BA/latex/tex/kapitel/k4_algorithms.tex

%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. Methods from both fields, that aquired reputation, are described in detail below \cite{cc14, moffat20, moffat_arith, alok17}.\\

\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:
% demo substrings
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 demo
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.\\


% unuseable due to the lack of probability
% - 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.
\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.\\

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.\\

\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.

\begin{equation}\label{eq:entropy}
%\resizebox{.9 \textwidth}{!}
%{$
  H(X):=\sum\limits_{x\in X, prob(x)\neq 0} prob(x) \cdot log_2(\frac{1}{prob(x)})
 \equiv-\sum\limits_{x\in X, prob(x)\neq 0} prob(x) \cdot log_2 (prob(x)).
%$}
\end{equation}

%\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 \eqref{eq: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{eq:info-in-bits}\cite{delfs_knebl,Shannon_1948}:

\begin{equation}\label{eq:info-in-bits}
 log_2(\frac{1}{prob(x)}) \equiv - log_2(prob(x)).
\end{equation} 

%\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}
This coding method 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. For example: 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 and thinking a step further: Less storage would be required, if there would be a possibility to encode more than one letter in two bit.\\
Dr. Jorma Rissanen described arithmetic coding in a publication in 1976 \autocite{ris76}. % Besides information theory and math, he also published stuff about dna
This works goal was to define an algorithm that requires no blocking. Meaning the input text could be encoded as one instead of splitting it and encoding the smaller texts or single symbols. He stated that the coding speed of arithmetic coding is comparable to that of conventional coding methods \cite{ris76}.  

% unusable because equation is only half correct
\mycomment{
The coding algorithm works with probabilities for symbols in an alphabet. From any text, the alphabet is defined by the set of individual symbols, used in the text. The probability for a single symbol is defined as its distribution. In a \texttt{n} symbol long text, with the first letter in the alphabet occuring \texttt{o} times, its probability is $\frac{o}{n}$.\\

% todo rethink this equation stuff (and compare it to the original work <-compl.)
\begin{itemize}
  \item $p_{i}$ represents the probability of the symbol, at position \texttt{i} in the alphabet.
  \item L will contain the bit sequence with which the text is encoded. This sequence can be seen as a single value. A fraction between 0 and 1 which gets more precise with every processed symbol.
  \item R represents the product of probabilities of processed symbols. 
\end{itemize}

\begin{equation}
  L_{i} = L_{i-1} + R \cdot \sum{i=1}^{j-1} \cdot p_{i}
\end{equation}
\begin{equation}
  R_{i} = R_{i-1} \cdot p_{j}
\end{equation}
}

This is possible by projecting the input text on a binary encoded fraction between 0 and 1. To get there, each character in the alphabet is represented by an interval between two floating point numbers in the space between 0.0 and 1.0 (exclusively). This interval is determined by the symbols distribution in the input text (interval start) and the the start of the next character (interval end). The sum of all intervals will result in one.\\
To encode a text, subdividing is used, step by step on the text symbols from start to the end. The interval that represents the current character will be subdivided. Meaning the choosen interval will be divided into subintervals with the proportional size of the intervals calculated in the beginning.\\
To store as few informations as possible and due to the fact that fractions in binary have limited accuracity, only a single number, that lays between upper and lower end of the last intervall will be stored. To encode in binary, the binary floating point representation of any number inside the interval, for the last character is calculated, by using a similar process, described above.
 To summarize the encoding process in short: 
\begin{itemize}
	\item The interval representing the first character is noted. 
	\item Its interval is split into smaller intervals, with the ratios of the initial intervals between 0.0 and 1.0. 
	\item The interval representing the second character is choosen.
	\item This process is repeated, until a interval for the last character is determined.
	\item A binary floating point number is determined wich lays in between the interval that represents the represents the last symbol. 
\end{itemize}
% its finite subdividing because of the limitation that comes with processor architecture

For the decoding process to work, the \ac{EOF} symbol must be be present as the last symbol in the text. The compressed file will store the probabilies of each alphabet symbol as well as the floatingpoint number. The decoding process executes in a simmilar procedure as the encoding. The stored probabilies determine intervals. Those will get subdivided, by using the encoded floating point as guidance, until the \ac{EOF} symbol is found. By noting in which interval the floating point is found, for every new subdivision, and projecting the probabilies associated with the intervals onto the alphabet, the origin text can be read.\\
% sclaing
In computers, arithmetic operations on floating point numbers are processed with integer representations of given floating point number \cite{ieee-float}. The number 0.4 + would be represented by $4\cdot 10^-1$.\\
Intervals for the first symbol would be represented by natural numbers between 0 and 100 and $... \cdot 10^-x$. \texttt{x} starts with the value 2 and grows as the intgers grow in length, meaning only if a uneven number is divided. For example: Dividing a uneven number like $5\cdot 10^-1$ by two, will result in $25\cdot 10^-2$. On the other hand, subdividing $4\cdot 10^y$ by two, with any negativ real number as y would not result in a greater \texttt{x} the length required to display the result will match the length required to display the input number.\\
% finite percission
The described coding is only feasible on machines with infinite percission. As soon as finite precission comes into play, the algorithm must be extendet, so that a certain length in the resulting number will not be exceeded. Since digital datatypes are limited in their capacity, like unsigned 64-bit integers which can store up to $2^64-1$ bits or any number between 0 and 18.446.744.073.709.551.615. That might seem like a great ammount at first, but considering a unfavorable alphabet, that extends the results lenght by one on each symbol that is read, only texts with the length of 63 can be encoded (62 if \acs{EOF} is exclued).

\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 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 \cite{alok17}:
% 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. This might be a fraction between 0 and 1 or an integer. In this scenario it will described as the first.
  \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
% 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 \cite{moffat20, alok17}.\\

Given \texttt{K(W,L)} as a node structure, with the weigth or probability as \texttt{$W_{i}$} and codeword length as \texttt{$L_{i}$} for the node \texttt{$K_{i}$}. Then will \texttt{$L_{av}$} be the average length for \texttt{L} in a finite chain of symbols, with a distribution that is mapped onto \texttt{W} \cite{huf}.
\begin{equation}\label{eq:huf}
  L_{av}=\sum_{i=0}^{n-1}w_{i}\cdot l_{i}
\end{equation}
The equation \eqref{eq:huf} describes the path, to the desired state, for the tree. The upper bound \texttt{n} is assigned the length of the input text. The touple in any node \texttt{K} consists of a weight \texttt{$w_{i}$}, that also references a symbol, and the length of a codeword \texttt{$l_{i}$}. This codeword will later encode a single symbol from the alphabet. Working with digital codewords, an element in \texttt{l} contains a sequence of zeros and ones. Since there in this coding method, there is no fixed length for codewords, the premise of \texttt{prefix free code} must be adhered to. This means there can be no codeword that match the sequence of any prefix of another codeword. To illustrate this: 0, 10, 11 would be a set of valid codewords but adding a codeword like 01 or 00 would make the set invalid because of the prefix 0, which is already a single codeword.\\
With all important elements described: the sum that results from this equation is the average length a symbol in the encoded input text will require to be stored \cite{huf52, moffat20}.

% example
% todo illustrate
For this example a four letter alphabet, containing \texttt{A, C, G and T} will be used. The exact input text is not relevant, since only the resulting probabilities are needed. With a distribution like \texttt{<A, $100\frac11=0.11$>, <C, $100\frac71=0.71$>, <G, $100\frac13=0.13$>, <T, $100\frac5=0.05$>}, a probability for each symbol is calculated by dividing the message length by the times the symbol occured.\\ 
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}. The average length for any symbol encoded in \acs{ASCII} is eight, while only using four of the available $2^8$ symbols, a overhead of 252 unused bit combinations. For this example it is more vivid, using a imaginary encoding format, without overhead. It would result in a average codeword length of two, because four symbols need a minimum of $2^2$ bits.\\
So starting with the two lowest weightened symbols, a node is added to connect both.\\
\texttt{<A, T>, <C>, <G>}\\
With the added, blank node the count of available nodes got down by one. The new node weights as much as the sum of weights of its child nodes so the probability of 0.16 is assigned to \texttt{<A,T>}. From there on, the two leafs will only get rearranged through the rearrangement of their temporary root node. Now the two lowest weights are paired as described, until there are only two subtrees or nodes left which can be combined by a root.\\
\texttt{<C, <A, T>>, <G>}\\
The \texttt{<C, <A, T>>} has a probability of 0.29. Adding the last node \texttt{G} results in a root node with the probability of 1.0.\\
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. With a corresponding tree, created from with the weights, the binary sequences to encode the alphabet would look like this:\\
\texttt{A -> 0, C -> 11, T -> 100, G -> 101}.\\ 
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 codeword. Applying \eqref{eq:huf} to this example, results in 1.45 bits per encoded symbol. In this example the text would require over one bit less storage for every second symbol.\\
% impacting the dark ground called reality
Leaving the theory and entering the practice, brings some details that lessen this improvement by a bit. A few bytes are added through the need of storing the information contained in the tree. Also, like described in % todo add ref to k3 formats
most formats, used for persisting \acs{DNA}, store more than just nucleotides and therefore require more characters. What compression ratios implementations of huffman coding provide, will be discussed in \ref{k5:results}.\\

\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)
}