1 \begin{figure}[h!]

     2 \centering

     3 \subfigure[First split]{\label{first_split}

     4 \scalebox{0.7}{

     5 \begin{tikzpicture}

     6     \node[draw,minimum height=3.8ex] (xa) {$\hspace{2em}xa\hspace{2em}$};

     7     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{5em}x - xa\hspace{5em}$ };

     8     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (z) { $\hspace{21em}$ };

     9 

    10     \draw[decoration={brace,transform={yscale=3}},decorate]

    11            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

    12                node[midway, above=0.5em]{$x$};

    13 

    14     \draw[decoration={brace,transform={yscale=3}},decorate]

    15            (z.north west) -- ($(z.north east)+(0em,0em)$)

    16                node[midway, above=0.5em]{$z$};

    17 

    18     \draw[decoration={brace,transform={yscale=3}},decorate]

    19            ($(xa.north west)+(0em,3ex)$) -- ($(z.north east)+(0em,3ex)$)

    20                node[midway, above=0.6em]{$x @ z \in L_1*$};

    21 

    22     \draw[decoration={brace,transform={yscale=3}},decorate]

    23            ($(z.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

    24                node[midway, below=0.5em]{$(x - xa) @ z \in L_1*$};

    25 

    26     \draw[decoration={brace,transform={yscale=3}},decorate]

    27            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

    28                node[midway, below=0.5em]{$xa \in L_1*$};

    29 \end{tikzpicture}}}

    30 

    31 \subfigure[Max split]{\label{max_split}

    32 \scalebox{0.7}{

    33 \begin{tikzpicture}

    34     \node[draw,minimum height=3.8ex] (xa) { $\hspace{4em}xa\_max\hspace{4em}$ };

    35     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{0.5em}x - xa\_max\hspace{0.5em}$ };

    36     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (z) { $\hspace{21em}$ };

    37 

    38     \draw[decoration={brace,transform={yscale=3}},decorate]

    39            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

    40                node[midway, above=0.5em]{$x$};

    41 

    42     \draw[decoration={brace,transform={yscale=3}},decorate]

    43            (z.north west) -- ($(z.north east)+(0em,0em)$)

    44                node[midway, above=0.5em]{$z$};

    45 

    46     \draw[decoration={brace,transform={yscale=3}},decorate]

    47            ($(xa.north west)+(0em,3ex)$) -- ($(z.north east)+(0em,3ex)$)

    48                node[midway, above=0.8em]{$x @ z \in L_1*$};

    49 

    50     \draw[decoration={brace,transform={yscale=3}},decorate]

    51            ($(z.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

    52                node[midway, below=0.5em]{$(x - xa\_max) @ z \in L_1*$};

    53 

    54     \draw[decoration={brace,transform={yscale=3}},decorate]

    55            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

    56                node[midway, below=0.5em]{$xa \in L_1*$};

    57 \end{tikzpicture}}}

    58 

    59 \subfigure[Max split with $a$ and $b$ (the right situation)]{\label{ab_split}

    60 \scalebox{0.7}{

    61 \begin{tikzpicture}

    62     \node[draw,minimum height=3.8ex] (xa) { $\hspace{4em}xa\_max\hspace{4em}$ };

    63     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{0.5em}x - xa\_max\hspace{0.5em}$ };

    64     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (z) { $\hspace{21em}$ };

    65 

    66     \draw[decoration={brace,transform={yscale=3}},decorate]

    67            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

    68                node[midway, above=0.5em]{$x$};

    69 

    70     \draw[decoration={brace,transform={yscale=3}},decorate]

    71            (z.north west) -- ($(z.north east)+(0em,0em)$)

    72                node[midway, above=0.5em]{$z$};

    73 

    74     \draw[decoration={brace,transform={yscale=3}},decorate]

    75            ($(xa.north west)+(0em,3ex)$) -- ($(z.north east)+(0em,3ex)$)

    76                node[midway, above=0.8em]{$x @ z \in L_1*$};

    77 

    78     \draw[decoration={brace,transform={yscale=3}},decorate]

    79            ($(z.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

    80                node[midway, below=0.5em]{$(x - xa\_max) @ z \in L_1*$};

    81 

    82     \draw[decoration={brace,transform={yscale=3}},decorate]

    83            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

    84                node[midway, below=0.5em]{$xa \in L_1*$};

    85 

    86     \draw[decoration={brace,transform={yscale=3}},decorate]

    87            ($(xxa.south east)+(6em,-5ex)$) -- ($(xxa.south west)+(0em,-5ex)$)

    88                node[midway, below=0.5em]{$a \in L_1$};

    89 

    90     \draw[decoration={brace,transform={yscale=3}},decorate]

    91            ($(z.south east)+(0em,-5ex)$) -- ($(xxa.south east)+(6em,-5ex)$)

    92                node[midway, below=0.5em]{$b \in L_1*$};

    93 \end{tikzpicture}}}

    94 

    95 

    96 \subfigure[Max split with $a$ and $b$ (the wrong situation)]{\label{ab_split_wrong}

    97 \scalebox{0.7}{

    98 \begin{tikzpicture}

    99     \node[draw,minimum height=3.8ex] (xa) { $\hspace{4em}xa\_max\hspace{4em}$ };

   100     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{0.5em}x - xa\_max\hspace{0.5em}$ };

   101     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (z) { $\hspace{21em}$ };

   102 

   103     \draw[decoration={brace,transform={yscale=3}},decorate]

   104            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

   105                node[midway, above=0.5em]{$x$};

   106 

   107     \draw[decoration={brace,transform={yscale=3}},decorate]

   108            (z.north west) -- ($(z.north east)+(0em,0em)$)

   109                node[midway, above=0.5em]{$z$};

   110 

   111     \draw[decoration={brace,transform={yscale=3}},decorate]

   112            ($(xa.north west)+(0em,3ex)$) -- ($(z.north east)+(0em,3ex)$)

   113                node[midway, above=0.8em]{$x @ z \in L_1*$};

   114 

   115     \draw[decoration={brace,transform={yscale=3}},decorate]

   116            ($(z.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

   117                node[midway, below=0.5em]{$(x - xa\_max) @ z \in L_1*$};

   118 

   119     \draw[decoration={brace,transform={yscale=3}},decorate]

   120            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

   121                node[midway, below=0.5em]{$xa \in L_1*$};

   122 

   123     \draw[decoration={brace,transform={yscale=3}},decorate]

   124            ($(xxa.south east)+(-3em,-5ex)$) -- ($(xxa.south west)+(0em,-5ex)$)

   125                node[midway, below=0.5em]{$a \in L_1$};

   126 

   127     \draw[decoration={brace,transform={yscale=3}},decorate]

   128            ($(z.south east)+(0em,-5ex)$) -- ($(xxa.south east)+(-3em,-5ex)$)

   129                node[midway, below=0.5em]{$b \in L_1*$};

   130 \end{tikzpicture}}}

   131 

   132 

   133 \subfigure[Last split]{\label{last_split}

   134 \scalebox{0.7}{

   135 \begin{tikzpicture}

   136     \node[draw,minimum height=3.8ex] (xa) { $\hspace{4em}xa\_max\hspace{4em}$ };

   137     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{0.5em}x - xa\_max\hspace{0.5em}$ };

   138     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (za) { $\hspace{2em}za\hspace{2em}$ };

   139     \node[draw,minimum height=3.8ex, right=-0.03em of za] (zb) { $\hspace{7em}zb\hspace{7em}$ };

   140 

   141     \draw[decoration={brace,transform={yscale=3}},decorate]

   142            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

   143                node[midway, above=0.5em]{$x$};

   144 

   145     \draw[decoration={brace,transform={yscale=3}},decorate]

   146            (za.north west) -- ($(zb.north east)+(0em,0em)$)

   147                node[midway, above=0.5em]{$z$};

   148 

   149     \draw[decoration={brace,transform={yscale=3}},decorate]

   150            ($(xa.north west)+(0em,3ex)$) -- ($(zb.north east)+(0em,3ex)$)

   151                node[midway, above=0.8em]{$x @ z \in L_1*$};

   152 

   153     \draw[decoration={brace,transform={yscale=3}},decorate]

   154            ($(za.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

   155                node[midway, below=0.5em]{$(x - xa\_max) @ za \in L_1$};

   156 

   157     \draw[decoration={brace,transform={yscale=3}},decorate]

   158            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

   159                node[midway, below=0.5em]{$xa\_max \in L_1*$};

   160 

   161     \draw[decoration={brace,transform={yscale=3}},decorate]

   162            ($(zb.south east)+(0em,0ex)$) -- ($(zb.south west)+(0em,0ex)$)

   163                node[midway, below=0.5em]{$zb \in L_1*$};

   164 

   165     \draw[decoration={brace,transform={yscale=3}},decorate]

   166            ($(zb.south east)+(0em,-4ex)$) -- ($(xxa.south west)+(0em,-4ex)$)

   167                node[midway, below=0.5em]{$(x - xa\_max)@z \in L_1*$};

   168 \end{tikzpicture}}}

   169 

   170 

   171 \subfigure[Structure transferred to $y$]{\label{trans_split}

   172 \scalebox{0.7}{

   173 \begin{tikzpicture}

   174     \node[draw,minimum height=3.8ex] (xa) { $\hspace{5em}ya\hspace{5em}$ };

   175     \node[draw,minimum height=3.8ex, right=-0.03em of xa] (xxa) { $\hspace{2em}y - ya\hspace{2em}$ };

   176     \node[draw,minimum height=3.8ex, right=-0.03em of xxa] (za) { $\hspace{2em}za\hspace{2em}$ };

   177     \node[draw,minimum height=3.8ex, right=-0.03em of za] (zb) { $\hspace{7em}zb\hspace{7em}$ };

   178 

   179     \draw[decoration={brace,transform={yscale=3}},decorate]

   180            (xa.north west) -- ($(xxa.north east)+(0em,0em)$)

   181                node[midway, above=0.5em]{$y$};

   182 

   183     \draw[decoration={brace,transform={yscale=3}},decorate]

   184            (za.north west) -- ($(zb.north east)+(0em,0em)$)

   185                node[midway, above=0.5em]{$z$};

   186 

   187     \draw[decoration={brace,transform={yscale=3}},decorate]

   188            ($(xa.north west)+(0em,3ex)$) -- ($(zb.north east)+(0em,3ex)$)

   189                node[midway, above=0.8em]{$y @ z \in L_1*$};

   190 

   191     \draw[decoration={brace,transform={yscale=3}},decorate]

   192            ($(za.south east)+(0em,0ex)$) -- ($(xxa.south west)+(0em,0ex)$)

   193                node[midway, below=0.5em]{$(y - ya) @ za \in L_1$};

   194 

   195     \draw[decoration={brace,transform={yscale=3}},decorate]

   196            ($(xa.south east)+(0em,0ex)$) -- ($(xa.south west)+(0em,0ex)$)

   197                node[midway, below=0.5em]{$ya \in L_1*$};

   198 

   199     \draw[decoration={brace,transform={yscale=3}},decorate]

   200            ($(zb.south east)+(0em,0ex)$) -- ($(zb.south west)+(0em,0ex)$)

   201                node[midway, below=0.5em]{$zb \in L_1*$};

   202 

   203     \draw[decoration={brace,transform={yscale=3}},decorate]

   204            ($(zb.south east)+(0em,-4ex)$) -- ($(xxa.south west)+(0em,-4ex)$)

   205                node[midway, below=0.5em]{$(y - ya)@z \in L_1*$};

   206 \end{tikzpicture}}}

   207 

   208 \caption{The case for $STAR$}

   209 \end{figure}