tex: NFBP impl detail

latex/content.tex

@@ -114,7 +114,42 @@ we store each item in the next bin if it fits, otherwise we open a new bin.
 will be of random sizes between $ 0 $ and $ 1 $. We will run X simulations % TODO
 with 10 packets.
 
-\subsubsection{Variables used in models}
+\subsection{Variables used in models}
+
+We use the following variables in our algorithms and models :
+
+\begin{itemize}
+
+	\item $ U_n $ : the size of the $ n $-th item. $ (U_n)_{n \in \mathbb{N}} $
+	      denotes the mathematical sequence of random variables of uniform
+	      distribution on $ [0, 1] $ representing the items' sizes.
+
+	\item $ T_i $ : the number of items in the $ i $-th bin.
+
+	\item $ V_i $ : the size of the first item in the $ i $-th bin.
+
+	\item $ H_n $ : the number of bins required to store $ n $ items.
+
+\end{itemize}
+
+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
+\end{align*}
+
+
+\subsection{Implementation and results}
+
+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.
+
+\footnotetext{The code is available in Annex \ref{annex:probabilistic}}
+
+
 
 
 
@@ -123,6 +158,8 @@ with 10 packets.
 
 \section{Next Fit Dual Bin Packing algorithm}
 
+The variables used are the same as for NFBP.
+
 \subsection{La giga demo}
 
 Let $ k \in \mathbb{N} $. Let $ (U_n)_{n \in \mathbb{N}} $ be a sequence of
@@ -225,8 +262,10 @@ $ f_{U_{k-1}}(y) = 1 $.
 
 \section{Complexity and implementation optimization}
 
-The NFBP algorithm has a linear complexity $ O(n) $, as we only need to iterate
-over the items once.
+Both the NFBP and NFDBP algorithms have a linear complexity $ O(n) $, as we
+only need to iterate over the items once. While the algorithms themselves are
+linear, calculating the statistics may not not be. In this section, we will
+discuss how to optimize the implementation of the statistical analysis.
 
 \subsection{Performance optimization}