143 lines
6.6 KiB
TeX
143 lines
6.6 KiB
TeX
%root: main.tex
|
|
\usetikzlibrary{shapes.geometric}%for cylinder
|
|
\usetikzlibrary{shapes.arrows}%for arrow shape
|
|
\usetikzlibrary{shapes.misc}
|
|
%rid of vertical spacing for booktabs rules
|
|
\renewcommand{\aboverulesep}{0pt}
|
|
\renewcommand{\belowrulesep}{0pt}
|
|
|
|
|
|
\begin{figure*}[t!]
|
|
\centering
|
|
\resizebox{\textwidth}{!}{%
|
|
\begin{tikzpicture}
|
|
%pdb cylinder
|
|
\node[cylinder, text width=0.28\textwidth, align=center, draw=black, text=black, cylinder uses custom fill, cylinder body fill=blue!10, aspect=0.12, minimum height=2.5cm, minimum width=2.5cm, cylinder end fill=blue!50, shape border rotate=90] (cylinder) at (0, 0) {
|
|
\tabcolsep=0.1cm
|
|
\begin{tabular}[t]{>{\small}c | >{\small}c | >{\small}c}
|
|
\multicolumn{2}{c}{$\boldsymbol{T}$}\\
|
|
%\toprule
|
|
Point & $\Phi$ \\%& $\semN$\\
|
|
\midrule
|
|
$e_1$ & $A$ \\%& 1 \\
|
|
$e_2$ & $B$ \\%& 1\\
|
|
$e_3$ & $C$ \\%& 1\\
|
|
$e_4$ & $E$ \\%& 1\\
|
|
\end{tabular}\hspace{0.15cm}
|
|
\tabcolsep=0.05cm
|
|
%\captionof{table}{Route}
|
|
\begin{tabular}[t]{>{\footnotesize}c | >{\footnotesize}c | >{\footnotesize}c | >{\footnotesize}c}
|
|
\multicolumn{3}{c}{$\boldsymbol{R$}}\\
|
|
%\toprule
|
|
$\text{Point}_1$ & $\text{Point}_2$ & $\Phi$\\% & $\semN$ \\
|
|
\midrule
|
|
$e_1$ & $e_2$ & $U$\\% & 2 \\
|
|
$e_2$ & $e_4$ & $Y$\\% & 4 \\
|
|
%& $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
|
|
$e_2$ & $e_3$ & $Z$\\% & 3 \\
|
|
\end{tabular}};
|
|
%label below cylinder
|
|
\node[below=0.2 cm of cylinder]{{\LARGE$ \pdb$}};
|
|
%First arrow
|
|
\node[single arrow, right=0.25 of cylinder, draw=black, fill=black!65, text=white, minimum height=0.75cm, minimum width=0.25cm](arrow1) {\textbf{\abbrStepOne}};
|
|
\node (arrow1Label) at (3, 1.4) {$\query_2$};
|
|
\usetikzlibrary{arrows.meta}%for the following arrow configurations
|
|
\draw[line width=0.5mm, dashed, arrows = -{Latex[length=3mm, open]}] ([yshift=0cm, xshift=-0.75cm]arrow1Label)->([yshift=0cm, xshift=0cm]arrow1);
|
|
%Query output (output of step 1)
|
|
\node[rectangle, right=0.175 of arrow1, draw=black, text=black, fill=purple!10, minimum height=2.5cm, minimum width=2cm](rect) {
|
|
\tabcolsep=0.075cm
|
|
%\captionof{table}{Q}
|
|
%\setlength{\cellspacetoplimit}{4pt}
|
|
%\setlength\extrarowheight{10pt}
|
|
% \setlength{\cellspacetoplimit}{10mm}
|
|
% \setlength{\cellspacebottomlimit}{10mm}
|
|
%\renewcommand{\arraystretch}{1.5}
|
|
\begin{tabular}{>{\footnotesize}c | >{\centering\arraybackslash\footnotesize}m{1.95cm} | >{\centering\arraybackslash\footnotesize}m{3.95cm}}
|
|
%\multicolumn{3}{c}{$\boldsymbol{\query_2(\pdb)}$}\\[1mm]
|
|
%\toprule
|
|
Point & $\Phi$ & Circuit\\% & $\expct_{\idb \sim \probDist}[\query_2(\db)(t)]$ \\ \hline
|
|
\midrule
|
|
%\hline
|
|
%\\\\[-3.5\medskipamount]
|
|
$e_1$ & $AU$ &\adjustbox{valign=b}{\resizebox{!}{9mm}{
|
|
\begin{tikzpicture}[thick]
|
|
\node[gen_tree_node](sink) at (0.5, 0.8){$\boldsymbol{\circmult}$};
|
|
\node[gen_tree_node](source1) at (0, 0){$A$};
|
|
\node[gen_tree_node](source2) at (1, 0){$U$};
|
|
\draw[->](source1)--(sink);
|
|
\draw[->] (source2)--(sink);
|
|
\end{tikzpicture}$\inparen{1}$% & $0.5 \cdot 1.0 + 0.5 \cdot 1.0 = 1.0$
|
|
}}\\% & $0.9$ \\
|
|
$e_2$ & $B(Y + Z)$ Or $BY+ BZ$&
|
|
\adjustbox{valign=m}{
|
|
\resizebox{!}{14mm} {
|
|
\begin{tikzpicture}[thick]
|
|
\node[gen_tree_node] (a1) at (1, 0){$Y$};
|
|
\node[gen_tree_node] (b1) at (2, 0){$Z$};
|
|
%level 1
|
|
\node[gen_tree_node] (a2) at (0.75, 0.8){$B$};
|
|
\node[gen_tree_node] (b2) at (1.5, 0.8){$\boldsymbol{\circplus}$};
|
|
%level 0
|
|
\node[gen_tree_node] (a3) at (1.1, 1.6){$\boldsymbol{\circmult}$};
|
|
%edges
|
|
\draw[->] (a1) -- (b2);
|
|
\draw[->] (b1) -- (b2);
|
|
\draw[->] (a2) -- (a3);
|
|
\draw[->] (b2) -- (a3);
|
|
\end{tikzpicture}$\inparen{2}$
|
|
}} \adjustbox{valign=m}{Or}
|
|
%%%%%%%%%%%
|
|
%Non factorized circuit%
|
|
%%%%%%%%%%%
|
|
\adjustbox{valign=m}{
|
|
\resizebox{!}{14mm} {
|
|
\begin{tikzpicture}[thick]
|
|
\node[gen_tree_node] (a2) at (0, 0){$Y$};
|
|
\node[gen_tree_node] (b2) at (1, 0){$B$};
|
|
\node[gen_tree_node] (c2) at (2, 0){$Z$};
|
|
%level 1
|
|
\node[gen_tree_node] (a1) at (0.5, 0.8){$\boldsymbol{\circmult}$};
|
|
\node[gen_tree_node] (b1) at (1.5, 0.8){$\boldsymbol{\circmult}$};
|
|
%level 0
|
|
\node[gen_tree_node] (a0) at (1.0, 1.6){$\boldsymbol{\circplus}$};
|
|
%edges
|
|
\draw[->] (a2) -- (a1);
|
|
\draw[->] (b2) -- (a1);
|
|
\draw[->] (b2) -- (b1);
|
|
\draw[->] (c2) -- (b1);
|
|
\draw[->] (a1) -- (a0);
|
|
\draw[->] (b1) -- (a0);
|
|
\end{tikzpicture}
|
|
}}\\
|
|
\end{tabular}
|
|
};
|
|
%label below rectangle
|
|
\node[below=0.2cm of rect]{{\LARGE $\query_2(\pdb)\inparen{\tup}\equiv \poly\inparen{\vct{X}}$}};
|
|
%Second arrow
|
|
\node[single arrow, right=0.25 of rect, draw=black, fill=black!65, text=white, minimum height=0.75cm, minimum width=0.25cm](arrow2) {\textbf{\abbrStepTwo}};
|
|
%Expectation computation; (output of step 2)
|
|
\node[rectangle, right=0.25 of arrow2, rounded corners, draw=black, fill=red!10, text=black, minimum height=2.5cm, minimum width=2cm](rrect) {
|
|
\tabcolsep=0.09cm
|
|
%\captionof{table}{Q}
|
|
\begin{tabular}{>{\small}c | >{\arraybackslash\normalsize}c}%m{1.95cm}}
|
|
%\multicolumn{2}{c}{$\expct\pbox{\poly(\vct{X})}$}\\[1mm]
|
|
%\toprule
|
|
Point & $\mathbb{E}[\poly(\vct{X})]$\\
|
|
\midrule%[0.05pt]
|
|
$e_1$ & $\inparen{\prob_{A_1} +\prob_{A_2}}\cdot\left(\prob_{U_1} + 2\prob_{U_2}\right)$\\%$1.0 \cdot 0.9 = 0.9$\\[3mm]
|
|
$e_2$ & $\inparen{\prob_{B_1} + \prob_{B_2}}\inparen{\prob_{Y_1}+2\prob_{Y_2} + \prob_{Z_1} + 2\prob_{Z_2}}$\\%$(0.5 \cdot 1.0) + $\newline $\hspace{0.2cm}(0.5 \cdot 1.0)$\newline $= 1.0$\\
|
|
\end{tabular}
|
|
};
|
|
%label of rounded rectangle
|
|
\node[below=0.2cm of rrect]{{\LARGE $\expct\pbox{\poly(\vct{X})}$}};
|
|
\end{tikzpicture}
|
|
}
|
|
%\setlength{\abovecaptionskip}{-0.35cm}
|
|
\savecaptionspace{
|
|
\caption{Intensional Query Evaluation Model $(\query_2 = \project_{\text{Point}}$ $\inparen{T\join_{\text{Point} = \text{Point}_1}R}$ where, for table $R,~\bound = 2$, while for $T,~\bound = 1.)$}
|
|
\label{fig:two-step}
|
|
}{-0.35cm}{-0.43cm}
|
|
%\vspace{-0.43cm}
|
|
\end{figure*}
|
|
|