% vim: set ts=2 sw=2 tw=80 et:
\documentclass[12pt]{article}

\usepackage[margin=3cm]{geometry}
\usepackage{xcolor}
\usepackage{lmodern}
\usepackage{listings}

\title{Graded Assignment 3 -- DSA}
\author{Claudio Maggioni}

% listings configuration
\lstset{
  basicstyle=\small\ttfamily,
  frame=shadowbox,
  rulesepcolor=\color{black},
  columns=fullflexible,
  commentstyle=\color{gray},
  keywordstyle=\bfseries,
  keywords={,while,if,elif,else,FUNCTION,return,for,from,to,TRUE,FALSE},
  mathescape=true,
  aboveskip=2em,
  captionpos=b,
  abovecaptionskip=1em,
  belowcaptionskip=1em
}

\begin{document}

\maketitle
\tableofcontents
\lstlistoflistings
\newpage

\section{Exercise 1}

{\small\ttfamily
\begin{verbatim}
12

Insert 6:
  12
 /
6

Insert 1:
    12
   /
  6
 /
1

Insert 7:
      12
   ///
  6
 / \
1   7

Insert 4:
        12
     ///
    6
 /// \
1     7
 \
  4

Insert 11:
           12
     //////
    6
 /// \
1     7
 \     \
  4     11

Insert 15:
           12
     //////  \
    6         15
 /// \
1     7
 \     \
  4     11

Insert 9:
             12
     ////////  \
    6           15
 /// \
1     7
 \     \\\
  4       11
         /
        9

Insert 5:
               12
       ////////  \
      6           15
 ///// \
1       7
 \       \\\
  4         11
   \       /
    5     9

Insert 13:
               12
       ////////  \\\\
      6              15
 ///// \            /
1       7         13
 \       \\\
  4         11
   \       /
    5     9

Insert 8:
                 12
       //////////  \\\\
      6                15
 ///// \              /
1       7           13
 \       \\\\\
  4           11
   \         /
    5       9
           /
          8

Insert 14:
                 12
       //////////  \\\\\\\
      6                   15
 ///// \              ////
1       7           13
 \       \\\\\        \
  4           11       14
   \         /
    5       9
           /
          8

Insert 3:
                   12
         //////////  \\\\\\\
        6                   15
 /////// \              ////
1         7           13
 \\\       \\\\\        \
    4           11       14
   / \         /
  3   5       9
             /
            8

Insert 10:
                      12
         /////////////  \\\\\\\
        6                      15
 /////// \                 ////
1         7              13
 \\\       \\\\\\\\        \
    4              11       14
   / \         ////
  3   5       9
             / \
            8   10

Insert 2:
                        12
           /////////////  \\\\\\\
          6                      15
 ///////// \                 ////
1           7              13
 \\\\\       \\\\\\\\        \
      4              11       14
     / \         ////
    3   5       9
   /           / \
  2           8   10

Delete 6:
                      12
           ///////////  \\\\\\\
          7                    15
 ///////// \\\\\\\\        ////
1                  11    13
 \\\\\         ////        \
      4       9             14
     / \     / \
    3   5   8   10
   /
  2

Delete 2:
                    12
         ///////////  \\\\\\\
        7                    15
 /////// \\\\\\\\        ////
1                11    13
 \\\         ////        \
    4       9             14
   / \     / \
  3   5   8   10

Delete 12:
                          15
                      ////
                    13
         ///////////  \
        7              14
 /////// \\\\\\\\
1                11
 \\\         ////
    4       9
   / \     / \
  3   5   8   10
\end{verbatim}
}

\section{Exercise 2}

\subsection{Point A}

{\small\ttfamily
\begin{verbatim}
12B

Insert 6:
   12B
  /
6R

Insert 1:
   6B
  /  \
1R    12R

Insert 7:
   6B
  /  \\\\
1B       12B
        /
      7R

Insert 4:
      6B
  ////  \\\\
1B          12B
  \        /
   4R    7R

Insert 11:
      6B
  ////  \\\\
1B          11B
  \        /   \
   4R    7R     12R

Insert 15:
            11B
        ////   \
      6R        12B
  ////  \          \
1B       7B         15R
  \
   4R

Insert 9:
               11B
        ///////   \
      6R           12B
  ////  \             \
1B       7B            15R
  \        \
   4R       9R

Insert 5:
         6B
     ////  \\\\\\\
   4B             11R
  /  \        ////   \
1R    5R    7B        12B
              \          \
               9R         15R

Insert 13:
                  11B
           ///////   \\\\\
         6R               13B
     ////  \             /   \
   4B       7B        12R     15R
  /  \        \
1R    5R       9R

Insert 8:
               8B
           ////  \\\\
         6R          11R
     ////  \        /   \\\\\
   4B       7R    9R         13B
  /  \                      /   \
1R    5R                 12R     15R

Insert 14:
               8B
           ////  \\\\
         6B          11B
     ////  \        /   \\\\\
   4B       7R    9B         13B
  /  \                      /   \\\\\
1R    5R                 12B         15B
                                    /
                                 14R

Insert 3:
            6B
        ////  \\\\
      4B          8R
  ////  \        /  \\\\
1B       5B    7B       11B
  \                    /   \\\\\
   3R                9B         13B
                               /   \\\\\
                            12B         15B
                                       /
                                    14R

Insert 10:
            6B
        ////  \\\\
      4B          8R
  ////  \        /  \\\\\\\\
1B       5B    7B           11B
  \                    /////   \\\\\
   3R                9B             13B
                       \           /   \\\\\
                        10R     12B         15B
                                           /
                                        14R

Insert 2:
               6B
           ////  \\\\
         4B          8R
     ////  \        /  \\\\\\\\
   2B       5B    7B           11B
  /  \                    /////   \\\\\
1R    3R                9B             13B
                          \           /   \\\\\
                           10R     12B         15B
                                              /
                                           14R
\end{verbatim}
}

\subsection{Point B}

A red-black tree of n distinct elements has an height between $\log(n)$ 
and $2\log(n)$ thanks to the red-black tree invariant. The worst-case insertion
complexity is $\log(n)$ since finding the right place to insert is as complex as
a regular tree (i.e. logarithmic) and the ``fixup'' operation is logarithmic as
well (the tree traversal is logarithmic while operations in each iteration are constant).
In asymptotic terms, the uneven height of leaves in the tree does not make a difference
since it is a constant factor (since the maximum height is $2\log(n)$ and the minimum one is $\log(n)$.

\section{Exercise 3}

\begin{lstlisting}[caption=Solution for exercise 3, label={lst:ex3}]
FUNCTION JOIN-INTERVALS(X)
  if X.length == 0:
    return
 
  for i from 1 to X.length:
    if i % 2 == 1:
      X[i] $\gets$ (X[i], 1)
    else:
      X[i] $\gets$ (X[i], -1)
  
  $\textit{sort X in place by 1st tuple element in O(n log(n))}$
  
  c $\gets$ 1
  n $\gets$ 0
  start $\gets$ X[i][1]
  for i from 2 to X.length:
    c $\gets$ c + X[i][2]
    if c == 0:
      X[2 * n + 1] $\gets$ start
      X[2 * n + 2] $\gets$ X[i][1]
      start $\gets$ X[i+1][1]
      n $\gets$ n + 1
  X.length $\gets$ 2 * n    
\end{lstlisting}
\\
The complexity of this algorithm is $O(n\log(n))$ since the sorting is in $\Theta(n\log(n))$ and the
union operation afterwards is $\Theta(n)$.

\end{document}