<commit_before>\chapter{Supplementary Material}
\label{chap:supp-material}
\centerline{\rule{149mm}{.02in}}
\vspace{2cm}

Supplementary diagrams or tables which are mentioned in the main report are present here.

\section{Schedule}

This section contains diagrams which illustrate the project's schedule, milestones and overall process. See Section \ref{sec:schedule} for a description of the project schedule.

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.45]{figures/schedule/initial_project_schedule.pdf}
	}%
	\caption{Initial Project Schedule Created on 31/01/14}
	\label{fig:initial-schedule}
\end{figure}

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.45]{figures/schedule/revised_project_schedule.pdf}
	}%
	\caption{Revised Project Schedule Created on 20/02/14}
	\label{fig:revised-schedule}
\end{figure}

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.6]{figures/schedule/full-project-process.pdf}
	}%
	\caption{Flow Chart of Full Project Schedule}
	\label{fig:full-project-process}
\end{figure}

\begin{landscape}	
	\begin{figure}
		\centering
		\centerline{ \includegraphics[scale=0.475]{figures/schedule/initial_schedule_timeline.pdf} }
		\caption{Initial Milestone Timeline Created on 20/02/14}
		\label{fig:initial-milestone-timeline}
	\end{figure}

	\begin{figure}
		\centering
		\centerline{ \includegraphics[scale=0.475]{figures/schedule/revised_schedule_timeline.pdf} }
		\caption{Revised Milestone Timeline Created on 20/02/14}
		\label{fig:revised-milestone-timeline}
	\end{figure}	
\end{landscape}



\begin{landscape}
	\section{Multi-dimensional Search Structures}

	Supplementary information gathered from the background research in Section \ref{chap:background_research} is presented here.

	\begin{table}[h]
		\centering
		\begin{tabular}{|p{2.8cm}|p{5cm}|p{5cm}|p{5cm}|p{2cm}|}
			\hline
			\textbf{Index Structure} &
			\textbf{Memory Overhead} &
			\textbf{Bucket Method?} &
			\textbf{High-Dimensional Data} &
			\textbf{Complexity} \\
			\hline
			Sequential Scan & Low & No (but since data is stored contiguously, there are minimal I/O operations due to sequential access) & Often better than other structures with high $d$ (but significantly poorer performance with low $d$) & Low \\		
			B${}^{+}$-Tree & Low & Yes & One-dimensional & Low \\
			R-Tree & Moderate & Yes & Poor for $d > 10$ \cite{pyramid-tree} & Moderate \\
			Quadtree & Low with uniformly distributed data, high for skewed data due to unnecessary nodes caused by splitting sparse regions of data space & No & Poor because it tries to use balanced splits \cite{pyramid-tree} & Low \\
			Pyramid Tree & Low & Yes (based on B${}^{+}$-tree) & Good & High \\
			PK-Tree & Moderate & No (but uses similar method to reduce I/O operations) & Good & High \\
			Skip Quadtree & Moderate & No & Untested & Moderate \\
			Quadtreap & Low & No & Untested & Moderate \\
			Splay Quadtree & Moderate & No & Untested & Moderate \\
			\hline<commit_msg>Changed ratings for complexity of structures<commit_after>\chapter{Supplementary Material}
\label{chap:supp-material}
\centerline{\rule{149mm}{.02in}}
\vspace{2cm}

Supplementary diagrams or tables which are mentioned in the main report are present here.

\section{Schedule}

This section contains diagrams which illustrate the project's schedule, milestones and overall process. See Section \ref{sec:schedule} for a description of the project schedule.

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.45]{figures/schedule/initial_project_schedule.pdf}
	}%
	\caption{Initial Project Schedule Created on 31/01/14}
	\label{fig:initial-schedule}
\end{figure}

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.45]{figures/schedule/revised_project_schedule.pdf}
	}%
	\caption{Revised Project Schedule Created on 20/02/14}
	\label{fig:revised-schedule}
\end{figure}

\begin{figure}[h]
	\centering
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.6]{figures/schedule/full-project-process.pdf}
	}%
	\caption{Flow Chart of Full Project Schedule}
	\label{fig:full-project-process}
\end{figure}

\begin{landscape}	
	\begin{figure}
		\centering
		\centerline{ \includegraphics[scale=0.475]{figures/schedule/initial_schedule_timeline.pdf} }
		\caption{Initial Milestone Timeline Created on 20/02/14}
		\label{fig:initial-milestone-timeline}
	\end{figure}

	\begin{figure}
		\centering
		\centerline{ \includegraphics[scale=0.475]{figures/schedule/revised_schedule_timeline.pdf} }
		\caption{Revised Milestone Timeline Created on 20/02/14}
		\label{fig:revised-milestone-timeline}
	\end{figure}	
\end{landscape}



\begin{landscape}
	\section{Multi-dimensional Search Structures}

	Supplementary information gathered from the background research in Section \ref{chap:background_research} is presented here.

	\begin{table}[h]
		\centering
		\begin{tabular}{|p{2.8cm}|p{5cm}|p{5cm}|p{5cm}|p{2cm}|}
			\hline
			\textbf{Index Structure} &
			\textbf{Memory Overhead} &
			\textbf{Bucket Method?} &
			\textbf{High-Dimensional Data} &
			\textbf{Complexity} \\
			\hline
			Sequential Scan & Low & No (but since data is stored contiguously, there are minimal I/O operations due to sequential access) & Often better than other structures with high $d$ (but significantly poorer performance with low $d$) & Low \\		
			B${}^{+}$-Tree & Low & Yes & One-dimensional & Low \\
			R-Tree & Moderate & Yes & Poor for $d > 10$ \cite{pyramid-tree} & Moderate \\
			Quadtree & Low with uniformly distributed data, high for skewed data due to unnecessary nodes caused by splitting sparse regions of data space & No & Poor because it tries to use balanced splits \cite{pyramid-tree} & Low \\
			Pyramid Tree & Low & Yes (based on B${}^{+}$-tree) & Good & Moderate \\
			PK-Tree & Moderate & No (but uses similar method to reduce I/O operations) & Good & High \\
			Skip Quadtree & Moderate & No & Untested & Moderate \\
			Quadtreap & Low & No & Untested & High \\
			Splay Quadtree & Moderate & No & Untested & High \\
			\hline
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