,7 @@ element in the whole array.
 The operations can be implemented like previously,
 but instead of sums, minima are calculated.
 
-\subsubsection{Binary search in tree}
+\subsubsection{Binary search in a tree}
 
 The structure of the segment tree allows us
 to use binary search for finding elements in the array.

luku16.tex

@@ -492,7 +492,7 @@ The reason for this is that any successor graph
 corresponds to a function $f$ that defines
 the edges in the graph.
 The parameter for the function is a node in the graph,
-and the function returns the successor of the node.
+and the function gives the successor of the node.
 
 \begin{samepage}
 For example, the function

luku19.tex

@@ -18,7 +18,7 @@ find a such path if it exists.
 On the contrary, checking the existence of a Hamiltonian path is a NP-hard
 problem and no efficient algorithm is known for solving the problem.
 
-\section{Eulerian path}
+\section{Eulerian paths}
 
 \index{Eulerian path}
 
@@ -391,7 +391,7 @@ to the circuit:
 Now all edges are included in the circuit,
 so we have successfully constructed an Eulerian circuit.
 
-\section{Hamiltonian path}
+\section{Hamiltonian paths}
 
 \index{Hamiltonian path}
 
@@ -521,7 +521,7 @@ The function indicates whether there is a Hamiltonian path
 that visits the nodes in $s$ and ends at node $x$.
 It is possible to implement this solution in $O(2^n n^2)$ time.
 
-\section{De Bruijn sequence}
+\section{De Bruijn sequences}
 
 \index{De Bruijn sequence}
 
@@ -573,7 +573,7 @@ The starting node has $n-1$ characters
 and there are $k^n$ characters in the edges,
 so the length of the string is $k^n+n-1$.
 
-\section{Knight's tour}
+\section{Knight's tours}
 
 \index{knight's tour}
 

luku20.tex

@@ -89,7 +89,7 @@ It is easy see that this flow is maximum,
 because the total capacity of the edges
 leading to the sink is $7$.
 
-\subsubsection{Minimum cut}
+\subsubsection{Minimum cuts}
 
 \index{cut}
 \index{minimum cut}

luku22.tex

@@ -323,7 +323,7 @@ There are $k+1$ positions,
 so the number of solutions is
 ${n-2k+1+k+1-1 \choose n-2k+1} = {n-k+1 \choose n-2k+1}$.
 
-\subsubsection{Multinomial coefficient}
+\subsubsection{Multinomial coefficients}
 
 \index{multinomial coefficient}
 

luku25.tex

@@ -386,7 +386,7 @@ sticks in a nim heap.
 When we know the Grundy numbers for all states,
 we can play the game like the nim game.
 
-\subsubsection{Grundy number}
+\subsubsection{Grundy numbers}
 
 \index{Grundy number}
 \index{mex function}

luku26.tex

@@ -108,7 +108,7 @@ It means that $x<y$ if either $x \neq y$ and $x$ is a prefix of $y$,
 or there is a position $k$ such that
 $x[i]=y[i]$ when $i<k$ and $x[k]<y[k]$.
 
-\section{Trie structure}
+\section{Trie structures}
 
 \index{trie}
 

luku28.tex

@@ -199,7 +199,7 @@ Both operations can be implemented using
 similar ideas, and it is even possible to construct
 a tree that supports both operations at the same time.
 
-\subsubsection{Lazy segment tree}
+\subsubsection{Lazy segment trees}
 
 Let us consider an example where our goal is to
 construct a segment tree that supports
@@ -506,7 +506,7 @@ In this problem, it is easy to combine lazy updates,
 because the combination of updates $z_1$ and $z_2$
 corresponds to an update $z_1+z_2$.
 
-\subsubsection{Polynomial update}
+\subsubsection{Polynomial updates}
 
 Lazy updates can be generalized so that it is
 possible to update ranges using polynomials of the form
@@ -584,7 +584,7 @@ node *u = new node(0, 0, 15);
 u->s = 5;
 \end{lstlisting}
 
-\subsubsection{Sparse segment tree}
+\subsubsection{Sparse segment trees}
 
 \index{sparse segment tree}
 
@@ -645,7 +645,7 @@ index compression (Chapter 9.4).
 However, this is not possible if the indices
 are generated during the algorithm.
 
-\subsubsection{Persistent segment tree}
+\subsubsection{Persistent segment trees}
 
 \index{persistent segment tree}
 

luku29.tex

@@ -527,7 +527,7 @@ For example, the area of the polygon
 \end{tikzpicture}
 \end{center}
 is
-\[\frac{|(2\cdot5-4\cdot5)+(5\cdot3-5\cdot7)+(7\cdot1-3\cdot4)+(4\cdot3-1\cdot4)+(4\cdot4-3\cdot2)|}{2} = 17/2.\]
+\[\frac{|(2\cdot5-5\cdot4)+(5\cdot3-7\cdot5)+(7\cdot1-4\cdot3)+(4\cdot3-4\cdot1)+(4\cdot4-2\cdot3)|}{2} = 17/2.\]
 
 The idea in the formula is to go through trapezoids
 whose one side is a side of the polygon,
@@ -565,7 +565,7 @@ all such trapezoids, which yields the formula
 Note that the absolute value of the sum is taken,
 because the value of the sum may be positive or negative
 depending on whether we walk clockwise or counterclockwise
-along the perimeter of the polygon.
+along the boundary of the polygon.
 
 \subsubsection{Pick's theorem}