Connect tables with arrows

Question

I'm trying to do something similar to what is shown in the picture, but do not even know where to start. I can get something similar with three tables and minipage environment. But I have no idea how to connect tables rows with arrows. Maybe I should use tikz for that, but I can't found similar examples with tables.

\documentclass{beamer}

\usepackage{tabu}
\usepackage{color}
\usepackage{xcolor, colortbl}

\begin{document}
\begin{table}[!htb]
    Interleaving example \\
    \begin{minipage}{.3\linewidth}
      \centering
        \begin{tabu}{|c|c|}
          \hline
          \rowcolor{green!25} 1 & $Document_a$ \\
          \hline
          \rowcolor{green!25} 2 & $Document_b$ \\
          \hline
          \rowcolor{green!25} 3 & $Document_c$ \\
          \hline
        \end{tabu}
        Ranking A
    \end{minipage}
    \begin{minipage}{.3\linewidth}
      \centering
        \begin{tabu}{|c|c|}
          \hline
            \rowcolor{green!25} 1 & $Document_a$ \\
          \hline
            \rowcolor{red!25} 2 & $Document_e$ \\
          \hline
           \rowcolor{green!25} 3 & $Document_b$ \\
          \hline
        \end{tabu}
        Shown to user
    \end{minipage}
    \begin{minipage}{.3\linewidth}
      \centering
        \begin{tabu}{|c|c|}
          \hline
          \rowcolor{red!25} 1 & $Document_e$ \\
          \hline
          \rowcolor{red!25} 2 & $Document_d$ \\
          \hline
          \rowcolor{red!25} 3 & $Document_f$ \\
          \hline
        \end{tabu}
        Ranking B
    \end{minipage} 
\end{table}

\end{document}

Answer 1

I would use a tikz matrix of (math) nodes, together with some fancy arrows from the shapes.arrows library of tikz:

Here is the code:

\documentclass[border=5mm,tikz]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix,shapes.arrows}

\begin{document}
    \begin{tikzpicture}[>=stealth,->,shorten >=2pt,looseness=.5,auto,
        Red/.style={fill=red,text=white},
        fat arrow/.style={single arrow,shape border rotate=90,
                          thick,draw=blue!70,fill=blue!30,
                          minimum height=10mm},
        empty node/.style={minimum height=10mm,fill=none},
      ]
      \matrix (M)[matrix of math nodes,row sep=2mm,column sep=10mm,
                        nodes={text width=20mm,rectangle,fill=green!20}
      ]{
          \text{Document}_A & \text{Document}_a & \text{Document}_e\\
          |[Red]|\text{Document}_B & \text{Document}_b & |[Red]|\text{Document}_f\\
          \text{Document}_C & \text{Document}_c & |[Red]|\text{Document}_g\\
          |[empty node]|& |[empty node]|& |[empty node]|\\
          \text{Ranking A}& \text{Ranking A}& \text{Ranking A}\\
       };
       \draw[red,->](M-2-1.east)--(M-2-2.west);
       \draw[red,->](M-2-3.west)--(M-2-2.east|-M-2-3.west);
       \draw[red,->](M-3-3.west)--(M-2-2.south east);
       \node at (M-4-1) [fat arrow]{};
       \node at (M-4-2) [fat arrow]{};
       \node at (M-4-3) [fat arrow]{};
    \end{tikzpicture}
\end{document}

A few words of explanation:

• The (M) in \matrix (M) means that the nodes in the matrix can be referred to as M-<row>-<col>
• If you use matrix of math nodes then it is better to write \text{Document}_A etc, (the \text command is from the amsmath package), because otherwise it looks like D*o*c*u*m*e*n*t (the OP's MWE has this problem). If there is more text than mathematics in the nodes then use matrix of nodes instead of matrix of math nodes
• I have set the default fill colour of the cells to green!20, assuming that this is the most common background
• For the red cells I have overridden the default background with |[Red]| using the Red style
• The empty node is used to control the (minimum) height of each of these nodes and to allow us to put a fat arrow there later
• You can use east, west, south east, ... modifiers to control where the arrows leave and enter the nodes
• The cells are of slightly different heights because of the different sizes of the subscripts. This is the reason for the specification of (M-2-2.east|-M-2-3.west) for the arrow from Document_f to Document_b. All this does is make the arrow perpendicular to the sides of the rectangle.
• the Red, empty node and fat arrow styles make the code easier to read and easier to modify.

Answer 2

Not complet yet, but shows idea how to use multipart nodes:

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{positioning,shapes.arrows,shapes.multipart}
    \begin{document}
%\begin{table}[!htb]
%    Interleaving example \\
    \begin{tikzpicture}[
    node distance=3mm and 9mm,
mytable/.style 2 args={name=mt#1,
    shape=rectangle split, draw=white, line width=2mm, 
    rectangle split parts=#2,
    text width = 33mm, font=\scriptsize,
    inner ysep=1mm, inner xsep=2mm, outer sep=0mm, align=left},
myArrow/.style={single arrow, draw=blue, fill=blue!30,
                minimum height=7mm,shape border rotate=90}
                        ],
\node[mytable={1}{3},fill=red!20]   
    {\nodepart{one}     Document$_a$
     \nodepart{two}     Document$_b$
     \nodepart{three}   Document$_c$
     };
\node[mytable={2}{4},
      rectangle split part fill={red!20,red!20,green!20,red!20},
      below right=0mm and 9mm of mt1.north east]
    {\nodepart{one}     Document$_a$
     \nodepart{two}     Document$_c$
     \nodepart{three}   Document$_b$
     \nodepart{four}    Document$_d$
     };
\node[myArrow,
      below=of mt1 |- mt2.south,
      label=below:Ranking A] {};
\node[myArrow,
      below=of mt2,
      label=below:Randomized] {};
\path[ultra thick, shorten >=1mm, shorten <=1mm,draw=gray,->]
    (mt1.one east) edge (mt2.one west)
    (mt1.two east) edge (mt2.three west)
    (mt1.three east) -- (mt2.two west);
\end{tikzpicture}
    \end{document}

names and arrows not follows your MWE.