From 5573a59348ae1f582b7974d61d1e86aeabbdb004 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Mon, 30 Jan 2017 20:39:11 +0200 Subject: [PATCH] Corrections --- johdanto.tex | 6 +-- luku01.tex | 148 ++++++++++++++++++++++++++------------------------- 2 files changed, 78 insertions(+), 76 deletions(-) diff --git a/johdanto.tex b/johdanto.tex index 6ceed11..2714fd1 100644 --- a/johdanto.tex +++ b/johdanto.tex @@ -4,12 +4,12 @@ The purpose of this book is to give you a thorough introduction to competitive programming. -The book assumes that you already +It is assumed that you already know the basics of programming, but previous background on competitive programming is not needed. The book is especially intended for -high school students who want to learn +secondary school students who want to learn algorithms and possibly participate in the International Olympiad in Informatics (IOI). The book is also suitable for university students @@ -19,7 +19,7 @@ It takes a long time to become a good competitive programmer, but it is also an opportunity to learn a lot. You can be sure that you will learn a great deal about algorithms if you spend time reading the book -and solving exercises. +and solving problems. The book is under continuous development. You can always send feedback about the book to diff --git a/luku01.tex b/luku01.tex index 9acf312..637612c 100644 --- a/luku01.tex +++ b/luku01.tex @@ -1,12 +1,12 @@ \chapter{Introduction} Competitive programming combines two topics: -(1) design of algorithms and (2) implementation of algorithms. +(1) the design of algorithms and (2) the implementation of algorithms. The \key{design of algorithms} consists of problem solving and mathematical thinking. Skills for analyzing problems and solving them -using creativity is needed. +creatively is needed. An algorithm for solving a problem has to be both correct and efficient, and the core of the problem is often @@ -63,23 +63,24 @@ large collection of data structures and algorithms. On the other hand, it is good to -master several languages and know the -benefits of them. -For example, if big integers are needed +master several languages and know +their strengths. +For example, if large numbers are needed in the problem, Python can be a good choice because it contains a built-in library for handling -big integers. -Still, usually the goal is to write the problems so that -the use of a specific programming language -is not an unfair advantage in the contest. +large numbers. +Still, most problems in programming contests +are set so that +using a specific programming language +is not an unfair advantage. -All examples in this book are written in C++, -and the data structures and algorithms in -the standard library are often used. -The book follows the C++11 standard, +All example programs in this book are written in C++, +and the standard library's +data structures and algorithms are often used. +The programs follow the C++11 standard, that can be used in most contests nowadays. -If you can't program in C++ yet, +If you cannot program in C++ yet, now it is a good time to start learning. \subsubsection{C++ template} @@ -155,21 +156,21 @@ or one newline between each element in the input. For example, the above code accepts both the following inputs: \begin{lstlisting} -123 456 apina +123 456 monkey \end{lstlisting} \begin{lstlisting} 123 456 -apina +monkey \end{lstlisting} The \texttt{cout} stream is used for output as follows: \begin{lstlisting} int a = 123, b = 456; -string x = "apina"; +string x = "monkey"; cout << a << " " << b << " " << x << "\n"; \end{lstlisting} -Handling input and output is sometimes +Input and output is sometimes a bottleneck in the program. The following lines at the beginning of the code make input and output more efficient: @@ -181,7 +182,7 @@ cin.tie(0); Note that the newline \texttt{"\textbackslash n"} works faster than \texttt{endl}, -becauses \texttt{endl} always causes +because \texttt{endl} always causes a flush operation. The C functions \texttt{scanf} @@ -214,7 +215,7 @@ If the amount of data is unknown, the following loop can be handy: \begin{lstlisting} while (cin >> x) { - // koodia + // code } \end{lstlisting} This loop reads elements from the input @@ -240,14 +241,14 @@ After this, the code reads the input from the file \subsubsection{Integers} -The most popular integer type in competitive programming -is \texttt{int}. This is a 32-bit type with -value range $-2^{31} \ldots 2^{31}-1$, -i.e., about $-2 \cdot 10^9 \ldots 2 \cdot 10^9$. +The most used integer type in competitive programming +is \texttt{int}, that is a 32-bit type with +value range $-2^{31} \ldots 2^{31}-1$ +or about $-2 \cdot 10^9 \ldots 2 \cdot 10^9$. If the type \texttt{int} is not enough, the 64-bit type \texttt{long long} can be used, -with value range $-2^{63} \ldots 2^{63}-1$, -i.e., about $-9 \cdot 10^{18} \ldots 9 \cdot 10^{18}$. +with value range $-2^{63} \ldots 2^{63}-1$ +or about $-9 \cdot 10^{18} \ldots 9 \cdot 10^{18}$. The following code defines a \texttt{long long} variable: @@ -257,7 +258,7 @@ long long x = 123456789123456789LL; The suffix \texttt{LL} means that the type of the number is \texttt{long long}. -A typical error when using the type \texttt{long long} +A common error when using the type \texttt{long long} is that the type \texttt{int} is still used somewhere in the code. For example, the following code contains @@ -279,13 +280,13 @@ The problem can be solved by changing the type of \texttt{a} to \texttt{long long} or by changing the expression to \texttt{(long long)a*a}. -Usually, the problems are written so that the +Usually contest problems are set so that the type \texttt{long long} is enough. Still, it is good to know that -the \texttt{g++} compiler also features +the \texttt{g++} compiler also provides an 128-bit type \texttt{\_\_int128\_t} with value range -$-2^{127} \ldots 2^{127}-1$, i.e., $-10^{38} \ldots 10^{38}$. +$-2^{127} \ldots 2^{127}-1$ or $-10^{38} \ldots 10^{38}$. However, this type is not available in all contest systems. \subsubsection{Modular arithmetic} @@ -299,12 +300,12 @@ For example, $17 \bmod 5 = 2$, because $17 = 3 \cdot 5 + 2$. Sometimes, the answer for a problem is a -very big integer but it is enough to -print it ''modulo $m$'', i.e., +very large number but it is enough to +output it ''modulo $m$'', i.e., the remainder when the answer is divided by $m$ (for example, ''modulo $10^9+7$''). The idea is that even if the actual answer -may be very big, +may be very large, it is enough to use the types \texttt{int} and \texttt{long long}. @@ -333,12 +334,12 @@ for (int i = 2; i <= n i++) { cout << x << "\n"; \end{lstlisting} -Usually, the answer should be always given so -that the remainder is between $0\ldots m-1$. +Usually the remainder should be always +be between $0\ldots m-1$. However, in C++ and other languages, -the remainder of a negative number can -be negative. -An easy way to make sure that this will +the remainder of a negative number +is either zero or negative. +An easy way to make sure this will not happen is to first calculate the remainder as usual and then add $m$ if the result is negative: @@ -363,10 +364,11 @@ In most cases, \texttt{double} is enough, but \texttt{long double} is more accurate. The required precision of the answer -is usually given. -The easiest way is to use +is usually given in the problem statement. +An easy way to output the answer is to use the \texttt{printf} function -that can be given the number of decimal places. +and give the number of decimal places +in the formatting string. For example, the following code prints the value of $x$ with 9 decimal places: @@ -376,7 +378,7 @@ printf("%.9f\n", x); A difficulty when using floating point numbers is that some numbers cannot be represented -accurately, but there will be rounding errors. +accurately but there will be rounding errors. For example, the result of the following code is surprising: @@ -417,7 +419,7 @@ integers having absolute value at most $2^{53}$. \section{Shortening code} Short code is ideal in competitive programming, -because the algorithm should be implemented +because algorithms should be implemented as fast as possible. Because of this, competitive programmers often define shorter names for datatypes and other parts of code. @@ -485,9 +487,9 @@ v.PB(MP(y2,x2)); int d = v[i].F+v[i].S; \end{lstlisting} -It is also possible to define a macro with parameters +A macro can also have parameters which makes it possible to shorten loops and other -structures in the code. +structures. For example, we can define the following macro: \begin{lstlisting} #define REP(i,a,b) for (int i = a; i <= b; i++) @@ -495,13 +497,13 @@ For example, we can define the following macro: After this, the code \begin{lstlisting} for (int i = 1; i <= n; i++) { - haku(i); + search(i); } \end{lstlisting} can be shortened as follows: \begin{lstlisting} REP(i,1,n) { - haku(i); + search(i); } \end{lstlisting} @@ -566,7 +568,7 @@ This formula can be derived as follows. Let By multiplying both sides by $x$, we get \[ xS = ax + ax^2 + ax^3 + \cdots + bx,\] and solving the equation -\[ xS-S = bx-a.\] +\[ xS-S = bx-a\] yields the formula. A special case of a geometric sum is the formula @@ -578,9 +580,9 @@ A \key{harmonic sum} is a sum of the form \[ \sum_{x=1}^n \frac{1}{x} = 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}.\] An upper bound for the harmonic sum is $\log_2(n)+1$. -The reason for this is that we can -change each term $1/k$ so that $k$ becomes -a power of two that doesn't exceed $k$. +Namely, we can +modify each term $1/k$ so that $k$ becomes +the nearest power of two that does not exceed $k$. For example, when $n=6$, we can estimate the sum as follows: \[ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6} \le @@ -605,18 +607,18 @@ For example, the set \[X=\{2,4,7\}\] contains elements 2, 4 and 7. The symbol $\emptyset$ denotes an empty set, -and $|S|$ denotes the size of set $S$, +and $|S|$ denotes the size of the set $S$, i.e., the number of elements in the set. For example, in the above set, $|X|=3$. -If set $S$ contains element $x$, +If a set $S$ contains an element $x$, we write $x \in S$, and otherwise we write $x \notin S$. For example, in the above set \[4 \in X \hspace{10px}\textrm{and}\hspace{10px} 5 \notin X.\] \begin{samepage} -New sets can be constructed as follows using set operations: +New sets can be constructed using set operations: \begin{itemize} \item The \key{intersection} $A \cap B$ consists of elements that are both in $A$ and $B$. @@ -631,7 +633,7 @@ that are not in $A$. The interpretation of a complement depends on the \key{universal set} that contains all possible elements. For example, if $A=\{1,2,5,7\}$ and the universal set is -$P=\{1,2,\ldots,10\}$, then $\bar A = \{3,4,6,8,9,10\}$. +$\{1,2,\ldots,10\}$, then $\bar A = \{3,4,6,8,9,10\}$. \item The \key{difference} $A \setminus B = A \cap \bar B$ consists of elements that are in $A$ but not in $B$. Note that $B$ can contain elements that are not in $A$. @@ -643,12 +645,12 @@ then $A \setminus B = \{2,7\}$. If each element of $A$ also belongs to $S$, we say that $A$ is a \key{subset} of $S$, denoted by $A \subset S$. -Set $S$ always has $2^{|S|}$ subsets, +A set $S$ always has $2^{|S|}$ subsets, including the empty set. For example, the subsets of the set $\{2,4,7\}$ are \begin{center} $\emptyset$, -$\{2\}$, $\{4\}$, $\{7\}$, $\{2,4\}$, $\{2,7\}$, $\{4,7\}$ ja $\{2,4,7\}$. +$\{2\}$, $\{4\}$, $\{7\}$, $\{2,4\}$, $\{2,7\}$, $\{4,7\}$ and $\{2,4,7\}$. \end{center} Often used sets are @@ -669,7 +671,7 @@ or $\mathbb{N}=\{1,2,3,...\}$. We can also construct a set using a rule of the form \[\{f(n) : n \in S\},\] where $f(n)$ is some function. -This set contains all elements $f(n)$ +This set contains all elements of the form $f(n)$, where $n$ is an element in $S$. For example, the set \[X=\{2n : n \in \mathbb{Z}\}\] @@ -692,7 +694,7 @@ $\land$ (\key{conjunction}), $\lor$ (\key{disjunction}), $\Rightarrow$ (\key{implication}) and $\Leftrightarrow$ (\key{equivalence}). -The following table shows the meaning of the operators: +The following table shows the meaning of these operators: \begin{center} \begin{tabular}{rr|rrrrrrr} @@ -705,7 +707,7 @@ $A$ & $B$ & $\lnot A$ & $\lnot B$ & $A \land B$ & $A \lor B$ & $A \Rightarrow B$ \end{tabular} \end{center} -The negation $\lnot A$ reverses the value of an expression. +The expression $\lnot A$ has the reverse value of $A$. The expression $A \land B$ is true if both $A$ and $B$ are true, and the expression $A \lor B$ is true if $A$ or $B$ or both @@ -741,10 +743,10 @@ but false in the set of natural numbers. Using the notation described above, we can express many kinds of logical propositions. For example, -\[\forall x ((x>2 \land \lnot P(x)) \Rightarrow (\exists a (\exists b (x = ab \land a > 1 \land b > 1))))\] -means that if a number $x$ is larger than 2 +\[\forall x ((x>1 \land \lnot P(x)) \Rightarrow (\exists a (\exists b (x = ab \land a > 1 \land b > 1))))\] +means that if a number $x$ is larger than 1 and not a prime number, -there are numbers $a$ and $b$ +then there exist numbers $a$ and $b$ that are larger than $1$ and whose product is $x$. This proposition is true in the set of integers. @@ -758,14 +760,14 @@ 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 the largest of values +return the smallest and largest of values $x_1,x_2,\ldots,x_n$. For example, \[ \min(1,2,3)=1 \hspace{10px} \textrm{and} \hspace{10px} \max(1,2,3)=3.\] \index{factorial} -The \key{factorial} $n!$ is defined +The \key{factorial} $n!$ can be defined \[\prod_{x=1}^n x = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\] or recursively \[ @@ -777,7 +779,7 @@ n! & = & n \cdot (n-1)! \\ \index{Fibonacci number} -The \key{Fibonacci numbers} arise in several situations. +The \key{Fibonacci numbers} arise in many situations. They can be defined recursively as follows: \[ \begin{array}{lcl} @@ -797,12 +799,12 @@ for calculating Fibonacci numbers: \index{logarithm} The \key{logarithm} of a number $x$ -is denoted $\log_k(x)$ where $k$ is the base +is denoted $\log_k(x)$, where $k$ is the base of the logarithm. -The logarithm is defined so that +According to the definition, $\log_k(x)=a$ exactly when $k^a=x$. -A useful interpretation in algorithmics is +A useful interpretation in algorithm design is that $\log_k(x)$ equals the number of times we have to divide $x$ by $k$ before we reach the number 1. @@ -812,10 +814,10 @@ because 5 divisions are needed: \[32 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \] Logarithms are often needed in the analysis of -algorithms because many efficient algorithms +algorithms, because many efficient algorithms divide in half something at each step. -Thus, we can estimate the efficiency of those algorithms -using the logarithm. +Hence, we can estimate the efficiency of such algorithms +using logarithms. The logarithm of a product is \[\log_k(ab) = \log_k(a)+\log_k(b),\] @@ -834,7 +836,7 @@ calculate logarithms to some fixed base. The \key{natural logarithm} $\ln(x)$ of a number $x$ is a logarithm whose base is $e \approx 2{,}71828$. -Another property of the logarithm is that +Another property of logarithms is that the number of digits of a number $x$ in base $b$ is $\lfloor \log_b(x)+1 \rfloor$. For example, the representation of