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tex: minor word corrections

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Paul ALNET 2023-06-04 13:39:59 +02:00
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@ -108,7 +108,7 @@ items of different heights in a linear bin.
Our goal is to study the number of bins $ H_n $ required to store $ n $ items 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 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. we store each item in the current bin if it fits, otherwise we open a new bin.
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -146,7 +146,7 @@ we store each item in the next bin if it fits, otherwise we open a new bin.
NFBP algorithm. The yellow item is stored in bin 2, while it could fit in bin 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. 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 \paragraph{} Each bin will have a fixed capacity of $ 1 $ and items
will be of random sizes between $ 0 $ and $ 1 $. will be of random sizes between $ 0 $ and $ 1 $.
\subsection{Variables used in models} \subsection{Variables used in models}
@ -155,7 +155,7 @@ We use the following variables in our algorithms and models :
\begin{itemize} \begin{itemize}
\item $ U_n $ : the size of the $ n $-th item. $ (U_n)_{n \in \mathbb{N}} $ \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 denotes the mathematical sequence of random variables of uniform
distribution on $ [0, 1] $ representing the items' sizes. distribution on $ [0, 1] $ representing the items' sizes.
@ -203,7 +203,7 @@ were needed to store $ n $ items.
\cite{hofri:1987} \cite{hofri:1987}
% TODO mettre de l'Histoire % TODO mettre de l'Histoire
\section{Next Fit Dual Bin Packing algorithm} \section{Next Fit Dual Bin Packing algorithm (NFDBP)}
Next Fit Dual Bin Packing is a variation of NFBP in which we allow the bins to 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. overflow. A bin must be fully filled, unless it is the last bin.
@ -365,8 +365,8 @@ 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 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. 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 While the mean can easily be calculated by summing then dividing, the empirical
can be calculated using the following formula: variance can be calculated using the following formula:
\begin{align*} \begin{align*}
{S_N}^2 & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X})^2 \\ {S_N}^2 & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X})^2 \\