tex: move and write performance part

latex/content.tex

@@ -117,11 +117,19 @@ with 10 packets.
 \subsubsection{Variables used in models}
 
 
-\subsubsection{Complexity and implementation optimization}
+
+\cite{hofri:1987}
+% TODO mettre de l'Histoire
+
+\section{Next Fit Dual Bin Packing algorithm}
+
+\section{Complexity and implementation optimization}
 
 The NFBP algorithm has a linear complexity $ O(n) $, as we only need to iterate
 over the items once.
 
+\subsection{Performance optimization}
+
 When implementing the statistical analysis, the intuitive way to do it is to
 run $ R $ simulations, store the results, then conduct the analysis. However,
 when running a large number of simulations, this can be very memory
@@ -129,17 +137,34 @@ consuming. We can optimize the process by computing the statistics on the fly,
 by using sum formulae. This uses nearly constant memory, as we only need to
 store the current sum and the current sum of squares for different variables.
 
+While the mean can easily be calculated by summing then dividing, the variance
+can be calculated using the following formula:
+
+\begin{align}
+	{S_N}^2 & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X})^2               \\
+	        & = \frac{1}{N-1} \sum_{i=1}^{N} X_i^2 - \frac{N}{N-1} \overline{X}^2
+\end{align}
+
+The sum $ \frac{1}{N-1} \sum_{i=1}^{N} X_i^2 $ can be calculated iteratively
+after each simulation.
+
+\subsection{Effective resource consumption}
+
+We set out to study the resource consumption of the algorithms. We implemented
+the above formulae to calculate the mean and variance of $ N = 10^6 $ random
+numbers. We wrote the following algorithms :
+
+\begin{lstlisting}[language=python]
+N = 10**6
+values = [random() for _ in range(N)]
+mean = mean(values)
+variance = variance(values)
+\end{lstlisting}
+
+
 % TODO : code
-% TODO : move this somewhere else ?
 % TODO : add a graph
 
-\cite{hofri:1987}
-% TODO mettre de l'Histoire
-
-\section{Next Fit Dual Bin Packing algorithm}
-
-\section{Algorithm comparisons and optimization}
-
 \subsection{NFBP vs NFDBP}
 
 \subsection{Optimal algorithm}