993 lines
29 KiB
TeX
993 lines
29 KiB
TeX
\chapter{Introduction}
|
|
|
|
Competitive programming combines two topics:
|
|
(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
|
|
creatively are needed.
|
|
An algorithm for solving a problem
|
|
has to be both correct and efficient,
|
|
and the core of the problem is often
|
|
about inventing an efficient algorithm.
|
|
|
|
Theoretical knowledge of algorithms
|
|
is very important to competitive programmers.
|
|
Typically, a solution to a problem is
|
|
a combination of well-known techniques and
|
|
new insights.
|
|
The techniques that appear in competitive programming
|
|
also form the basis for the scientific research
|
|
of algorithms.
|
|
|
|
The \key{implementation of algorithms} requires good
|
|
programming skills.
|
|
In competitive programming, the solutions
|
|
are graded by testing an implemented algorithm
|
|
using a set of test cases.
|
|
Thus, it is not enough that the idea of the
|
|
algorithm is correct, but the implementation also
|
|
has to be correct.
|
|
|
|
A good coding style in contests is
|
|
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
|
|
hundreds of lines) and it is not needed to
|
|
maintain them after the contest.
|
|
|
|
\section{Programming languages}
|
|
|
|
\index{programming language}
|
|
|
|
At the moment, the most popular programming
|
|
languages used in contests are C++, Python and Java.
|
|
For example, in Google Code Jam 2016,
|
|
among the best 3,000 participants,
|
|
73 \% used C++,
|
|
15 \% used Python and
|
|
10 \% used Java \cite{goo16}.
|
|
Some participants also used several languages.
|
|
|
|
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
|
|
it is a very efficient language and
|
|
its standard library contains a
|
|
large collection
|
|
of data structures and algorithms.
|
|
|
|
On the other hand, it is good to
|
|
master several languages and understand
|
|
their strengths.
|
|
For example, if large integers are needed
|
|
in the problem,
|
|
Python can be a good choice, because it
|
|
contains built-in operations for
|
|
calculating with large integers.
|
|
Still, most problems in programming contests
|
|
are set so that
|
|
using a specific programming language
|
|
is not an unfair advantage.
|
|
|
|
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,
|
|
which can be used in most contests nowadays.
|
|
If you cannot program in C++ yet,
|
|
now is a good time to start learning.
|
|
|
|
\subsubsection{C++ code template}
|
|
|
|
A typical C++ code template for competitive programming
|
|
looks like this:
|
|
|
|
\begin{lstlisting}
|
|
#include <bits/stdc++.h>
|
|
|
|
using namespace std;
|
|
|
|
int main() {
|
|
// solution comes here
|
|
}
|
|
\end{lstlisting}
|
|
|
|
The \texttt{\#include} line at the beginning
|
|
of the code is a feature of the \texttt{g++} compiler
|
|
that allows us to include the entire standard library.
|
|
Thus, it is not needed to separately include
|
|
libraries such as \texttt{iostream},
|
|
\texttt{vector} and \texttt{algorithm},
|
|
but rather they are available automatically.
|
|
|
|
The \texttt{using} line declares
|
|
that the classes and functions
|
|
of the standard library can be used directly
|
|
in the code.
|
|
Without the \texttt{using} line we would have
|
|
to write, for example, \texttt{std::cout},
|
|
but now it suffices to write \texttt{cout}.
|
|
|
|
The code can be compiled using the following command:
|
|
|
|
\begin{lstlisting}
|
|
g++ -std=c++11 -O2 -Wall code.cpp -o bin
|
|
\end{lstlisting}
|
|
|
|
This command produces a binary file \texttt{bin}
|
|
from the source code \texttt{code.cpp}.
|
|
The compiler follows the C++11 standard
|
|
(\texttt{-std=c++11}),
|
|
optimizes the code (\texttt{-O2})
|
|
and shows warnings about possible errors (\texttt{-Wall}).
|
|
|
|
\section{Input and output}
|
|
|
|
\index{input and output}
|
|
|
|
In most contests, standard streams are used for
|
|
reading input and writing output.
|
|
In C++, the standard streams are
|
|
\texttt{cin} for input and \texttt{cout} for output.
|
|
In addition, the C functions
|
|
\texttt{scanf} and \texttt{printf} can be used.
|
|
|
|
The input for the program usually consists of
|
|
numbers and strings that are separated with
|
|
spaces and newlines.
|
|
They can be read from the \texttt{cin} stream
|
|
as follows:
|
|
|
|
\begin{lstlisting}
|
|
int a, b;
|
|
string x;
|
|
cin >> a >> b >> x;
|
|
\end{lstlisting}
|
|
|
|
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:
|
|
\begin{lstlisting}
|
|
123 456 monkey
|
|
\end{lstlisting}
|
|
\begin{lstlisting}
|
|
123 456
|
|
monkey
|
|
\end{lstlisting}
|
|
The \texttt{cout} stream is used for output
|
|
as follows:
|
|
\begin{lstlisting}
|
|
int a = 123, b = 456;
|
|
string x = "monkey";
|
|
cout << a << " " << b << " " << x << "\n";
|
|
\end{lstlisting}
|
|
|
|
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:
|
|
|
|
\begin{lstlisting}
|
|
ios_base::sync_with_stdio(0);
|
|
cin.tie(0);
|
|
\end{lstlisting}
|
|
|
|
Note that the newline \texttt{"\textbackslash n"}
|
|
works faster than \texttt{endl},
|
|
because \texttt{endl} always causes
|
|
a flush operation.
|
|
|
|
The C functions \texttt{scanf}
|
|
and \texttt{printf} are an alternative
|
|
to the C++ standard streams.
|
|
They are usually a bit faster,
|
|
but they are also more difficult to use.
|
|
The following code reads two integers from the input:
|
|
\begin{lstlisting}
|
|
int a, b;
|
|
scanf("%d %d", &a, &b);
|
|
\end{lstlisting}
|
|
The following code prints two integers:
|
|
\begin{lstlisting}
|
|
int a = 123, b = 456;
|
|
printf("%d %d\n", a, b);
|
|
\end{lstlisting}
|
|
|
|
Sometimes the program should read a whole line
|
|
from the input, possibly containing spaces.
|
|
This can be accomplished by using the
|
|
\texttt{getline} function:
|
|
|
|
\begin{lstlisting}
|
|
string s;
|
|
getline(cin, s);
|
|
\end{lstlisting}
|
|
|
|
If the amount of data is unknown, the following
|
|
loop is useful:
|
|
\begin{lstlisting}
|
|
while (cin >> x) {
|
|
// code
|
|
}
|
|
\end{lstlisting}
|
|
This loop reads elements from the input
|
|
one after another, until there is no
|
|
more data available in the input.
|
|
|
|
In some contest systems, files are used for
|
|
input and output.
|
|
An easy solution for this is to write
|
|
the code as usual using standard streams,
|
|
but add the following lines to the beginning of the code:
|
|
\begin{lstlisting}
|
|
freopen("input.txt", "r", stdin);
|
|
freopen("output.txt", "w", stdout);
|
|
\end{lstlisting}
|
|
After this, the program reads the input from the file
|
|
''input.txt'' and writes the output to the file
|
|
''output.txt''.
|
|
|
|
\section{Working with numbers}
|
|
|
|
\index{integer}
|
|
|
|
\subsubsection{Integers}
|
|
|
|
The most used integer type in competitive programming
|
|
is \texttt{int}, which is a 32-bit type with
|
|
a value range of $-2^{31} \ldots 2^{31}-1$
|
|
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.
|
|
It has a value range of $-2^{63} \ldots 2^{63}-1$
|
|
or about $-9 \cdot 10^{18} \ldots 9 \cdot 10^{18}$.
|
|
|
|
The following code defines a
|
|
\texttt{long long} variable:
|
|
\begin{lstlisting}
|
|
long long x = 123456789123456789LL;
|
|
\end{lstlisting}
|
|
The suffix \texttt{LL} means that the
|
|
type of the number is \texttt{long long}.
|
|
|
|
A common mistake 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
|
|
a subtle error:
|
|
|
|
\begin{lstlisting}
|
|
int a = 123456789;
|
|
long long b = a*a;
|
|
cout << b << "\n"; // -1757895751
|
|
\end{lstlisting}
|
|
|
|
Even though the variable \texttt{b} is of type \texttt{long long},
|
|
both numbers in the expression \texttt{a*a}
|
|
are of type \texttt{int} and the result is
|
|
also of type \texttt{int}.
|
|
Because of this, the variable \texttt{b} will
|
|
contain a wrong result.
|
|
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 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 provides
|
|
a 128-bit type \texttt{\_\_int128\_t}
|
|
with a value range of
|
|
$-2^{127} \ldots 2^{127}-1$ or about $-10^{38} \ldots 10^{38}$.
|
|
However, this type is not available in all contest systems.
|
|
|
|
\subsubsection{Modular arithmetic}
|
|
|
|
\index{remainder}
|
|
\index{modular arithmetic}
|
|
|
|
We denote by $x \bmod m$ the remainder
|
|
when $x$ is divided by $m$.
|
|
For example, $17 \bmod 5 = 2$,
|
|
because $17 = 3 \cdot 5 + 2$.
|
|
|
|
Sometimes, the answer to a problem is a
|
|
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
|
|
is very large,
|
|
it suffices to use the types
|
|
\texttt{int} and \texttt{long long}.
|
|
|
|
An important property of the remainder is that
|
|
in addition, subtraction and multiplication,
|
|
the remainder can be taken before the operation:
|
|
|
|
\[
|
|
\begin{array}{rcr}
|
|
(a+b) \bmod m & = & (a \bmod m + b \bmod m) \bmod m \\
|
|
(a-b) \bmod m & = & (a \bmod m - b \bmod m) \bmod m \\
|
|
(a \cdot b) \bmod m & = & (a \bmod m \cdot b \bmod m) \bmod m
|
|
\end{array}
|
|
\]
|
|
|
|
Thus, we can take the remainder after every operation
|
|
and the numbers will never become too large.
|
|
|
|
For example, the following code calculates $n!$,
|
|
the factorial of $n$, modulo $m$:
|
|
\begin{lstlisting}
|
|
long long x = 1;
|
|
for (int i = 2; i <= n i++) {
|
|
x = (x*i)%m;
|
|
}
|
|
cout << x%m << "\n";
|
|
\end{lstlisting}
|
|
|
|
Usually the remainder should always
|
|
be between $0\ldots m-1$.
|
|
However, in C++ and other languages,
|
|
the remainder of a negative number
|
|
is either zero or negative.
|
|
An easy way to make sure there
|
|
are no negative remainders is to first calculate
|
|
the remainder as usual and then add $m$
|
|
if the result is negative:
|
|
\begin{lstlisting}
|
|
x = x%m;
|
|
if (x < 0) x += m;
|
|
\end{lstlisting}
|
|
However, this is only needed when there
|
|
are subtractions in the code and the
|
|
remainder may become negative.
|
|
|
|
\subsubsection{Floating point numbers}
|
|
|
|
\index{floating point number}
|
|
|
|
The usual floating point types in
|
|
competitive programming are
|
|
the 64-bit \texttt{double}
|
|
and, as an extension in the \texttt{g++} compiler,
|
|
the 80-bit \texttt{long double}.
|
|
In most cases, \texttt{double} is enough,
|
|
but \texttt{long double} is more accurate.
|
|
|
|
The required precision of the answer
|
|
is usually given in the problem statement.
|
|
An easy way to output the answer is to use
|
|
the \texttt{printf} function
|
|
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:
|
|
|
|
\begin{lstlisting}
|
|
printf("%.9f\n", x);
|
|
\end{lstlisting}
|
|
|
|
A difficulty when using floating point numbers
|
|
is that some numbers cannot be represented
|
|
accurately as floating point numbers,
|
|
and there will be rounding errors.
|
|
For example, the result of the following code
|
|
is surprising:
|
|
|
|
\begin{lstlisting}
|
|
double x = 0.3*3+0.1;
|
|
printf("%.20f\n", x); // 0.99999999999999988898
|
|
\end{lstlisting}
|
|
|
|
Due to a rounding error,
|
|
the value of \texttt{x} is a bit smaller than 1,
|
|
while the correct value would be 1.
|
|
|
|
It is risky to compare floating point numbers
|
|
with the \texttt{==} operator,
|
|
because it is possible that the values should be
|
|
equal but they are not because of precision errors.
|
|
A better way to compare floating point numbers
|
|
is to assume that two numbers are equal
|
|
if the difference between them is less than $\varepsilon$,
|
|
where $\varepsilon$ is a small number.
|
|
|
|
In practice, the numbers can be compared
|
|
as follows ($\varepsilon=10^{-9}$):
|
|
|
|
\begin{lstlisting}
|
|
if (abs(a-b) < 1e-9) {
|
|
// a and b are equal
|
|
}
|
|
\end{lstlisting}
|
|
|
|
Note that while floating point numbers are inaccurate,
|
|
integers up to a certain limit can still be
|
|
represented accurately.
|
|
For example, using \texttt{double},
|
|
it is possible to accurately represent all
|
|
integers whose absolute value is at most $2^{53}$.
|
|
|
|
\section{Shortening code}
|
|
|
|
Short code is ideal in competitive programming,
|
|
because programs should be written
|
|
as fast as possible.
|
|
Because of this, competitive programmers often define
|
|
shorter names for datatypes and other parts of code.
|
|
|
|
\subsubsection{Type names}
|
|
\index{tuppdef@\texttt{typedef}}
|
|
Using the command \texttt{typedef}
|
|
it is possible to give a shorter name
|
|
to a datatype.
|
|
For example, the name \texttt{long long} is long,
|
|
so we can define a shorter name \texttt{ll}:
|
|
\begin{lstlisting}
|
|
typedef long long ll;
|
|
\end{lstlisting}
|
|
After this, the code
|
|
\begin{lstlisting}
|
|
long long a = 123456789;
|
|
long long b = 987654321;
|
|
cout << a*b << "\n";
|
|
\end{lstlisting}
|
|
can be shortened as follows:
|
|
\begin{lstlisting}
|
|
ll a = 123456789;
|
|
ll b = 987654321;
|
|
cout << a*b << "\n";
|
|
\end{lstlisting}
|
|
|
|
The command \texttt{typedef}
|
|
can also be used with more complex types.
|
|
For example, the following code gives
|
|
the name \texttt{vi} for a vector of integers
|
|
and the name \texttt{pi} for a pair
|
|
that contains two integers.
|
|
\begin{lstlisting}
|
|
typedef vector<int> vi;
|
|
typedef pair<int,int> pi;
|
|
\end{lstlisting}
|
|
|
|
\subsubsection{Macros}
|
|
\index{macro}
|
|
Another way to shorten code is to define
|
|
\key{macros}.
|
|
A macro means that certain strings in
|
|
the code will be changed before the compilation.
|
|
In C++, macros are defined using the
|
|
\texttt{\#define} keyword.
|
|
|
|
For example, we can define the following macros:
|
|
\begin{lstlisting}
|
|
#define F first
|
|
#define S second
|
|
#define PB push_back
|
|
#define MP make_pair
|
|
\end{lstlisting}
|
|
After this, the code
|
|
\begin{lstlisting}
|
|
v.push_back(make_pair(y1,x1));
|
|
v.push_back(make_pair(y2,x2));
|
|
int d = v[i].first+v[i].second;
|
|
\end{lstlisting}
|
|
can be shortened as follows:
|
|
\begin{lstlisting}
|
|
v.PB(MP(y1,x1));
|
|
v.PB(MP(y2,x2));
|
|
int d = v[i].F+v[i].S;
|
|
\end{lstlisting}
|
|
|
|
A macro can also have parameters
|
|
which makes it possible to shorten loops and other
|
|
structures.
|
|
For example, we can define the following macro:
|
|
\begin{lstlisting}
|
|
#define REP(i,a,b) for (int i = a; i <= b; i++)
|
|
\end{lstlisting}
|
|
After this, the code
|
|
\begin{lstlisting}
|
|
for (int i = 1; i <= n; i++) {
|
|
search(i);
|
|
}
|
|
\end{lstlisting}
|
|
can be shortened as follows:
|
|
\begin{lstlisting}
|
|
REP(i,1,n) {
|
|
search(i);
|
|
}
|
|
\end{lstlisting}
|
|
|
|
Sometimes macros cause bugs that may be difficult
|
|
to detect. For example, consider the following macro
|
|
that calculates the square of a number:
|
|
\begin{lstlisting}
|
|
#define SQ(a) a*a
|
|
\end{lstlisting}
|
|
This macro \emph{does not} always work as expected.
|
|
For example, the code
|
|
\begin{lstlisting}
|
|
cout << SQ(3+3) << "\n";
|
|
\end{lstlisting}
|
|
corresponds to the code
|
|
\begin{lstlisting}
|
|
cout << 3+3*3+3 << "\n"; // 15
|
|
\end{lstlisting}
|
|
|
|
A better version of the macro is as follows:
|
|
\begin{lstlisting}
|
|
#define SQ(a) (a)*(a)
|
|
\end{lstlisting}
|
|
Now the code
|
|
\begin{lstlisting}
|
|
cout << SQ(3+3) << "\n";
|
|
\end{lstlisting}
|
|
corresponds to the code
|
|
\begin{lstlisting}
|
|
cout << (3+3)*(3+3) << "\n"; // 36
|
|
\end{lstlisting}
|
|
|
|
|
|
\section{Mathematics}
|
|
|
|
Mathematics plays an important role in competitive
|
|
programming, and it is not possible to become
|
|
a successful competitive programmer without
|
|
having good mathematical skills.
|
|
This section discusses some important
|
|
mathematical concepts and formulas that
|
|
are needed later in the book.
|
|
|
|
\subsubsection{Sum formulas}
|
|
|
|
Each sum of the form
|
|
\[\sum_{x=1}^n x^k = 1^k+2^k+3^k+\ldots+n^k,\]
|
|
where $k$ is a positive integer,
|
|
has a closed-form formula that is a
|
|
polynomial of degree $k+1$.
|
|
For example\footnote{\index{Faulhaber's formula}
|
|
There is even a general formula for such sums, called \key{Faulhaber's formula},
|
|
but it is too complex to be presented here.},
|
|
\[\sum_{x=1}^n x = 1+2+3+\ldots+n = \frac{n(n+1)}{2}\]
|
|
and
|
|
\[\sum_{x=1}^n x^2 = 1^2+2^2+3^2+\ldots+n^2 = \frac{n(n+1)(2n+1)}{6}.\]
|
|
|
|
An \key{arithmetic progression} is a \index{arithmetic progression}
|
|
sequence of numbers
|
|
where the difference between any two consecutive
|
|
numbers is constant.
|
|
For example,
|
|
\[3, 7, 11, 15\]
|
|
is an arithmetic progression with constant 4.
|
|
The sum of an arithmetic progression can be calculated
|
|
using the formula
|
|
\[\frac{n(a+b)}{2}\]
|
|
where $a$ is the first number,
|
|
$b$ is the last number and
|
|
$n$ is the amount of numbers.
|
|
For example,
|
|
\[3+7+11+15=\frac{4 \cdot (3+15)}{2} = 36.\]
|
|
The formula is based on the fact
|
|
that the sum consists of $n$ numbers and
|
|
the value of each number is $(a+b)/2$ on average.
|
|
|
|
\index{geometric progression}
|
|
A \key{geometric progression} is a sequence
|
|
of numbers
|
|
where the ratio between any two consecutive
|
|
numbers is constant.
|
|
For example,
|
|
\[3,6,12,24\]
|
|
is a geometric progression with constant 2.
|
|
The sum of a geometric progression can be calculated
|
|
using the formula
|
|
\[\frac{bx-a}{x-1}\]
|
|
where $a$ is the first number,
|
|
$b$ is the last number and the
|
|
ratio between consecutive numbers is $x$.
|
|
For example,
|
|
\[3+6+12+24=\frac{24 \cdot 2 - 3}{2-1} = 45.\]
|
|
|
|
This formula can be derived as follows. Let
|
|
\[ S = a + ax + ax^2 + \cdots + b .\]
|
|
By multiplying both sides by $x$, we get
|
|
\[ xS = ax + ax^2 + ax^3 + \cdots + bx,\]
|
|
and solving the equation
|
|
\[ xS-S = bx-a\]
|
|
yields the formula.
|
|
|
|
A special case of a sum of a geometric progression is the formula
|
|
\[1+2+4+8+\ldots+2^{n-1}=2^n-1.\]
|
|
|
|
\index{harmonic sum}
|
|
|
|
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 a harmonic sum is $\log_2(n)+1$.
|
|
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:
|
|
\[ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6} \le
|
|
1+\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}.\]
|
|
This upper bound consists of $\log_2(n)+1$ parts
|
|
($1$, $2 \cdot 1/2$, $4 \cdot 1/4$, etc.),
|
|
and the value of each part is at most 1.
|
|
|
|
\subsubsection{Set theory}
|
|
|
|
\index{set theory}
|
|
\index{set}
|
|
\index{intersection}
|
|
\index{union}
|
|
\index{difference}
|
|
\index{subset}
|
|
\index{universal set}
|
|
\index{complement}
|
|
|
|
A \key{set} is a collection of elements.
|
|
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 a set $S$,
|
|
i.e., the number of elements in the set.
|
|
For example, in the above set, $|X|=3$.
|
|
|
|
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 using set operations:
|
|
\begin{itemize}
|
|
\item The \key{intersection} $A \cap B$ consists of elements
|
|
that are in both $A$ and $B$.
|
|
For example, if $A=\{1,2,5\}$ and $B=\{2,4\}$,
|
|
then $A \cap B = \{2\}$.
|
|
\item The \key{union} $A \cup B$ consists of elements
|
|
that are in $A$ or $B$ or both.
|
|
For example, if $A=\{3,7\}$ and $B=\{2,3,8\}$,
|
|
then $A \cup B = \{2,3,7,8\}$.
|
|
\item The \key{complement} $\bar A$ consists of elements
|
|
that are not in $A$.
|
|
The interpretation of a complement depends on
|
|
the \key{universal set}, which contains all possible elements.
|
|
For example, if $A=\{1,2,5,7\}$ and the universal set is
|
|
$\{1,2,\ldots,10\}$, 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 $A$.
|
|
For example, if $A=\{2,3,7,8\}$ and $B=\{3,5,8\}$,
|
|
then $A \setminus B = \{2,7\}$.
|
|
\end{itemize}
|
|
\end{samepage}
|
|
|
|
If each element of $A$ also belongs to $S$,
|
|
we say that $A$ is a \key{subset} of $S$,
|
|
denoted by $A \subset S$.
|
|
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}
|
|
$\emptyset$,
|
|
$\{2\}$, $\{4\}$, $\{7\}$, $\{2,4\}$, $\{2,7\}$, $\{4,7\}$ and $\{2,4,7\}$.
|
|
\end{center}
|
|
|
|
Some often used sets are
|
|
$\mathbb{N}$ (natural numbers),
|
|
$\mathbb{Z}$ (integers),
|
|
$\mathbb{Q}$ (rational numbers) and
|
|
$\mathbb{R}$ (real numbers).
|
|
The set $\mathbb{N}$
|
|
can be defined in two ways, depending
|
|
on the situation:
|
|
either $\mathbb{N}=\{0,1,2,\ldots\}$
|
|
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 of the form $f(n)$,
|
|
where $n$ is an element in $S$.
|
|
For example, the set
|
|
\[X=\{2n : n \in \mathbb{Z}\}\]
|
|
contains all even integers.
|
|
|
|
\subsubsection{Logic}
|
|
|
|
\index{logic}
|
|
\index{negation}
|
|
\index{conjuction}
|
|
\index{disjunction}
|
|
\index{implication}
|
|
\index{equivalence}
|
|
|
|
The value of a logical expression is either
|
|
\key{true} (1) or \key{false} (0).
|
|
The most important logical operators are
|
|
$\lnot$ (\key{negation}),
|
|
$\land$ (\key{conjunction}),
|
|
$\lor$ (\key{disjunction}),
|
|
$\Rightarrow$ (\key{implication}) and
|
|
$\Leftrightarrow$ (\key{equivalence}).
|
|
The following table shows the meanings of these operators:
|
|
|
|
\begin{center}
|
|
\begin{tabular}{rr|rrrrrrr}
|
|
$A$ & $B$ & $\lnot A$ & $\lnot B$ & $A \land B$ & $A \lor B$ & $A \Rightarrow B$ & $A \Leftrightarrow B$ \\
|
|
\hline
|
|
0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
|
|
0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\
|
|
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
|
|
1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
The expression $\lnot A$ has the opposite 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
|
|
are true.
|
|
The expression $A \Rightarrow B$ is true
|
|
if whenever $A$ is true, also $B$ is true.
|
|
The expression $A \Leftrightarrow B$ is true
|
|
if $A$ and $B$ are both true or both false.
|
|
|
|
\index{predicate}
|
|
|
|
A \key{predicate} is an expression that is true or false
|
|
depending on its parameters.
|
|
Predicates are usually denoted by capital letters.
|
|
For example, we can define a predicate $P(x)$
|
|
that is true exactly when $x$ is a prime number.
|
|
Using this definition, $P(7)$ is true but $P(8)$ is false.
|
|
|
|
\index{quantifier}
|
|
|
|
A \key{quantifier} connects a logical expression
|
|
to the elements of a set.
|
|
The most important quantifiers are
|
|
$\forall$ (\key{for all}) and $\exists$ (\key{there is}).
|
|
For example,
|
|
\[\forall x (\exists y (y < x))\]
|
|
means that for each element $x$ in the set,
|
|
there is an element $y$ in the set
|
|
such that $y$ is smaller than $x$.
|
|
This is true in the set of integers,
|
|
but false in the set of natural numbers.
|
|
|
|
Using the notation described above,
|
|
we can express many kinds of logical propositions.
|
|
For example,
|
|
\[\forall x ((x>1 \land \lnot P(x)) \Rightarrow (\exists a (\exists b (x = ab \land a > 1 \land b > 1))))\]
|
|
means that if a number $x$ is larger than 1
|
|
and not a prime number,
|
|
then there are numbers $a$ and $b$
|
|
that are larger than $1$ and whose product is $x$.
|
|
This proposition is true in the set of integers.
|
|
|
|
\subsubsection{Functions}
|
|
|
|
The function $\lfloor x \rfloor$ rounds the number $x$
|
|
down to an integer, and the function
|
|
$\lceil x \rceil$ rounds the number $x$
|
|
up to an integer. For example,
|
|
\[ \lfloor 3/2 \rfloor = 1 \hspace{10px} \textrm{and} \hspace{10px} \lceil 3/2 \rceil = 2.\]
|
|
|
|
The functions $\min(x_1,x_2,\ldots,x_n)$
|
|
and $\max(x_1,x_2,\ldots,x_n)$
|
|
give the smallest and largest of values
|
|
$x_1,x_2,\ldots,x_n$.
|
|
For example,
|
|
\[ \min(1,2,3)=1 \hspace{10px} \textrm{and} \hspace{10px} \max(1,2,3)=3.\]
|
|
|
|
\index{factorial}
|
|
|
|
The \key{factorial} $n!$ can be defined
|
|
\[\prod_{x=1}^n x = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\]
|
|
or recursively
|
|
\[
|
|
\begin{array}{lcl}
|
|
0! & = & 1 \\
|
|
n! & = & n \cdot (n-1)! \\
|
|
\end{array}
|
|
\]
|
|
|
|
\index{Fibonacci number}
|
|
|
|
The \key{Fibonacci numbers}
|
|
%\footnote{Fibonacci (c. 1175--1250) was an Italian mathematician.}
|
|
arise in many situations.
|
|
They can be defined recursively as follows:
|
|
\[
|
|
\begin{array}{lcl}
|
|
f(0) & = & 0 \\
|
|
f(1) & = & 1 \\
|
|
f(n) & = & f(n-1)+f(n-2) \\
|
|
\end{array}
|
|
\]
|
|
The first Fibonacci numbers are
|
|
\[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots\]
|
|
There is also a closed-form formula
|
|
for calculating Fibonacci numbers\footnote{This formula is sometimes called
|
|
\index{Binet's formula} \key{Binet's formula}.}:
|
|
\[f(n)=\frac{(1 + \sqrt{5})^n - (1-\sqrt{5})^n}{2^n \sqrt{5}}.\]
|
|
|
|
\subsubsection{Logarithms}
|
|
|
|
\index{logarithm}
|
|
|
|
The \key{logarithm} of a number $x$
|
|
is denoted $\log_k(x)$, where $k$ is the base
|
|
of the logarithm.
|
|
According to the definition,
|
|
$\log_k(x)=a$ exactly when $k^a=x$.
|
|
|
|
A useful property of logarithms is
|
|
that $\log_k(x)$ equals the number of times
|
|
we have to divide $x$ by $k$ before we reach
|
|
the number 1.
|
|
For example, $\log_2(32)=5$
|
|
because 5 divisions are needed:
|
|
|
|
\[32 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \]
|
|
|
|
Logarithms are often used in the analysis of
|
|
algorithms, because many efficient algorithms
|
|
halve something at each step.
|
|
Hence, we can estimate the efficiency of such algorithms
|
|
using logarithms.
|
|
|
|
The logarithm of a product is
|
|
\[\log_k(ab) = \log_k(a)+\log_k(b),\]
|
|
and consequently,
|
|
\[\log_k(x^n) = n \cdot \log_k(x).\]
|
|
In addition, the logarithm of a quotient is
|
|
\[\log_k\Big(\frac{a}{b}\Big) = \log_k(a)-\log_k(b).\]
|
|
Another useful formula is
|
|
\[\log_u(x) = \frac{\log_k(x)}{\log_k(u)},\]
|
|
and using this, it is possible to calculate
|
|
logarithms to any base if there is a way to
|
|
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 $e \approx 2.71828$.
|
|
|
|
Another property of logarithms is that
|
|
the number of digits of an integer $x$ in base $b$ is
|
|
$\lfloor \log_b(x)+1 \rfloor$.
|
|
For example, the representation of
|
|
$123$ in base $2$ is 1111011 and
|
|
$\lfloor \log_2(123)+1 \rfloor = 7$.
|
|
|
|
\section{Contests and resources}
|
|
|
|
\subsubsection{IOI}
|
|
|
|
The International Olympiad in Informatics (IOI)
|
|
is an annual programming contest for
|
|
secondary school students.
|
|
Each country is allowed to send a team of
|
|
four students to the contest.
|
|
There are usually about 300 participants
|
|
from 80 countries.
|
|
|
|
The IOI consists of two five-hour long contests.
|
|
In both contests, the participants are asked to
|
|
solve three algorithm tasks of various difficulty.
|
|
The tasks are divided into subtasks,
|
|
each of which has an assigned score.
|
|
Even if the contestants are divided into teams,
|
|
they compete as individuals.
|
|
|
|
The IOI syllabus \cite{iois} regulates the topics
|
|
that may appear in IOI tasks.
|
|
Almost all the topics in the IOI syllabus
|
|
are covered by this book.
|
|
|
|
Participants for the IOI are selected through
|
|
national contests.
|
|
Before the IOI, many regional contests are organized,
|
|
such as the Baltic Olympiad in Informatics (BOI),
|
|
the Central European Olympiad in Informatics (CEOI)
|
|
and the Asia-Pacific Informatics Olympiad (APIO).
|
|
|
|
Some countries organize online practice contests
|
|
for future IOI participants,
|
|
such as the Croatian Open Competition in Informatics \cite{coci}
|
|
and the USA Computing Olympiad \cite{usaco}.
|
|
In addition,
|
|
many problems from Polish contests
|
|
are available online \cite{main}.
|
|
|
|
\subsubsection{ICPC}
|
|
|
|
The International Collegiate Programming Contest (ICPC)
|
|
is an annual programming contest for university students.
|
|
Each team in the contest consists of three students,
|
|
and unlike in the IOI, the students work together;
|
|
there is only one computer available for each team.
|
|
|
|
The ICPC consists of several stages, and finally the
|
|
best teams are invited to the World Finals.
|
|
While there are tens of thousands of participants
|
|
in the contest, there are only a small number\footnote{The exact number of final
|
|
slots varies from year to year; in 2016, there were 128 final slots.} of final slots available,
|
|
so even advancing to the finals
|
|
is a great achievement in some regions.
|
|
|
|
In each ICPC contest, the teams have five hours of time to
|
|
solve about ten algorithm problems.
|
|
A solution to a problem is accepted only if it solves
|
|
all test cases efficiently.
|
|
During the contest, competitors may view the results of other teams,
|
|
but for the last hour the scoreboard is frozen and it
|
|
is not possible to see the results of the last submissions.
|
|
|
|
The topics that may appear at the ICPC are not so well
|
|
specified as those at the IOI.
|
|
In any case, it is clear that more knowledge is needed
|
|
at the ICPC, especially more mathematical skills.
|
|
|
|
\subsubsection{Online contests}
|
|
|
|
There are also many online contests that are open for everybody.
|
|
At the moment, the most active contest site is Codeforces,
|
|
which organizes contests about weekly.
|
|
In Codeforces, participants are divided into two divisions:
|
|
beginners compete in Div2 and more experienced programmers in Div1.
|
|
Other contest sites include AtCoder, CS Academy, HackerRank and Topcoder.
|
|
|
|
Some companies organize online contests with onsite finals.
|
|
Examples of such contests are Facebook Hacker Cup,
|
|
Google Code Jam and Yandex.Algorithm.
|
|
Of course, companies also use those contests for recruiting:
|
|
performing well in a contest is a good way to prove one's skills.
|
|
|
|
\subsubsection{Books}
|
|
|
|
There are already some books (besides this book) that
|
|
concentrate on competitive programming and algorithmic problem solving:
|
|
|
|
\begin{itemize}
|
|
\item S. Halim and F. Halim:
|
|
\emph{Competitive Programming 3: The New Lower Bound of Programming Contests} \cite{hal13}
|
|
\item S. S. Skiena and M. A. Revilla:
|
|
\emph{Programming Challenges: The Programming Contest Training Manual} \cite{ski03}
|
|
\item K. Diks et al.: \emph{Looking for a Challenge? The Ultimate Problem Set from
|
|
the University of Warsaw Programming Competitions} \cite{dik12}
|
|
\end{itemize}
|
|
|
|
The first two books are intended for beginners,
|
|
whereas the last book contains advanced material.
|
|
|
|
Of course, general algorithm books are also suitable for
|
|
competitive programmers.
|
|
Some popular books are:
|
|
|
|
\begin{itemize}
|
|
\item T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein:
|
|
\emph{Introduction to Algorithms} \cite{cor09}
|
|
\item J. Kleinberg and É. Tardos:
|
|
\emph{Algorithm Design} \cite{kle05}
|
|
\item S. S. Skiena:
|
|
\emph{The Algorithm Design Manual} \cite{ski08}
|
|
\end{itemize}
|