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+\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 $log(n)$. + +\end{document} \ No newline at end of file