tex: add NFDP math spec

latex/content.tex

@@ -110,9 +110,44 @@ Our goal is to study the number of bins $ H_n $ required to store $ n $ items
 for each algorithm. We first consider the Next Fit Bin Packing algorithm, where
 we store each item in the next bin if it fits, otherwise we open a new bin.
 
+\begin{figure}[h]
+	\centering
+	\begin{tikzpicture}[scale=0.8]
+		% Bins
+		\draw[thick] (0,0) rectangle (2,6);
+		\draw[thick] (3,0) rectangle (5,6);
+		\draw[thick] (6,0) rectangle (8,6);
+
+		% Items
+		\draw[fill=red] (0.5,0.5) rectangle (1.5,3.25);
+		\draw[fill=blue] (0.5,3.5) rectangle (1.5,5.5);
+		\draw[fill=green] (3.5,0.5) rectangle (4.5,1.5);
+		\draw[fill=orange] (3.5,1.75) rectangle (4.5,3.75);
+		\draw[fill=purple] (6.5,0.5) rectangle (7.5,2.75);
+		\draw[fill=yellow] (6.5,3) rectangle (7.5,4);
+
+		% arrow
+		\draw[->, thick] (8.6,3.5) -- (7.0,3.5);
+		\draw[->, thick] (8.6,1.725) -- (7.0,1.725);
+
+		% Labels
+		\node at (1,-0.75) {Bin 0};
+		\node at (4,-0.75) {Bin 1};
+		\node at (7,-0.75) {Bin 2};
+		\node at (10.0,3.5) {Yellow item};
+		\node at (10.0,1.725) {Purple item};
+
+	\end{tikzpicture}
+	\label{fig:nfbp}
+	\caption{Next Fit Bin Packing example}
+\end{figure}
+
+\paragraph{} The example in figure \ref{fig:nfbp} shows the limitations of the
+NFBP algorithm. The yellow item is stored in bin 2, while it could fit in bin
+1, because the purple item is considered first and is too large to fit.
+
 \paragraph{} Each bin will have a fixed capacity of $ 1 $ and items and items
-will be of random sizes between $ 0 $ and $ 1 $. We will run X simulations % TODO
-with 10 packets.
+will be of random sizes between $ 0 $ and $ 1 $.
 
 \subsection{Variables used in models}
 
@@ -135,8 +170,8 @@ We use the following variables in our algorithms and models :
 Mathematically, the NFBP algorithm imposes the following constraint on the first box :
 
 \begin{align*}
-	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k-1}  < 1                                  \\
-	\text{ and } & U_1 + U_2 + \ldots + U_{k}    \geq 1 \qquad \text{ with } k \geq 2
+	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k}  < 1                                      \\
+	\text{ and } & U_1 + U_2 + \ldots + U_{k+1}    \geq 1 \qquad \text{ with } k \geq 1 \\
 \end{align*}
 
 
@@ -145,10 +180,22 @@ Mathematically, the NFBP algorithm imposes the following constraint on the first
 We implemented the NFBP algorithm in Python \footnotemark, for its ease of use
 and broad recommendation. We used the \texttt{random} library to generate
 random numbers between $ 0 $ and $ 1 $ and \texttt{matplotlib} to plot the
-results.
+results in the form of histograms. We ran $ R = 10^6 $ simulations with
+$ N = 10 $ different items each.
 
 \footnotetext{The code is available in Annex \ref{annex:probabilistic}}
 
+\paragraph{Distribution of $ T_i $} We first studied how many items were
+present per bin.
+
+\paragraph{Distribution of $ V_i $} We then looked at the size of the first
+item in each bin.
+
+\paragraph{Asymptotic behavior of $ H_n $} Finally, we analyzed how many bins
+were needed to store $ n $ items.
+
+% TODO histograms
+% TODO analysis histograms
 
 
 
@@ -158,7 +205,49 @@ results.
 
 \section{Next Fit Dual Bin Packing algorithm}
 
-The variables used are the same as for NFBP.
+Next Fit Dual Bin Packing is a variation of NFBP in which we allow the bins to
+overflow. A bin must be fully filled, unless it is the last bin.
+
+\begin{figure}[h]
+	\centering
+	\begin{tikzpicture}[scale=0.8]
+		% Bins
+		\draw[thick] (0,0) rectangle (2,6);
+		\draw[thick] (3,0) rectangle (5,6);
+
+		% Transparent Tops
+		\fill[white,opacity=1.0] (0,5.9) rectangle (2,6.5);
+		\fill[white,opacity=1.0] (3,5.9) rectangle (5,6.5);
+
+		% Items
+		\draw[fill=red] (0.5,0.5) rectangle (1.5,3.25);
+		\draw[fill=blue] (0.5,3.5) rectangle (1.5,5.5);
+		\draw[fill=green] (0.5,5.75) rectangle (1.5,6.75);
+		\draw[fill=orange] (3.5,0.5) rectangle (4.5,2.5);
+		\draw[fill=purple] (3.5,2.75) rectangle (4.5,5.0);
+		\draw[fill=yellow] (3.5,5.25) rectangle (4.5,6.25);
+
+		% Labels
+		\node at (1,-0.75) {Bin 0};
+		\node at (4,-0.75) {Bin 1};
+
+	\end{tikzpicture}
+	\caption{Next Fit Dual Bin Packing example}
+	\label{fig:nfdbp}
+\end{figure}
+
+\paragraph{} The example in figure \ref{fig:nfdbp} shows how NFDBP utilizes
+less bins than NFBP, due to less stringent constraints. The top of the bin is
+effectively removed, allowing for an extra item to be stored in the bin. We can
+easily see how with NFDBP each bin can at least contain two items.
+
+\paragraph{} The variables used are the same as for NFBP. Mathematically, the
+new constraints on the first bin can be expressed as follows :
+
+\begin{align*}
+	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k-1}  < 1                                  \\
+	\text{ and } & U_1 + U_2 + \ldots + U_{k}    \geq 1 \qquad \text{ with } k \geq 2 \\
+\end{align*}
 
 \subsection{La giga demo}
 
@@ -206,7 +295,7 @@ $ U_1 $ and $ U_2 $ are independent, so
 	                   & = \begin{cases}
 		1 & \text{if } x \in [0, 1] \text{ and } y \in [0, 1] \\
 		0 & \text{otherwise}                                  \\
-	\end{cases}   \\
+	\end{cases}  \\
 \end{align*}
 
 Hence,

latex/main.tex

@@ -14,6 +14,7 @@
 \usepackage[english]{babel}
 \usepackage{eso-pic} % for background on cover
 \usepackage{listings}
+\usepackage{tikz}
 
 % Define colors for code
 \definecolor{codegreen}{rgb}{0,0.4,0}