tex: final commit I suppose (I wish)

latex/content.tex

@@ -258,12 +258,10 @@ of $ T_i $. Our calculations have yielded that $ \overline{T_1} = 1.72 $ and $
 	{S_N}^2 = 0.88 $. Our Student coefficient is $ t_{0.95, 2} = 2 $.
 
 We can now calculate the Confidence Interval for $ T_1 $ for $ R = 10^5 $ simulations :
-
 \begin{align*}
 	IC_{95\%}(T_1) & = \left[ 1.72 \pm 1.96 \frac{\sqrt{0.88}}{\sqrt{10^5}} \cdot 2 \right] \\
 	               & = \left[ 172 \pm 0.012 \right]                                         \\
 \end{align*}
-
 We can see that the Confidence Interval is very small, thanks to the large number of iterations.
 This results in a steady curve in figure \ref{fig:graphic-NFBP-Ti-105-sim}.
 
@@ -274,12 +272,24 @@ This results in a steady curve in figure \ref{fig:graphic-NFBP-Ti-105-sim}.
 	\label{fig:graphic-NFBP-Vi-105-sim}
 \end{figure}
 
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Hn-105-sim}
+	\caption{Histogram of $ H_n $ for $ R = 10^5 $ simulations and $ N = 50 $ items (number of bins required to store $n$ items)}
+	\label{fig:graphic-NFBP-Hn-105-sim}
+\end{figure}
+
 \paragraph{Asymptotic behavior of $ H_n $} Finally, we analyzed how many bins
 were needed to store $ n $ items. We used the numbers from the $ R = 10^5 $ simulations.
 
 
+We can see in figure \ref{fig:graphic-NFBP-Hn-105-sim} that $ H_n $ is
+asymptotically linear. The expected value and the variance are also displayed.
+The variance also increases linearly.
 
 
+\paragraph{} The Next Fit Bin Packing algorithm is a very simple algorithm
+with predictable results. It is very fast, but it is not optimal.
 
 
 \section{Next Fit Dual Bin Packing algorithm (NFDBP)}
@@ -328,7 +338,10 @@ new constraints on the first bin can be expressed as follows :
 	\text{ and } & U_1 + U_2 + \ldots + U_{k}    \geq 1 \qquad \text{ with } k \geq 2 \\
 \end{align*}
 
-\subsection{La giga demo}
+
+\subsection{Building a mathematical model}
+
+In this section we will try to determine the probabilistic law followed by $ T_i $.
 
 Let $ k \geq 2 $. Let $ (U_n)_{n \in \mathbb{N}^*} $ be a sequence of
 independent random variables with uniform distribution on $ [0, 1] $, representing
@@ -343,12 +356,12 @@ bin. We have that
 
 Let $ A_k = \{ U_1 + U_2 + \ldots + U_{k} < 1 \}$. Hence,
 
-\begin{align}
+\begin{align*}
 	\label{eq:prob}
 	P(T_i = k)
 	 & = P(A_{k-1} \cap A_k^c)                                           \\
 	 & = P(A_{k-1}) - P(A_k) \qquad \text{ (as $ A_k \subset A_{k-1} $)} \\
-\end{align}
+\end{align*}
 
 We will try to show that $ \forall k \geq 1 $, $ P(A_k) = \frac{1}{k!} $. To do
 so, we will use induction to prove the following proposition \eqref{eq:induction},
@@ -414,6 +427,18 @@ Finally, plugging this into \eqref{eq:prob} gives us
 	P(T_i = k) = P(A_{k-1}) - P(A_{k}) = \frac{1}{(k-1)!} - \frac{1}{k!} \qquad \forall k \geq 2
 \]
 
+\subsection{Empirical results}
+
+We ran $ R = 10^3 $ simulations for $ N = 10 $ items. The empirical results are
+similar to the mathematical model.
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=1.0\textwidth]{graphics/graphic-NFDBP-T1-103-sim}
+	\caption{Therotical and empiric histograms of $ T_1 $ for $ R = 10^3 $ simulations and $ N = 10 $ items (number of itens in the first bin)}
+	\label{fig:graphic-NFDBP-T1-103-sim}
+\end{figure}
+
 \subsection{Expected value of $ T_i $}
 
 We now compute the expected value $ \mu $ and variance $ \sigma^2 $ of $ T_i $.
@@ -441,6 +466,8 @@ We now compute the expected value $ \mu $ and variance $ \sigma^2 $ of $ T_i $.
 	\sigma^2 = E({T_i}^2) - E(T_i)^2 = 3e - 1 - e^2
 \end{align*}
 
+$ H_n $ is asymptotically normal, following a $ \mathcal{N}(\frac{N}{\mu}, \frac{N \sigma^2}{\mu^3}) $
+
 
 \section{Complexity and implementation optimization}
 
@@ -519,13 +546,43 @@ then calculate the statistics (which iterates multiple times over the array).
 	between devices. Execution time and memory usage do not include the import of
 	libraries.}
 
-\subsection{NFBP vs NFDBP}
-
 \subsection{Optimal algorithm}
 
+As we have seen, NFDBP algorithm is much better than NFBP algorithm. All the
+variables excluding V are showing this. More specifically, the most relevant
+variable is Hn which is growing slightly slower in the NFDBP algorithm than in
+the NFBP algorithm.
+
+
+Another algorithm that we did not explore in this project is the SUBP (Skim Up
+Bin Packing) algorithm. It works in the same way as the NFDBP algorithm.
+However, when an item exceeds the box size, it is removed from the current bin
+and placed into the next bin. This algorithm that we could not exploit is much
+more efficient than both of the previous algorithms. His main issue is that it
+takes a lot of storage and requires higher capacities.
+
+We redirect you towards this video which demonstrates why another algorithm is
+actually the most efficient that we can imagine. In this video we see that the
+mostoptimized of alrogithm is another version of NFBP where we sort the items
+in a decreasing order before sending them into the different bins.
+
+\clearpage
 
 \sectionnn{Conclusion}
 
+In this project, we explored many bin packing algorithms in 1 dimension. We
+discovered how some bin packing algorithms can be really simple to implement
+but also a strong data consumer as the NFBP algorithm.
+
+By modifying the conditions of bin packing we can upgrade our performances. For
+example, the NFDBP doest not permit to close the boxes (which depend of the
+context of this implementation). The performance analysis conclusions are the
+consequences of a precise statistical and probabilistic study that we have leaded
+on this project.
+
+To go further, we could now think about the best applications of different
+algorithms in real contexts, thanks to simulations.
+
 
 \nocite{bin-packing-approximation:2022}
 \nocite{hofri:1987}