\twocolumn[
  \title{\bf \acronym (High end, Advanced, Data driven, Enterprise grade Sock sorting algorithm) - An algorithm for faster sock sorting}
  \author{
    CelloClemens$^{1,2}$,
    Henri\emoji{duck}$^{1}$,
  }
  \date{\today}
  % List of institutions
  \maketitle
  $^{1}$Department for theoretical laundry science, Karlsruhe institute of suffering and sorrow (KISS), Karlsruhe, Germany \\
  $^{2}$Institute of laundry sorting, Department for socks, Karlsruhe institute of suffering and sorrow (KISS), Karlsruhe, Germany
  \begin{psummary}
    Sorting socks can often be a time consuming task. This paper introduces the fastest method known in the scientific community to tackle
    this challanging task. To be able to implement this new algorithm
    a new datastructure will be introduced and discussed. Abundant application of this novel algorithm may be able to
    reduce the time required for sorting socks considerably.
  \end{psummary}
  \vspace{2mm}
]
\fancypagestyle{firstpage}{%
  \lhead{Please help I am stuck in the basement sorting socks}
  \rhead{Journal of Immaterial Science}
}
\thispagestyle{firstpage}

% The introduction
\section{Introduction}
While sorting algorithms are one of the most discussed algorithms in the
computer science community, application of this field to laundry is still quite new.
In fact no research is known to the authors connecting the fields of computer science
and laundry sorting. A few definitions are required in order to establish a baseline
for the algorithm discussed in the following paper.

\subsection{Definitions}
In this section a few definitions, common in the field of theoretical laundry science shall be
introduced. These are required to understand the algorithm and its advantages.

\subsubsection{Sock}
Let $\Lambda_a$ be the Set of laundry. The set of socks, $\Sigma\subset\Lambda_a$
is defined as $\Sigma:=\{s\in\Lambda_a|\chi(s)=1\}$\footnote{Yes, some socks have holes. So what?!}, where $\chi(s)$ is the Euler
characteristic of $s$. For every sock $s$ there is an equal counterpart $s^{-1}$ giving rise
to the identity $s\cong s^{-1}$. The task commonly known as "sock sorting" is in fact the
search for this isomorphism $\eta$ and matching every sock $s$ to its inverse
$s^{-1}$.
\begin{figure}[h]
  \centering
  \includegraphics[width=\columnwidth]{\projectpath figs/Sock.jpeg}
  \vspace{0.1in}
  \caption{A pair of blue socks and a single orange sock.}
  \label{fig:Sock}
  \vspace{0.1in}
\end{figure}
\subsubsection{Laundry basket}
Let $\Lambda\subseteq\Lambda_a$ be a set of laundry items. Then a laundry basket is a
triplet $(\Lambda, +, -)$ representing a datastructure that implements
the following functions:
\begin{itemize}
  \item \texttt{get: }$\mathscr{L}\in\Lambda_a$, returns a uniformly random
        laundry item from the basket or $\mathscr{L}_0$, the Zero element of laundry, iff
        There are no items left.
  \item  \texttt{put($\mathscr{L}\in\Lambda_a$)}, deposits the given laundry item
        into the basket.
\end{itemize}
Note that both operations run in $\mathcal{O}(1)$. Because of the nature of
a laundry basket finding a unique item requires transferring the content of
the whole basket to a new basket thus requiring $\mathcal{O}(n)$ operations,
$n$ being the number of items currently inside the basket.
\subsection{Ongoing and latest research}
To fully appreciate the gravity of \acronym it has to first be discussed
how most resent research tackles the problem of sock sorting.
The following code describes the most recently developed sock sorting
algorithm from the paper by my colleague which is the current
industry standard. Notice the code has a runtime complexity of $\mathcal{O}(n^2)$.

\begin{algorithm}
  \caption{Conventional sock sorting}
  \begin{algorithmic}[1]
    \State\Comment initialize a new laundry basket with a given set of laundry
    \State $A\gets\Lambda$\Comment WLOG assume $\forall\mathscr{L}\in\Lambda|\mathscr{L}$ is a sock
    \State $\text{matches}\gets []$
    \Repeat
    \State $\mathscr{L}\gets$ A.get

    \Repeat\Comment find the inverse Sock by checking all other socks
    \State $\mathscr{L}^\prime\gets$ A.get
    \Until{$\mathscr{L}^\prime =\mathscr{L}^{-1}$}
    \State matches.append(($\mathscr{L}, \mathscr{L}^\prime$))
    \Until{$\mathscr{L}\ne\mathscr{L}_0$}
  \end{algorithmic}
\end{algorithm}

\section{Concepts}
The basis for every fast algorithm are  simple yet equally fast
datastructures. To enable the low runtime achieved by \acronym,
the introduction of a new datastructure, the "laundry rack" is integral.

\subsection{Laundry rack}
Let $\Lambda\subseteq\Lambda_a$ be a set of laundry. A laundry rack (See figure \ref{fig:Rack}) is a
triplet $(\Lambda, +, -)$ representing a datastructure that implements the
following methods:
\begin{itemize}
  \item \texttt{get($\mathscr{L}\in\Lambda$)}: $\mathscr{L}\in\Lambda$, gets
        a specific laundry item from the laundry rack.
  \item \texttt{put($\mathscr{L}\in\Lambda$)}, deposits a laundry item onto
        the laundry rack.
  \item \texttt{match($\mathscr{L}\in\Lambda$)}: $(\mathscr{P}\in\Lambda\times\Lambda)|\mathscr{L}_0$,
        returns a tuple $(\mathscr{L}, \mathscr{L}^{-1})$ representing
        a pair of socks iff $\mathscr{L}^{-1}$ is already on the laundry rack,
        $\mathscr{L}_0$ otherwise.
\end{itemize}
All these operations (especially \texttt{match}) run in $\mathcal{O}(1)$,
making iteration over all $n$ laundry items to find a pair $(\mathscr{L}, \mathscr{L}^{-1})$
obsolete.
\begin{figure}[h]
  \centering
  \includegraphics[width=\columnwidth]{\projectpath figs/rack.jpeg}
  \vspace{0.1in}
  \caption{A blue drying rack, found in many housholds.}
  \label{fig:Rack}
  \vspace{0.2in}
\end{figure}

\section{Algorithm}
Making use of the novel advanced features of a "drying rack" we are able to
implement the following algorithm in $\mathscr{O}(n)$:\\
\begin{algorithm}
  \caption{\acronym}
  \begin{algorithmic}[1]
    \State\Comment initialize a new laundry basket with a given set of laundry
    \State $A\gets\Lambda$\Comment WLOG assume $\forall\mathscr{L}\in\Lambda|\mathscr{L}$ is a sock
    \State $\text{matches}\gets []$
    \State $\daleth\gets []$ \Comment{Create a new empty drying rack}
    \Repeat
    \State $\mathscr{L}\gets $ A.get
    \State result $\gets$ $\daleth$.match($\mathscr{L}$)
    \If{result $\ne\mathscr{L}_0$}
    \State matches.append(result)
    \Else
    \State $\daleth$.put($\mathscr{L}$)
    \EndIf

    \Until{$\mathscr{L}\ne\mathscr{L}_0$}
  \end{algorithmic}
\end{algorithm}
As evident from the algorithm above, only one loop performing
operations which are all in $\mathscr{O}(1)$ is required thus
putting the algorithm in a $\mathscr{O}(n)$ runtime complexity
class. Assuming that $\forall\mathscr{L}\in\Lambda\exists\mathscr{L}^{-1}|\mathscr{L}\cong\mathscr{L}^{-1}$
the algorithm always yields a correct solution for the problem
(proof is left as an exercise to the reader).
\section{Discussion and Results}
To evaluate the algorithms performance it has been executed
on different platforms consisting of diverse hardware:\\
\resizebox*{\linewidth}{!}{
  \begin{tabular}[]{l||l|c}
    \textbf{Hardware} & \textbf{Algorithm}   & \textbf{Runtime [s]} \\
    \hline
    Myself            & Conventional (n=20)  & 352.7                \\
    Myself            & \acronym (n=20)      & 92.3                 \\
    Myself            & Conventional (n=100) & 42069                \\
    Myself            & \acronym (n=100)     & 420.69               \\
    \hline
    My roommate       & Conventional (n=10)  & 91.7                 \\
    My roommate       & \acronym (n=10)      & -2                   \\
    \hline
    Girlfriend        & n.a.                 & n.a.
  \end{tabular}
}\\
\begin{figure}[h]
  \begin{center}
    \includegraphics[width=\linewidth]{figs/graph.png}
    \bigskip
    \caption{Comparative statistical time analysis of both algorithms.
    The graph depicts algorithm runtime (y-axis) and graphs it against 
    input size (x-axis). Data for 
    \acronym in blue, for conventional sock sorting in green.}
  \end{center}
\end{figure}\\
From the above data it is evident that \acronym bears a clear advantage in comparison
to the conventional algorithm when it comes to sock sorting. Utilizing advanced statistical modelling we calculated a
speedup factor of about $3.1415926535897932384626433\cdot n$. The data also illustrates
the scalability of the algorithm and its adaptability to different hardware.
\section{Conclusion}
It can be concluded that the algorithm presented in this paper is
greatly superior to the conventional method of sorting socks.
It will probably revolutionize not only the field of laundry science but
also have great impact in the industry.\\
The datastructures outlined above may become abundantly used and be the future
industry standard. Although the field of laundry science is still rather new
there are still a lot of open questions to be answered. However it is unlikely
that a faster sock sorting algorithm than \acronym can be developed.
\section{Acknowledgements}
We shall use this historic opportunity to thank the Journal of immaterial science
for publishing great \st{memes} research. We also want to thank our university
for giving us this great opportunity for \st{depression and self loathing}
research and personal advancement.\\
It is also only appropriate to thank the air. Without it no laundry would be 
dry and we would not have written this paper.\\


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