\documentclass{beamer}
%include lhs2TeX.fmt
\author{Piyush P Kurur\\
  Office no: 224\\
  Dept. of Comp. Sci. and Engg.\\
  IIT Kanpur}


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\newcommand{\T}[1]{\ensuremath{T\left(#1\right)}}  
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\title{Fundamentals of Computing: Lecture 27}
\date{October 9, 2009}
\begin{document}
\begin{frame}
  \maketitle
\end{frame}

\begin{frame}
  \frametitle{Recursive data types}
  \begin{itemize}
  
  \item List.
    \pause
    A list is either an empty list or 
    an element followed by a list.
    \pause
  
  \item Binary trees.
    \pause     
    A binary tree is either an empty Tree or a root
    node with two children left subtree and right subtree.
    \pause

  \item General trees (some times called Rose trees).  \pause Either
    an empty tree or a node with a Forest of subtrees.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{List}
  \begin{spec}
    data List a = Empty | Cons a (List a)
  \end{spec}
  \pause

  \begin{code}
    typedef struct Cons Cons;
    typedef Cons *List
    struct Cons {
      int datum;
      List next;
    };
    List emptyList = (List) NULL;
  \end{code}

\end{frame}

\begin{frame}
  \begin{code}
    int head(List l)
    {
      if( l == NULL) {error("head of an empty list");}
      else return l -> datum
    }
    List tail (List l)
    {
      if( l == NULL) {error("tail of an empty list");}
      else return l -> next
    }
  \end{code}

\end{frame}

\begin{frame}
  \frametitle{Some list functions}
  Function |singleton(x)| creates a list of just one element.

  \pause
  \begin{code}
    List singleton(int x){
      List l;
      l = (List) malloc(sizeof(Cons))
      if( l != NULL)
      {
        l -> datum = x;
        l -> next  = NULL;
      }
      return l;
    }
  \end{code}
\end{frame}

\begin{frame}
     
  \begin{code}
    void appendTo(List *a, List b)
    {
      List ptr;
      if( *a == NULL){
        *a = b;
        return;
      }
      ptr = *a;
      while(ptr -> next != NULL)
      {
        ptr = ptr -> next;
      }
      ptr -> next = b;
      return;
    }
  \end{code}
\end{frame}

\begin{frame}
  \frametitle{Reverse a list}
  
  \only<1> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots, label=above:$\texttt{p}$](p){$42$};
      \node[redbox, right=of p, label=above:$\texttt{q}$](q){$15$};
      \node[redbox, right=of q, label=above:$\texttt{r}$](r){$100$};
      \node[redbox, right=of r](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q) -- (r);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
   \end{tikzpicture}
  }

  \only<2> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots, label=above:$p$](p){$42$};
      \node[redbox, right=of p, label=above:$q$](q){$15$};
      \node[redbox, right=of q, label=above:$r$](r){$100$};
      \node[redbox, right=of r](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q.east) to [out=0, in = 90] (p.north);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
    \end{tikzpicture}
  }

  \only<3> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots](p){$42$};
      \node[redbox, right=of p, label=above:$\texttt{p,q}$](q){$15$};
      \node[redbox, right=of q, label=above:$\texttt{r}$](r){$100$};
      \node[redbox, right=of r](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q.east) to [out=0, in = 90] (p.north);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
    \end{tikzpicture}
  }

  \only<4> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots](p){$42$};
      \node[redbox, right=of p, label=above:$\texttt{p}$](q){$15$};
      \node[redbox, right=of q, label=above:$\texttt{q,r}$](r){$100$};
      \node[redbox, right=of r](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q.east) to [out=0, in = 90] (p.north);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
    \end{tikzpicture}
  }


  \only<5> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots](p){$42$};
      \node[redbox, right=of p, label=above:$\texttt{p}$](q){$15$};
      \node[redbox, right=of q, label=above:$\texttt{q}$](r){$100$};
      \node[redbox, right=of r, label=above:$\texttt{r}$](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q.east) to [out=0, in = 90] (p.north);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
    \end{tikzpicture}
  }

  \only<6> {
    \begin{tikzpicture}[node distance=1.25em]
      \node[null](leftnull){};
      \node[redbox, right=of leftnull](a){2};
      \node[right=of a](leftdots){$\ldots$};
      \node[redbox, right=of leftdots](p){$42$};
      \node[redbox, right=of p, label=above:$\texttt{p}$](q){$15$};
      \node[redbox, right=of q, label=above:$\texttt{q}$](r){$100$};
      \node[redbox, right=of r, label=above:$\texttt{r}$](rnext){$34$};
      \node[right=of rnext](rightdots){$\ldots$};
      \node[redbox, right=of rightdots](b){$1$};
      \node[null, right=of b](rightnull){};
      
      \draw[->, thick] (a) -- (leftnull);
      \draw[->, thick] (leftdots) -- (a);
      \draw[->, thick](p) -- (leftdots);
      \draw[->,thick](q) -- (p);
      \draw[->,thick](r) -- (rnext);
      \draw[->, thick](rnext)--(rightdots);
      \draw[->, thick](rightdots) -- (b);
      \draw[->, thick](b) -- (rightnull);
    \end{tikzpicture}
  }

\end{frame}

\begin{frame}
  \begin{code}

    void reverse(List a)
    {
      List p,q,r;
      if( a == NULL) return;
      p = NULL;
      q = a;
      r = a -> next;
      while( r )
      {
        q -> next = p;
        p = q;
        q = r;
        r = r -> next;
      }
    }
  \end{code}
\end{frame}
\end{document}