tex: minor word corrections
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@ -108,7 +108,7 @@ items of different heights in a linear bin.
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Our goal is to study the number of bins $ H_n $ required to store $ n $ items
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for each algorithm. We first consider the Next Fit Bin Packing algorithm, where
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we store each item in the next bin if it fits, otherwise we open a new bin.
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we store each item in the current bin if it fits, otherwise we open a new bin.
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\begin{figure}[h]
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\centering
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@ -146,7 +146,7 @@ we store each item in the next bin if it fits, otherwise we open a new bin.
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NFBP algorithm. The yellow item is stored in bin 2, while it could fit in bin
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1, because the purple item is considered first and is too large to fit.
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\paragraph{} Each bin will have a fixed capacity of $ 1 $ and items and items
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\paragraph{} Each bin will have a fixed capacity of $ 1 $ and items
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will be of random sizes between $ 0 $ and $ 1 $.
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\subsection{Variables used in models}
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@ -155,7 +155,7 @@ We use the following variables in our algorithms and models :
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\begin{itemize}
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\item $ U_n $ : the size of the $ n $-th item. $ (U_n)_{n \in \mathbb{N}} $
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\item $ U_n $ : the size of the $ n $-th item. $ (U_n)_{n \in \mathbb{N^*}} $
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denotes the mathematical sequence of random variables of uniform
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distribution on $ [0, 1] $ representing the items' sizes.
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@ -203,7 +203,7 @@ were needed to store $ n $ items.
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\cite{hofri:1987}
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% TODO mettre de l'Histoire
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\section{Next Fit Dual Bin Packing algorithm}
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\section{Next Fit Dual Bin Packing algorithm (NFDBP)}
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Next Fit Dual Bin Packing is a variation of NFBP in which we allow the bins to
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overflow. A bin must be fully filled, unless it is the last bin.
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@ -365,8 +365,8 @@ consuming. We can optimize the process by computing the statistics on the fly,
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by using sum formulae. This uses nearly constant memory, as we only need to
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store the current sum and the current sum of squares for different variables.
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While the mean can easily be calculated by summing then dividing, the variance
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can be calculated using the following formula:
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While the mean can easily be calculated by summing then dividing, the empirical
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variance can be calculated using the following formula:
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\begin{align*}
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{S_N}^2 & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X})^2 \\
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