From 1c6db889a62fa245a9c9e9dda10131b450c2846b Mon Sep 17 00:00:00 2001 From: Paul ALNET Date: Sun, 4 Jun 2023 13:39:59 +0200 Subject: [PATCH] tex: minor word corrections --- latex/content.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/latex/content.tex b/latex/content.tex index 41dd42b..7d6d5cd 100644 --- a/latex/content.tex +++ b/latex/content.tex @@ -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 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] \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 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 $. \subsection{Variables used in models} @@ -155,7 +155,7 @@ We use the following variables in our algorithms and models : \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 distribution on $ [0, 1] $ representing the items' sizes. @@ -203,7 +203,7 @@ were needed to store $ n $ items. \cite{hofri:1987} % 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 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 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: +While the mean can easily be calculated by summing then dividing, the empirical +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 \\