frame-algorithm-split-merge-details.tex

% SPDX-License-Identifier: CC-BY-4.0
% Copyright 2018 Toni Dietze
\documentclass[beamer]{standalone}
\input{preamble.tex}
\begin{document}
\begin{standaloneframe}{\jobname}
\begin{algorithmic}[1]
	\Function{split}{$ℳ$}
		\State $\pi \gets$ $ℳ$-splitter splitting every state $A$ in $ℳ$ into $A^1$ and $A^2$
		\State \Return a proper $\pi$-split of $ℳ$
	\EndFunction
	\vspace{1em}
	\Function{merge}{$ℳ'$}
		\State $\pi \gets \text{identity mapping}$
		\ForAll{states $A$ s.t.\ $A^1$, $A^2$ in $ℳ'$}
			\State $\widehat{\pi} \gets \text{identity mapping}$
			\State $\widehat{\pi}(A^1) \gets A$ and $\widehat{\pi}(A^2) \gets A$
			\State $\lambda \gets \text{a good $\widehat{\pi}$-distributor}$
			\If{$\displaystyle\frac{\lklhd{c}{\mrg_{\widehat{\pi}}^\lambda(ℳ')}}{\lklhd{c}{ℳ'}} \geq \mu$}
				\label{algorithmic:lklhd-ratio}
				\State $\pi(A^1) \gets A$ and $\pi(A^2) \gets A$
			\EndIf
		\EndFor
		\State $\lambda \gets \text{a good $\pi$-distributor}$
		\State \Return $\mrg_\pi^\lambda(ℳ')$
	\EndFunction
\end{algorithmic}
\end{standaloneframe}
\end{document}