tex: NFBP

latex/content.tex

@@ -104,12 +104,34 @@ conduct our own analysis and study various algorithms and their probabilistic
 advantages, focusing on one-dimensional bin packing, where we try to store
 items of different heights in a linear bin.
 
-\section{Next Fit Bin Packing algorithm}
+\section{Next Fit Bin Packing algorithm (NFBP)}
+
+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.
 
 \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.
 
+\subsubsection{Variables used in models}
+
+
+\subsubsection{Complexity and implementation optimization}
+
+The NFBP algorithm has a linear complexity $ O(n) $, as we only need to iterate
+over the items once.
+
+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
+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.
+
+% TODO : code
+% TODO : move this somewhere else ?
+% TODO : add a graph
 
 \cite{hofri:1987}
 % TODO mettre de l'Histoire