%SUMMARY
%- ABSTRACT
%- INTRODUCTION
%# BASICS
%- \acs{DNA} STRUCTURE
%- DATA TYPES
% - BAM/FASTQ
% - NON STANDARD
%- COMPRESSION APPROACHES
% - SAVING DIFFERENCES WITH GIVEN BASE \acs{DNA}
% - HUFFMAN ENCODING
% - PROBABILITY APPROACHES (WITH BASE?)
%
%# COMPARING TOOLS
%- 
%# POSSIBLE IMPROVEMENT
%- \acs{DNA}S STOCHASTICAL ATTRIBUTES 
%- IMPACT ON COMPRESSION
\newcommand{\mycomment}[1]{}

% entropie fim doku grundlagen2 
% dna nucleotide zu einem kapitel -> structure of dna. auch kapitel wegstreichen (zu generisch)
% file structure/format <-> datatypes. länger beschreiben: e.g. File formats to store dna
% 3.2.1 raus


\section{Compression aproaches}
The process of compressing data serves the goal to generate an output that is smaller than its input data.\\
In many cases, like in gene compressing, the compression is idealy lossless. This means it is possible for every compressed data, to receive the whole information, which were available in the origin data, by decompressing it.\\
Before going on, the difference between information and data should be emphasized.\\
% excurs data vs information
Data contians information. In digital data  clear, physical limitations delimit what and how much of something can be stored. A bit can only store 0 or 1, eleven bits can store up to $2^11$ combinations of bits and a 1\acs{GB} drive can store no more than 1\acs{GB} data. Information on the other hand, is limited by the way how it is stored. In some cases the knowledge received in a earlier point in time must be considered too, but this can be neglected for reasons described in the subsection \ref{k4:dict}.\\
% excurs information vs data
The boundaries of information, when it comes to storing capabilities, can be illustrated by using the example mentioned above. A drive with the capacity of 1\acs{GB} could contain a book in form of images, where the content of each page is stored in a single image. Another, more ressourcefull way would be storing just the text of every page in \acs{UTF-16}. The information, the text would provide to a potential reader would not differ. Changing the text encoding to \acs{ASCII} and/or using compression techniques would reduce the required space even more, without loosing any information.\\
% excurs end
In contrast to lossless compression, lossy compression might excludes parts of data in the compression process, in order to increase the compression rate. The excluded parts are typicaly not necessary to persist the origin information. This works with certain audio and pictures formats, and in network protocols \cite{cnet13}.
For \acs{DNA} a lossless compression is needed. To be preceice a lossy compression is not possible, because there is no unnecessary data. Every nucleotide and its position is needed for the sequenced \acs{DNA} to be complete. For lossless compression two mayor approaches are known: the dictionary coding and the entropy coding. Both are described in detail below \cite{cc14}.\\

\subsection{Dictionary coding}
\label{k4:dict}
Dictionary coding, as the name suggest, uses a dictionary to eliminate redundand occurences of strings. Strings are a chain of characters representing a full word or just a part of it. For a better understanding this should be illustrated by a short example:
% exkurs
Looking at the string 'stationary' it might be smart to store 'station' and 'ary' as seperate dictionary enties. Which way is more efficient depents on the text that should get compressed. 
% end exkurs
The dictionary should only store strings that occour in the input data. Also storing a dictionary in addition to the (compressed) input data, would be a waste of resources. Therefore the dicitonary is made out of the input data. Each first occourence is left uncompressed. Every occurence of a string, after the first one, points to its first occurence. Since this 'pointer' needs less space than the string it points to, a decrease in the size is created.\\

Unfortunally, known implementations like the ones out of LZ Family, do not use probabilities to compress and are therefore out of scope for this work. Since finding repeting sections and their location might also be improved, this chapter will remain.


% unuseable due to the lack of probability
\mycomment{
% - known algo
\subsubsection{The LZ Family}
The computer scientist Abraham Lempel and the electrical engineere Jacob Ziv created multiple algorithms that are based on dictionary coding. They can be recognized by the substring \texttt{LZ} in its name, like \texttt{LZ77 and LZ78} which are short for Lempel Ziv 1977 and 1978. The number at the end indictates when the algorithm was published. Today LZ78 is widely used in unix compression solutions like gzip and bz2. Those tools are also used in compressing \ac{DNA}.\\
\ac{LZ77} basically works, by removing all repetition of a string or substring and replacing them with information where to find the first occurence and how long it is. Typically it is stored in two bytes, whereby more than one one byte can be used to point to the first occurence because usually less than one byte is required to store the length.\\
% example 
}

 % (genomic squeeze <- official | inofficial -> GDC, GRS). Further \ac{ANS} or rANS ... TBD.
\subsection{\ac{LZ77}}
 \ac{LZ77} basically works, by removing all repetition of a string or substring and replacing them with information where to find the first occurence and how long it is. Typically it is stored in two bytes, whereby more than one one byte can be used to point to the first occurence because usually less than one byte is required to store the length.


\subsection{Shannons Entropy}
The founder of information theory Claude Elwood Shannon described entropy and published it in 1948 \cite{Shannon_1948}. In this work he focused on transmitting information. His theorem is applicable to almost any form of communication signal. His findings are not only usefull for forms of information transmition. 

% todo insert Fig. 1 shannon_1948

Altering this figure shows how it can be used for other technology like compression.\\
The Information source and destination are left unchanged, one has to keep in mind, that it is possible that both are represented by the same phyiscal actor. 
transmitter and receiver are changed to compression/encoding and decompression/decoding and inbetween ther is no signal but any period of time \cite{Shannon_1948}.\\

Shannons Entropy provides a formular to determine the 'uncertainty of a probability distribution' in a finite field.

%H(X) \defd \Sum{x\in X, prob(x)\neq0}{}{prob(x) * log_2(frac{1}{prob(x)})} \equiv  - \Sum { x\in X, prob(x)\neq0 } {} {prob(x) * log_2 (prob(x))}.
\begin{figure}[H]
  \centering
  \includegraphics[width=12cm]{k4/shannon_entropy.png}
  \caption{Shannons definition of entropy.}
  \label{k4:entropy}
\end{figure}

He defined entropy as shown in figure \ref{f4:entropy}. Let X be a finite probability space. Then x in X are possible final states of an probability experimen over X. Every state that actually occours, while executing the experiment generates infromation which is meassured in \textit{Bits} with the part of the formular displayed in \ref{k4:info-in-bits}\cite{delfs_knebl,Shannon_1948}:

%\bein{math}
% log_2(frac{1}{prob(x)}) \equiv - log_2(prob(x)).
%\end{math} 
\begin{figure}[H]
  \centering
  \includegraphics[width=8cm]{k4/information_bits.png}
  \caption{The amount of information measured in bits, in case x is the end state of a probability experiment.}
  \label{f4:info-in-bits}
\end{figure}

%todo explain 2.2 second bulletpoint of delfs_knebl. Maybe read gumbl book

%This can be used to find the maximum amount of bits needed to store information.\\ 
% alphabet, chain of symbols, kurz entropy erklären

\label{k4:arith}
\subsection{Arithmetic coding}
Arithmetic coding is an approach to solve the problem of wasting memeory due to the overhead which is created by encoding certain lenghts of alphabets in binary. Encoding a three-letter alphabet requires at least two bit per letter. Since there are four possilbe combinations with two bits, one combination is not used, so the full potential is not exhausted. Looking at it from another perspective, less storage would be required, if there would be a possibility to encode two letters in the alphabet with one bit and the other one with a two byte combination. This approache is not possible because the letters would not be clearly distinguishable. The two bit letter could be interpreted either as the letter it should represent or as two one bit letters.
% check this wording 'simulating' with sources 
% this is called subdividing
Arithmetic coding works by translating a n-letter alphabet into a n-letter binary encoding. This is possible by projecting the input text on a floatingpoint number. Every character in the alphabet is represented by an intervall between two floating point numbers in the space between 0.0 and 1.0 (exclusively). This intervall is determined by its distribution in the input text (intervall start) and the the start of the next character (intervall end). To encode a sequence of characters subdividing is used.
% exkurs on subdividing?
This means the intervall start of the character is noted, its intervall is split into smaller intervalls with the ratios of the initial intervalls between 0.0 and 1.0. With this, the second character is choosen. This process is repeated for until a intervall for the last character is choosen.\\
To encode in binary, the binary floating point representation of a number inside the intervall, for the last character is calculated, by using a similar process, described above, called subdividing.
% its finite subdividing because processors bottleneck floatingpoints 

\subsection{Huffman encoding}
% list of algos and the tools that use them
D. A. Huffmans work focused on finding a method to encode messages with a minimum of redundance. He referenced a coding procedure developed by Shannon and Fano and named after its developers, which worked similar. The \ac{SF} coding is not used today, due to the superiority in both efficiency and effectivity, in comparison to Huffman. % todo any source to last sentence. Rethink the use of finite in the following text
Even though his work was released in 1952, the method he developed is in use  today. Not only tools for genome compression but in compression tools with a more general ussage \cite{rfcgzip}.\\ 
Compression with the Huffman algorithm works only on finite alphabets. It also provides a solution to the problem, described at the beginning of \ref{k4:arith}, on waste through unused bits, for certain alphabet lengths. Huffman did not save more than one symbol in one bit, like it is done in arithmetic coding, but he decreased the number of bits used per symbol in a message. This is possible by setting individual bit lengths for symbols, used in the text that should get compressed \cite{huf52}. 
As with other codings, a set of symbols must be defined. For any text constructed with symbols from mentioned alphabet, a binary tree is constructed, which will determine how the symbols will be encoded. As in arithmetic coding, the probability of a letter is calculated for given text. The binary tree will be constructed after following guidelines:
% greedy algo?
\begin{itemize}
  \item Every symbol of the alphabet is one leaf.
  \item The right branch from every knot is marked as a 1, the left one is marked as a 0.
  \item Every symbol got a weight, the weight is defined by the frequency the symbol occours in the input text.
  \item The less weight a leaf has, the higher the probability is, that this node is read next in the symbol sequence.
  \item The leaf with the lowest probability is most left and the one with the highest probability is most right in the tree. 
\end{itemize}
%todo tree building explanation
Constructing the tree begins with as many nodes as there are symbols, in the alphabet. 
% storytime might need to be rearranged
A often mentioned difference between \acs{FA} and Huffman coding, is that first is working top down while the latter is working bottom up. This means the tree starts with the lowest weights. The nodes that are not leafs have no value ascribed to them. They only need their weight, which is defined by the weights of their individual child nodes.\\
So starting with the two lowest weightened symbols, a node is added to connect both. from there on, the two leafs will only get rearranged through the rearrangement of their temporary root node. With the added, blank node the count of available nodes got down by one. The new node weights as much as the summ of weights of its child nodes. Now the two lowest weights are paired as described until there are only two subtrees left which can be combined by a root.\\
With the fact in mind, that left branches are assigned with 0 and right branches with 1, following a path until a leaf is reached reveals the encoding for this particular leaf. Since high weightened and therefore often occuring leafs are positioned to the left, short paths lead to them and so only few bits are needed to encode them. Following the tree on the other side, the symbols occur more rarely, paths get longer and so do the bit counts. % todo <- rewrite '...counts'


In our case a four letter alphabet, containing \texttt{A, C, G and T} is sufficient.
The process of compressing starts with the nodes with the lowest weight and buids up to the hightest. Each step adds nodes to a tree where the most left branch should be the shortest and the most right the longest. The most left branch ends with the symbol with the highest weight, therefore occours the most in the input data.
Following one path results in the binary representation for one symbol. For an alphabet like the one described above, the binary representation encoded in ASCI is shown here \texttt{A -> 01000001, C -> 01000011, G -> 01010100, T -> 00001010}. An imaginary sequence, that has this distribution of characters \texttt{A -> 10, C -> 8, G -> 4, T -> 2}. From this information a weighting would be calculated for each character by dividing one by the characters occurence. With a corresponding tree, created from with the weights, the binary data for each symbol would change to this \texttt{A -> 0, C -> 11, T -> 100, G -> 101}. Besides the compressed data, the information contained in the tree msut be saved for the decompression process.\\
% todo shannon fano mention. SF might be older than huffman and inspired it?
% -> yes


\section{DEFLATE}
% mix of huffman and lz77
The DEFLATE compression algorithm combines \ac{lz77} and huffman coding. It is used in well known tools like gzip.

\subsubsection{misc}

%check if (small) text coding is done with this:
Arithmetic Asymmetric numeral systems ?
Modified -> used in cram


\section{Implementations in Relevant Tools}
\subsection{} % geco
\subsection{} % genie
\subsection{} % samtools 

\mycomment{
\subsection{\ac{CABAC}}
% a form of entropy coding
% https://en.wikipedia.org/wiki/Context-adaptive_binary_arithmetic_coding


\section{Implementations}
% SAM - LZ4 src: https://github.com/lz4/lz4/blob/dev/doc/lz4_Block_format.md
% GeCo - arithmetic coding
% Genie - CABAC

% following text is irelevant. Just describe used algorithms in comparison chapter and refere to their base algo

% mix of Huffman and lz77
The DEFLATE compression algorithm combines \ac{LZ77} and Huffman coding. To get more specific, the raw data is compressed with \ac{LZ77} and remaining data is shortened by using Huffman coding. 
% huffman - little endian
% lz77 compressed - big endian (least significant byte first/most left)
}