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


\newcommand{\Token}[2][]{\node(#2#1)[token]{#2};}
\newcommand{\Symbol}[2][]{\node(#2#1)[symbol]{#2};}
\newcommand{\Point}[1]{\node(#1)[point]{#1};}

\usepackage{tikz}
\usetikzlibrary{positioning,shapes,chains,fit}
\usetikzlibrary{shapes.symbols}
\usetikzlibrary{matrix}
\usetikzlibrary{backgrounds}
\usetikzlibrary{shapes.geometric}

\usepackage{pgfkeys}
\usepackage{multicol}
\pgfkeys %
    {%
      /tikz/coin/.style={
        rectangle,
        minimum height=2.5mm,
        minimum width=1cm,
        draw=black,
        fill=black!20,
        rounded corners
      },
      /tikz/towercolor/.style={
        fill=black!90
      },
      /tikz/towerbase/.style={
        trapezium,
        trapezium angle=45,
        trapezium stretches=true,
        towercolor,
        minimum width=7.5mm,
        minimum height=2.5mm,
      },
      /tikz/tower/.style={
        rectangle,
        rounded corners,
        towercolor,
        minimum width=2.5mm,
        minimum height=17.5mm,
      }
    }

\title{Fundamentals of Computing: Lecture 8}
\date{August 12, 2009}
\begin{document}
\begin{frame}
\maketitle
\end{frame}

\begin{frame}
  \frametitle{Summary of last class}
  \pause
  \begin{itemize}
  \item We looked at functions.
    \pause
  \item For C the argument passing scheme is call by value
    \pause
  \item Functions can be recursive
    \pause
  \item The |main| function and its return type
    \pause
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Recursion: Tower of Hanoi}
  \begin{multicols}{2}
    \begin{tikzpicture}
      \pgfdeclarelayer{background}
      \pgfsetlayers{background,main}
      \node[towerbase] (baseA) at (0,0){};
      \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
      \node[towerbase] (baseC) at (3cm,0){};
      \node[tower,above] (towerA) at (baseA.south){};
      \node[tower,above] (towerB) at (baseB.south){};
      \node[tower,above] (towerC) at (baseC.south){};
      \node[above] at (towerA.north) {$A$}; 
      \node[above] at (towerB.north) {$B$}; 
      \node[above] at (towerC.north) {$C$}; 
      
      \begin{scope}[every node/.style={coin,above}]
        \node[minimum width=1.75cm] (d) at (baseA.north){};
        \node[coin,minimum width=1.5cm, above] (c) at (d.north) {};
        \node[coin,minimum width=1.25cm] (b) at (c.north) {};
        \node[coin,minimum width=1cm] (a) at (b.north) {};
      \end{scope}
      \node[above] at (towerA.north) {$A$}; 
      \begin{pgfonlayer}{background}
        \node[rectangle, fill=yellow!40, fit=(towerA) (baseB) (baseC) (d)] {};
      \end{pgfonlayer}
    \end{tikzpicture}
    \newpage
    \begin{tikzpicture}
      \pgfdeclarelayer{background}
      \pgfsetlayers{background,main}
      \node[towerbase] (baseA) at (0,0){};
      \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
      \node[towerbase] (baseC) at (3cm,0){};
      \node[tower,above] (towerA) at (baseA.south){};
      \node[tower,above] (towerB) at (baseB.south){};
      \node[tower,above] (towerC) at (baseC.south){};
      \node[above] at (towerA.north) {$A$}; 
      \node[above] at (towerB.north) {$B$}; 
      \node[above] at (towerC.north) {$C$}; 
      \begin{scope}[every node/.style={coin,above}]
        \node[minimum width=1.75cm] (d) at (baseC.north){};
        \node[coin,minimum width=1.5cm, above] (c) at (d.north) {};
        \node[coin,minimum width=1.25cm] (b) at (c.north) {};
        \node[coin,minimum width=1cm] (a) at (b.north) {};
      \end{scope}
      \begin{pgfonlayer}{background}
        \node[rectangle, fill=blue!40, fit=(baseA) (baseB) (towerC) (d)] {};
      \end{pgfonlayer}
    \end{tikzpicture}
  \end{multicols}


  %% \begin{multicols}{2}
  %%   \begin{tikzpicture}
  %%     \pgfdeclarelayer{background}
  %%     \pgfsetlayers{background,main}
  %%     \node[towerbase] (baseA) at (0,0){};
  %%     \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
  %%     \node[towerbase] (baseC) at (3cm,0){};
  %%     \node[tower,above] (towerA) at (baseA.south){};
  %%     \node[tower,above] (towerB) at (baseB.south){};
  %%     \node[tower,above] (towerC) at (baseC.south){};
  %%     \node[above] at (towerA.north) {$A$}; 
  %%     \node[above] at (towerB.north) {$B$}; 
  %%     \node[above] at (towerC.north) {$C$}; 
      
  %%     \begin{scope}[every node/.style={coin,above}]
  %%       \node[minimum width=1.75cm] (d) at (a.north){};
  %%       \node[coin,minimum width=1cm] (a1) at (baseA.north) {};
  %%     \end{scope}
  %%     \begin{pgfonlayer}{background}
  %%       \node[rectangle, fill=green!40, fit=(towerA) (baseB) (baseC) (d)] {};
  %%     \end{pgfonlayer}
  %%   \end{tikzpicture}

  %%   \newpage
  %%   \begin{tikzpicture}
  %%     \pgfdeclarelayer{background}
  %%     \pgfsetlayers{background,main}
  %%     \node[towerbase] (baseA) at (0,0){};
  %%     \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
  %%     \node[towerbase] (baseC) at (3cm,0){};
  %%     \node[tower,above] (towerA) at (baseA.south){};
  %%     \node[tower,above] (towerB) at (baseB.south){};
  %%     \node[tower,above] (towerC) at (baseC.south){};
  %%     \node[above] at (towerA.north) {$A$}; 
  %%     \node[above] at (towerB.north) {$B$}; 
  %%     \node[above] at (towerC.north) {$C$}; 
  %%     \begin{scope}[every node/.style={coin,above}]
  %%       \node[minimum width=1cm] (d) at (a1.north){};
  %%       \node[coin,minimum width=1.75cm] (a1) at (baseA.north) {};
  %%     \end{scope}
  %%     \begin{pgfonlayer}{background}
  %%       \node[rectangle, fill=red!40, fit=(baseA) (baseB) (towerC) (d)] {};
  %%     \end{pgfonlayer}
  %%   \end{tikzpicture}
  %% \end{multicols}

\end{frame}




\begin{frame}
  \begin{multicols}{2}
  \begin{tikzpicture}
    \pgfdeclarelayer{background}
    \pgfsetlayers{background,main}
    \node[towerbase] (baseA) at (0,0){};
    \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
    \node[towerbase] (baseC) at (3cm,0){};
    \node[tower,above] (towerA) at (baseA.south){};
    \node[tower,above] (towerB) at (baseB.south){};
    \node[tower,above] (towerC) at (baseC.south){};
    \node[above] (A) at (towerA.north) {$A$}; 
    \node[above] (B) at (towerB.north) {$B$}; 
    \node[above] (C) at (towerC.north) {$C$}; 
    
    \begin{scope}[every node/.style={coin,above}]
      \node[minimum width=1.75cm] (d) at (baseA.north){};
      \node[coin,minimum width=1.5cm, above] (c) at (d.north) {};
      \node[coin,minimum width=1.25cm] (b) at (c.north) {};
      \node[coin,minimum width=1cm] (a) at (b.north) {};
    \end{scope}
    \node[above] at (towerA.north) {$A$}; 
    \begin{pgfonlayer}{background}
      \node[rectangle, fill=yellow!30, fit=(towerA) (baseB) (baseC) (d)] {};
      \uncover<2->{
        \node[rectangle, rounded corners, fill=red!50, fit=(a)(b)(c)](abc){};
        \path (towerA.north) edge[->,bend left] (towerB.north);
      }
    \end{pgfonlayer}
  \end{tikzpicture}
  \newpage

  \uncover<3->{
    \begin{tikzpicture}
      \pgfdeclarelayer{background}
      \pgfsetlayers{background,main}
      \node[towerbase] (baseA) at (0,0){};
      \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
      \node[towerbase] (baseC) at (3cm,0){};
      \node[tower,above] (towerA) at (baseA.south){};
      \node[tower,above] (towerB) at (baseB.south){};
      \node[tower,above] (towerC) at (baseC.south){};
      \node[above] at (towerA.north) {$A$}; 
      \node[above] at (towerB.north) {$B$}; 
      \node[above] at (towerC.north) {$C$}; 
      \begin{scope}[every node/.style={coin,above}]
        \node[minimum width=1.75cm] (d) at (baseA.north){};
        \node[coin,minimum width=1.5cm, above] (c) at (baseB.north) {};
        \node[coin,minimum width=1.25cm] (b) at (c.north) {};
        \node[coin,minimum width=1cm] (a) at (b.north) {};
      \end{scope}
      \begin{pgfonlayer}{background}
        \node[rectangle, fill=yellow!30, fit=(baseA) (baseB) (towerC) (d)] {};
        \uncover<4->{
          \node[rectangle, rounded corners, fill=red!50, fit=(d)](abc){};
          \path (towerA.north) edge[->,bend left] (towerC.north);
        }   
      \end{pgfonlayer}
    \end{tikzpicture}
  }
  \end{multicols}
  \begin{multicols}{2}
    \uncover<5->{
      \begin{tikzpicture}
        \pgfdeclarelayer{background}
        \pgfsetlayers{background,main}
        \node[towerbase] (baseA) at (0,0){};
        \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
        \node[towerbase] (baseC) at (3cm,0){};
        \node[tower,above] (towerA) at (baseA.south){};
        \node[tower,above] (towerB) at (baseB.south){};
        \node[tower,above] (towerC) at (baseC.south){};
        \node[above] at (towerA.north) {$A$}; 
        \node[above] at (towerB.north) {$B$}; 
        \node[above] at (towerC.north) {$C$}; 
        \begin{scope}[every node/.style={coin,above}]
          \node[minimum width=1.75cm] (d) at (baseC.north){};
          \node[coin,minimum width=1.5cm, above] (c) at (baseB.north) {};
          \node[coin,minimum width=1.25cm] (b) at (c.north) {};
          \node[coin,minimum width=1cm] (a) at (b.north) {};
        \end{scope}
        \begin{pgfonlayer}{background}
          \node[rectangle, fill=yellow!30, fit=(baseA) (baseB) (towerC) (d)] {};
          \uncover<6->{
            \node[rectangle, rounded corners, fill=red!50, fit=(a)(b)(c)](abc){};
            \draw[->,out=90,in=180] (B.north) to (towerC.north);
          }   
        \end{pgfonlayer}
      \end{tikzpicture}
    }
    \newpage
    \vspace{1cm}
    \uncover<7->{
      \begin{tikzpicture}
        \pgfdeclarelayer{background}
        \pgfsetlayers{background,main}
        \node[towerbase] (baseA) at (0,0){};
        \node[towerbase] (baseB) at (1.5cm,-1.5cm){};
        \node[towerbase] (baseC) at (3cm,0){};
        \node[tower,above] (towerA) at (baseA.south){};
        \node[tower,above] (towerB) at (baseB.south){};
        \node[tower,above] (towerC) at (baseC.south){};
        \node[above] at (towerA.north) {$A$}; 
        \node[above] at (towerB.north) {$B$}; 
        \node[above] at (towerC.north) {$C$}; 
        \begin{scope}[every node/.style={coin,above}]
          \node[minimum width=1.75cm] (d) at (baseC.north){};
          \node[coin,minimum width=1.5cm, above] (c) at (d.north) {};
          \node[coin,minimum width=1.25cm] (b) at (c.north) {};
          \node[coin,minimum width=1cm] (a) at (b.north) {};
        \end{scope}
        \begin{pgfonlayer}{background}
          \node[rectangle, fill=blue!40, fit=(baseA) (baseB) (towerC) (d)] {};
        \end{pgfonlayer}
      \end{tikzpicture}
    }
  \end{multicols}
\end{frame}

\begin{frame}
  \begin{code}
#include <stdio.h>
    
void hanoi(int, char, char,char);
int main()
{
        int n;
        printf("enter the number of disks: "); 
        scanf("%d",&n);
        if( n < 0 ) return 1;
        hanoi(n,'A','B','C');
        return 0;
}

void hanoi(int n, char a, char b, char c)
{
        if ( n <= 0 ) return;
        hanoi(n-1 , a, c, b);
        printf("(%d) %c -> %c\n", n, a, c);
        hanoi( n-1, b, a, c);
}
        
  \end{code}
\end{frame}     

\end{document}





\begin{frame}
  \frametitle{How to design loops?}
  
  \begin{definition}
    An invarient of a loop is a condition that is
    true always during the execution of the loop.
  \end{definition}
  
  Eg.
  
  \begin{spec}
    int sum = 0;
    int i   = 0;
    while( i < n)
    {
      i++;
      sum += i;
    }
  \end{spec}

  A nice invarient is $sum = \sum_{j=0}^i i$.
  
\end{frame}