459 lines
19 KiB
TeX
459 lines
19 KiB
TeX
\section{Comparison}%
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\label{sec:Comparison}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Comparison of Simulation Results}%
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\label{sub:Comparison of Simulation Results}
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\begin{frame}[t]
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\frametitle{Decoding Performance}
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\vspace*{-5mm}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=5e-5,
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legend columns = 3,
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legend style={at={(0.5,-0.5)},anchor=south},
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width=0.45\textwidth,
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height=0.3375\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{Proximal decoding}
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={mu}{3.0},
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discard if gt={SNR}{4.0}]
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{res/admm/ber_2d_20433484.csv};
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\addlegendentry{LP Decoding}
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\addplot[PineGreen, line width=1pt, mark=triangle, solid]
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table [col sep=comma, x=SNR, y=FER,
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discard if gt={SNR}{3.0}]
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{res/generic/fer_ml_20433484.csv};
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\addlegendentry{ML}
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\end{axis}
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\end{tikzpicture}
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\caption{Comparison of FER of proximal decoding and LP decoding using ADMM%
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\protect\footnotemark{}}
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\end{figure}%
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\footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
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\cite[Code: 204.33.484]{mackay_enc}}
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\end{frame}
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\begin{frame}[t]
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\frametitle{Decoding Performance}
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\captionsetup[subfigure]{font=footnotesize}
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\hspace*{-0.4cm}
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\begin{minipage}[c]{0.9\textwidth}
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\centering
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\begin{figure}[H]
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\vspace*{-0.5cm}
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\centering
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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%{res/hybrid/2d_ber_fer_dfr_963965.csv};
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{res/admm/ber_2d_963965.csv};
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% \addplot[PineGreen, line width=1pt, mark=triangle]
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% table [col sep=comma, x=SNR, y=FER,]
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% {res/generic/fer_ml_9633965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_bch_31_26.csv};
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% \addplot[PineGreen, line width=1pt, mark=triangle*]
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% table [x=SNR, y=FER, col sep=comma,
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% discard if gt={SNR}{5.5},
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% discard if lt={SNR}{1},
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% ]
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% {res/generic/fer_ml_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={mu}{3.0},
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discard if gt={SNR}{5.5}]
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{res/admm/ber_2d_20433484.csv};
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% \addplot[PineGreen, line width=1pt, mark=triangle, solid]
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% table [col sep=comma, x=SNR, y=FER,
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% discard if gt={SNR}{5.5}]
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% {res/generic/fer_ml_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_20455187.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
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\cite[\text{204.55.187}]{mackay_enc}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_40833844.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.4]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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% width=1.1\textwidth,
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% height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_pegreg252x504.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
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\cite[\text{PEGReg252x504}]{mackay_enc}}
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\end{subfigure}%
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\end{figure}
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\end{minipage}%
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\begin{minipage}[c]{0.1\textwidth}
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\centering
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\begin{figure}
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\centering
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\vspace*{-1.2cm}
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\hspace*{-0.5cm}
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\begin{tikzpicture}[scale=0.7]
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\begin{axis}[hide axis,
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xmin=10, xmax=50,
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ymin=0, ymax=0.4,
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legend columns=1,
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legend style={draw=white!15!black}]
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\addlegendimage{RedOrange, line width=1pt, mark=*, solid}
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\addlegendentry{Proximal decoding}
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\addlegendimage{NavyBlue, line width=1pt, mark=triangle, densely dashed}
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\addlegendentry{LP decoding}
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% \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
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% \addlegendentry{ML}
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\end{minipage}
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\end{frame}
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\begin{frame}[t]
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\frametitle{Time Complexity}
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\vspace*{-8mm}
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\begin{itemize}
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\item Both algorithms are $\mathcal{O}\left( n \right)$ on average
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\item LP decoding implementation significantly faster
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\end{itemize}
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\begin{figure}[h]
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\centering
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\begin{tikzpicture}
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\begin{axis}[grid=both,
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xlabel={$n$}, ylabel={Time per frame (s)},
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width=0.45\textwidth,
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height=0.3375\textwidth,
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legend style={at={(0.03,0.96)},anchor=north west},
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%legend pos=outer north east,
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legend cell align={left},]
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\addplot[RedOrange, only marks, mark=*]
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table [col sep=comma, x=n, y=spf]
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{res/proximal/fps_vs_n.csv};
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\addlegendentry{Proximal decoding}
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\addplot[RoyalPurple, only marks, mark=triangle*]
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table [col sep=comma, x=n, y=spf]
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{res/admm/fps_vs_n.csv};
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\addlegendentry{LP decoding}
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\end{axis}
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\end{tikzpicture}
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\vspace*{-2mm}
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\caption{Timing requirements of the proximal decoding and LP decoding imlementations%
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\protect\footnotemark{}}
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\end{figure}%
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\footnotetext{The points shown were calculated by evaluating the metadata
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of BER simulation results for the following codes:
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BCH $\left( 31, 11 \right)$;
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BCH $\left( 31, 26 \right)$;
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\cite[\text{96.3.965; 204.33.484;
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204.55.187; 408.33.844; PEGReg252x504}]{mackay_enc}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theoretical Comparison}%
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\label{sub:Theoretical Comparison}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t, fragile]
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\frametitle{Comparison of Proximal Decoding and\\LP Decoding using ADMM}
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\vspace*{-1cm}
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\begin{figure}[h]
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\centering
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\begin{subfigure}[b]{0.47\textwidth}
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\centering
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\begin{align*}
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\text{minimize}\hspace{2mm} & \textcolor{KITblue}{\underbrace{L\left(
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\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)}_{\text{Likelihood}}}
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+ \textcolor{KITred}{\underbrace{\gamma h\left( \tilde{\boldsymbol{x}}
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\right)} _{\text{Constraints}}} \\
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\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
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\end{align*}
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\vspace*{-5mm}
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\begin{algorithm}[caption={}, label={},
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basicstyle=\fontsize{10}{18}\selectfont
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]
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Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
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while stopping critierion unfulfilled do
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$\textcolor{KITblue}{\boldsymbol{r} \leftarrow \boldsymbol{r}
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+ \omega \nabla L\left( \boldsymbol{y} \mid \boldsymbol{s} \right)} $
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$\textcolor{KITred}{\boldsymbol{s} \leftarrow
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\textbf{prox}_{\gamma h}\left( \boldsymbol{r} \right)} $|\Suppressnumber|
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|\Reactivatenumber|
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end while
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return $\boldsymbol{s}$
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\end{algorithm}
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\end{subfigure}\hfill%
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\begin{subfigure}[b]{0.5\textwidth}
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\centering
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\begin{align*}
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\text{minimize}\hspace{5mm} &
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\textcolor{KITblue}{\underbrace{\boldsymbol{\gamma}^\text{T}
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\tilde{\boldsymbol{c}}}_{\text{Likelihood}}}
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+ \textcolor{KITred}{\underbrace{\sum\nolimits_{j\in\mathcal{J}}
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g_j\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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\right)}_{\text{Constraints}}} \\
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\text{subject to}\hspace{5mm} &
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\tilde{\boldsymbol{c}} \in \mathbb{R}^n
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\end{align*}
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\vspace*{-5mm}
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\begin{algorithm}[caption={}, label={},
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basicstyle=\fontsize{10}{18}\selectfont
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]
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Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
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while stopping criterion unfulfilled do
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$\textcolor{KITblue}{\tilde{\boldsymbol{c}} \leftarrow \argmin_{\tilde{\boldsymbol{c}}}
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\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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+ \frac{\rho}{2}\sum_{j\in\mathcal{J}} \left\Vert
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
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+ \boldsymbol{u}_j \right\Vert \right)}$
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$\textcolor{KITred}{\boldsymbol{z}_j \leftarrow \textbf{prox}_{g_j}
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\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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+ \boldsymbol{u}_j \right), \hspace{5mm}\forall j\in\mathcal{J}}$
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$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j
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+ \tilde{\boldsymbol{c}} - \boldsymbol{z}_j, \hspace{14.5mm}\forall j\in\mathcal{J}$
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end while
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return $\tilde{\boldsymbol{c}}$
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\end{algorithm}
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\end{subfigure}%
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\end{figure}%
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t, fragile]
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\frametitle{Comparison of Proximal Decoding and\\LP Decoding using ADMM}
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\vspace*{-1cm}
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\begin{figure}[h]
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\centering
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\begin{subfigure}{0.48\textwidth}
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\centering
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\begin{algorithm}[caption={}, label={},]
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Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
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while stopping critierion unfulfilled do
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for j in $\mathcal{J}$ do
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$p_j \leftarrow \prod_{i\in N_c\left( j \right) } r_i $
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$\textcolor{KITblue}{M_{j\to} \leftarrow p_j^2 - p_j}$|\Suppressnumber|
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|\vspace{0.7mm}\Reactivatenumber|
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end for
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for i in $\mathcal{I}$ do
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$s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
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\phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
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|\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
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M_{j\to} \right] $
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$r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
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end for
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end while
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return $\boldsymbol{s}$
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\end{algorithm}
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\end{subfigure}%
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\hfill
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\begin{subfigure}{0.48\textwidth}
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\centering
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\begin{algorithm}[caption={}, label={},]
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Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
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while stopping criterion unfulfilled do
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for j in $\mathcal{J}$ do
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$\boldsymbol{z}_j \leftarrow \Pi_{P_{d_j}}\left(
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j\right)$
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$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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- \boldsymbol{z}_j$
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$\textcolor{KITblue}{M_{j\to i} \leftarrow \left( z_j \right)_i - \left( u_j \right)_i,
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\hspace{3mm} \forall i \in N_c\left( j \right)}$
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end for
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for i in $\mathcal{I}$ do
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$\tilde{c}_i \leftarrow \frac{1}{d_i}
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\left(\sum_{j\in N_v\left( i \right) } M_{j\to i}
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- \frac{\gamma_i}{\mu} \right)$|\Suppressnumber|
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|\vspace{7mm}\Reactivatenumber|
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end for
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end while
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return $\tilde{\boldsymbol{c}}$
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\end{algorithm}
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\end{subfigure}%
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\end{figure}%
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\end{frame}
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\begin{frame}[t]
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\frametitle{Conclusion}
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\end{frame}
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