169 lines
7.3 KiB
TeX
169 lines
7.3 KiB
TeX
\twocolumn[
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\title{\bf \acronym (High end, Advanced, Data driven, Enterprise grade Sock sorting algorithm) - An algorithm for faster sock sorting}
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\author{
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CelloClemens$^{1,2}$,
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Henri\emoji{duck}$^{1}$,
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}
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\date{\today}
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% List of institutions
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\maketitle
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$^{1}$Department for theoretical laundry science, Karlsruhe institute of suffering and sorrow (KISS), Karlsruhe, Germany \\
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$^{2}$Institute of laundry sorting, Department for socks, Karlsruhe institute of suffering and sorrow (KISS), Karlsruhe, Germany
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\begin{psummary}
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Sorting socks can often be a time consuming task. This paper introduces the fastest method known in the scientific community to tackle
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this challanging task. To be able to implement this new algorithm
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a new datastructure will be introduced and discussed. Abundant application of this novel algorithm may be able to
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reduce the time required for sorting socks considerably.
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\end{psummary}
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\vspace{2mm}
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]
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\fancypagestyle{firstpage}{%
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\lhead{Please help I am stuck in the basement sorting socks}
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\rhead{Journal of Immaterial Science}
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}
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\thispagestyle{firstpage}
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% The introduction
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\section{Introduction}
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While sorting algorithms are one of the most discussed algorithms in the
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computer science community, application of this field to laundry is still quite new.
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In fact no research is known to the authors connecting the fields of computer science
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and laundry sorting. A few definitions are required in order to establish a baseline
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for the algorithm discussed in the following paper.
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\subsection{Definitions}
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In this section a few definitions, common in the field of theoretical laundry science shall be
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introduced. These are required to understand the algorithm and its advantages.
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\subsubsection{Sock}
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Let $\Lambda_a$ be the Set of laundry. The set of socks, $\Sigma\subset\Lambda_a$
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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
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characteristic of $s$. For every sock $s$ there is an equal counterpart $s^{-1}$ giving rise
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to the identity $s\cong s^{-1}$. The task commonly known as "sock sorting" is in fact the
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search for this isomorphism $\eta$ and matching every sock $s$ to its inverse
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$s^{-1}$.
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\begin{figure}[h]
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\centering
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\includegraphics[width=\columnwidth]{\projectpath figs/Sock.jpeg}
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\vspace{0.1in}
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\caption{A pair of blue socks and a single orange sock.}
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\label{fig:Sock}
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\vspace{0.1in}
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\end{figure}
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\subsubsection{Laundry basket}
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Let $\Lambda\subseteq\Lambda_a$ be a set of laundry items. Then a laundry basket is a
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triplet $(\Lambda, +, -)$ representing a datastructure that implements
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the following functions:
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\begin{itemize}
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\item \texttt{get: }$\mathscr{L}\in\Lambda_a$, returns a uniformly random
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laundry item from the basket or $\mathscr{L}_0$, the Zero element of laundry, iff
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There are no items left.
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\item \texttt{put($\mathscr{L}\in\Lambda_a$)}, deposits the given laundry item
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into the basket.
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\end{itemize}
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Note that both operations run in $\mathcal{O}(1)$. Because of the nature of
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a laundry basket finding a unique item requires transferring the content of
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the whole basket to a new basket thus requiring $\mathcal{O}(n)$ operations,
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$n$ being the number of items currently inside the basket.
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\subsection{Ongoing and latest research}
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To fully appreciate the gravity of \acronym it has to first be discussed
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how most resent research tackles the problem of sock sorting.
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The following code describes the most recently developed sock sorting
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algorithm from the paper by \citeauthor{My colleague et al.} Which is the current
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industry standard. Notice the code has a runtime complexity of $\mathcal{O}(n^2)$.
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\begin{algorithm}
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\caption{Conventional sock sorting}
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\begin{algorithmic}[1]
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\State\Comment initialize a new laundry basket with a given set of laundry
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\State $A\gets\Lambda$\Comment WLOG assume $\forall\mathscr{L}\in\Lambda|\mathscr{L}$ is a sock
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\State $\text{matches}\gets []$
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\Repeat
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\State $\mathscr{L}\gets$ A.get
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\Repeat\Comment find the inverse Sock by checking all other socks
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\State $\mathscr{L}^\prime\gets$ A.get
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\Until{$\mathscr{L}^\prime =\mathscr{L}^{-1}$}
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\State matches.append(($\mathscr{L}, \mathscr{L}^\prime$))
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\Until{$\mathscr{L}\ne\mathscr{L}_0$}
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\end{algorithmic}
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\end{algorithm}
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\section{Concepts}
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The basis for every fast algorithm are simple yet equally fast
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datastructures. To enable the low runtime achieved by \acronym,
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the introduction of a new datastructure, the "laundry rack" is integral.
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\subsection{Laundry rack}
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Let $\Lambda\subseteq\Lambda_a$ be a set of laundry. A laundry rack (See figure \ref{fig:Rack}) is a
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triplet $(\Lambda, +, -)$ representing a datastructure that implements the
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following methods:
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\begin{itemize}
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\item \texttt{get($\mathscr{L}\in\Lambda$)}: $\mathscr{L}\in\Lambda$, gets
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a specific laundry item from the laundry rack.
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\item \texttt{put($\mathscr{L}\in\Lambda$)}, deposits a laundry item onto
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the laundry rack.
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\item \texttt{match($\mathscr{L}\in\Lambda$)}: $(\mathscr{P}\in\Lambda\times\Lambda)|\mathscr{L}_0$,
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returns a tuple $(\mathscr{L}, \mathscr{L}^{-1})$ representing
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a pair of socks iff $\mathscr{L}^{-1}$ is already on the laundry rack,
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$\mathscr{L}_0$ otherwise.
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\end{itemize}
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All these operations (especially \texttt{match}) run in $\mathcal{O}(1)$,
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making iteration over all $n$ laundry items to find a pair $(\mathscr{L}, \mathscr{L}^{-1})$
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obsolete.
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\begin{figure}[h]
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\centering
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\includegraphics[width=\columnwidth]{\projectpath figs/rack.jpeg}
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\vspace{0.1in}
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\caption{A blue drying rack, found in many housholds.}
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\label{fig:Rack}
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\vspace{0.1in}
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\end{figure}
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\section{Algorithm}
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Making use of the novel advanced features of a "drying rack" we are able to
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implement the following algorithm in $\mathscr{O}(n)$:\\
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\begin{algorithm}
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\caption{\acronym}
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\begin{algorithmic}[1]
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\State\Comment initialize a new laundry basket with a given set of laundry
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\State $A\gets\Lambda$\Comment WLOG assume $\forall\mathscr{L}\in\Lambda|\mathscr{L}$ is a sock
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\State $\text{matches}\gets []$
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\State $\daleth\gets []$ \Comment{Create a new empty drying rack}
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\Repeat
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\State $\mathscr{L}\gets $ A.get
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\State result $\gets$ $\daleth$.match($\mathscr{L}$)
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\If{result $\ne\mathscr{L}_0$}
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\State matches.append(result)
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\Else
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\State $\daleth$.put($\mathscr{L}$)
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\EndIf
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\Until{$\mathscr{L}\ne\mathscr{L}_0$}
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\end{algorithmic}
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\end{algorithm}
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As evident from the algorithm above, only one loop performing
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operations which are all in $\mathscr{O}(1)$ is required thus
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putting the algorithm in a $\mathscr{O}(n)$ runtime complexity
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class. Assuming that $\forall\mathscr{L}\in\Lambda\exists\mathscr{L}^{-1}|\mathscr{L}\cong\mathscr{L}^{-1}$
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the algorithm always yields a correct solution for the problem
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(proof is left as an exercise to the reader).
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\section{Discussion and Results}
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To evaluate the algorithms performance it has been executed
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on different platforms consisting of diverse hardware:\\
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\begin{tabular}[]{l||l|c}
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\textbf{Hardware} & \textbf{Algorithm} & \textbf{Runtime [s]}\\
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\hline
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Myself & Conventional (n=20) & 352.7\\
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\end{tabular}
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\section{Conclusion}
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\section{Acknowledgements}
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\begingroup
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\setlength\bibitemsep{0pt}
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\setlength\bibnamesep{0pt}
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\printbibliography[heading=subbibliography]
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\endgroup
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