From 5246b1c56b4f95d23980fdbef57229f49c13d639 Mon Sep 17 00:00:00 2001 From: Paul ALNET Date: Mon, 5 Jun 2023 00:08:26 +0200 Subject: [PATCH] tex: final commit I suppose (I wish) --- latex/content.tex | 71 ++++++++++++++++++-- latex/graphics/graphic-NFBP-Hn-105-sim.png | Bin 0 -> 29794 bytes latex/graphics/graphic-NFDBP-T1-103-sim.png | Bin 0 -> 38591 bytes 3 files changed, 64 insertions(+), 7 deletions(-) create mode 100644 latex/graphics/graphic-NFBP-Hn-105-sim.png create mode 100644 latex/graphics/graphic-NFDBP-T1-103-sim.png diff --git a/latex/content.tex b/latex/content.tex index 98d9a91..3ce936a 100644 --- a/latex/content.tex +++ b/latex/content.tex @@ -258,12 +258,10 @@ of $ T_i $. Our calculations have yielded that $ \overline{T_1} = 1.72 $ and $ {S_N}^2 = 0.88 $. Our Student coefficient is $ t_{0.95, 2} = 2 $. We can now calculate the Confidence Interval for $ T_1 $ for $ R = 10^5 $ simulations : - \begin{align*} IC_{95\%}(T_1) & = \left[ 1.72 \pm 1.96 \frac{\sqrt{0.88}}{\sqrt{10^5}} \cdot 2 \right] \\ & = \left[ 172 \pm 0.012 \right] \\ \end{align*} - We can see that the Confidence Interval is very small, thanks to the large number of iterations. This results in a steady curve in figure \ref{fig:graphic-NFBP-Ti-105-sim}. @@ -274,12 +272,24 @@ This results in a steady curve in figure \ref{fig:graphic-NFBP-Ti-105-sim}. \label{fig:graphic-NFBP-Vi-105-sim} \end{figure} +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Hn-105-sim} + \caption{Histogram of $ H_n $ for $ R = 10^5 $ simulations and $ N = 50 $ items (number of bins required to store $n$ items)} + \label{fig:graphic-NFBP-Hn-105-sim} +\end{figure} + \paragraph{Asymptotic behavior of $ H_n $} Finally, we analyzed how many bins were needed to store $ n $ items. We used the numbers from the $ R = 10^5 $ simulations. +We can see in figure \ref{fig:graphic-NFBP-Hn-105-sim} that $ H_n $ is +asymptotically linear. The expected value and the variance are also displayed. +The variance also increases linearly. +\paragraph{} The Next Fit Bin Packing algorithm is a very simple algorithm +with predictable results. It is very fast, but it is not optimal. \section{Next Fit Dual Bin Packing algorithm (NFDBP)} @@ -328,7 +338,10 @@ new constraints on the first bin can be expressed as follows : \text{ and } & U_1 + U_2 + \ldots + U_{k} \geq 1 \qquad \text{ with } k \geq 2 \\ \end{align*} -\subsection{La giga demo} + +\subsection{Building a mathematical model} + +In this section we will try to determine the probabilistic law followed by $ T_i $. Let $ k \geq 2 $. Let $ (U_n)_{n \in \mathbb{N}^*} $ be a sequence of independent random variables with uniform distribution on $ [0, 1] $, representing @@ -343,12 +356,12 @@ bin. We have that Let $ A_k = \{ U_1 + U_2 + \ldots + U_{k} < 1 \}$. Hence, -\begin{align} +\begin{align*} \label{eq:prob} P(T_i = k) & = P(A_{k-1} \cap A_k^c) \\ & = P(A_{k-1}) - P(A_k) \qquad \text{ (as $ A_k \subset A_{k-1} $)} \\ -\end{align} +\end{align*} We will try to show that $ \forall k \geq 1 $, $ P(A_k) = \frac{1}{k!} $. To do so, we will use induction to prove the following proposition \eqref{eq:induction}, @@ -414,6 +427,18 @@ Finally, plugging this into \eqref{eq:prob} gives us P(T_i = k) = P(A_{k-1}) - P(A_{k}) = \frac{1}{(k-1)!} - \frac{1}{k!} \qquad \forall k \geq 2 \] +\subsection{Empirical results} + +We ran $ R = 10^3 $ simulations for $ N = 10 $ items. The empirical results are +similar to the mathematical model. + +\begin{figure}[h] + \centering + \includegraphics[width=1.0\textwidth]{graphics/graphic-NFDBP-T1-103-sim} + \caption{Therotical and empiric histograms of $ T_1 $ for $ R = 10^3 $ simulations and $ N = 10 $ items (number of itens in the first bin)} + \label{fig:graphic-NFDBP-T1-103-sim} +\end{figure} + \subsection{Expected value of $ T_i $} We now compute the expected value $ \mu $ and variance $ \sigma^2 $ of $ T_i $. @@ -441,6 +466,8 @@ We now compute the expected value $ \mu $ and variance $ \sigma^2 $ of $ T_i $. \sigma^2 = E({T_i}^2) - E(T_i)^2 = 3e - 1 - e^2 \end{align*} +$ H_n $ is asymptotically normal, following a $ \mathcal{N}(\frac{N}{\mu}, \frac{N \sigma^2}{\mu^3}) $ + \section{Complexity and implementation optimization} @@ -519,13 +546,43 @@ then calculate the statistics (which iterates multiple times over the array). between devices. Execution time and memory usage do not include the import of libraries.} -\subsection{NFBP vs NFDBP} - \subsection{Optimal algorithm} +As we have seen, NFDBP algorithm is much better than NFBP algorithm. All the +variables excluding V are showing this. More specifically, the most relevant +variable is Hn which is growing slightly slower in the NFDBP algorithm than in +the NFBP algorithm. + + +Another algorithm that we did not explore in this project is the SUBP (Skim Up +Bin Packing) algorithm. It works in the same way as the NFDBP algorithm. +However, when an item exceeds the box size, it is removed from the current bin +and placed into the next bin. This algorithm that we could not exploit is much +more efficient than both of the previous algorithms. His main issue is that it +takes a lot of storage and requires higher capacities. + +We redirect you towards this video which demonstrates why another algorithm is +actually the most efficient that we can imagine. In this video we see that the +mostoptimized of alrogithm is another version of NFBP where we sort the items +in a decreasing order before sending them into the different bins. + +\clearpage \sectionnn{Conclusion} +In this project, we explored many bin packing algorithms in 1 dimension. We +discovered how some bin packing algorithms can be really simple to implement +but also a strong data consumer as the NFBP algorithm. + +By modifying the conditions of bin packing we can upgrade our performances. For +example, the NFDBP doest not permit to close the boxes (which depend of the +context of this implementation). The performance analysis conclusions are the +consequences of a precise statistical and probabilistic study that we have leaded +on this project. + +To go further, we could now think about the best applications of different +algorithms in real contexts, thanks to simulations. + \nocite{bin-packing-approximation:2022} \nocite{hofri:1987} diff --git a/latex/graphics/graphic-NFBP-Hn-105-sim.png b/latex/graphics/graphic-NFBP-Hn-105-sim.png new file mode 100644 index 0000000000000000000000000000000000000000..6232ddd0602cef1d3cee606ea8ef4a6594d772b1 GIT binary patch literal 29794 zcmdRWWmMJcy6;pJL> z&vfk*W88D@`Ec)-%NU!r7mN9iH=gHLFF^`&61Z2%uObi#Tq#ME5(0sNi9lRPx^fYI zve|N-5B}w~xtN%Ol$h8pI|o}6bE{_v1apK}gs^1G1Ijji(TWV>%YuRTjdSHl-Ur{e zAP!cb9lBa15U+DY%WW>Hq(gc6`%6EX5LGNvQ(Y>Z%MUj&T8G=D2E?UL^p(sAW|s?W9pW6 z{CtPSE$V{E3s?U|fAI!6p-S_`h;{Gfr&V1z@L-8QxW5R11udZC+${&3| zMq*fD@jhO1(ED?X=GL)Yv$g04e|MS+&zdqebKqDV77uwLr-rT$IBA2f6i%t+0K ztd5UIt0ay}ST{YtF^D<5N67lTLo}of 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