lacau/Projet_Boites
2
0
Fork 0
Code Issues Pull requests Releases Wiki Activity

Merge branch 'latex'

Browse source
This commit is contained in:
Paul ALNET 2023-06-04 13:40:26 +02:00
commit 7c1e115951
19 changed files with 935 additions and 0 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

21
.gitignore vendored Normal file
View file

@ -0,0 +1,21 @@
*.aux
*.log
*.swp
*.out
*.pdf
*.toc
*.maf
*.mtc
*.mtc0
*.stc
*.stc0
*.stc1
*.stc2
*.stc3
*.bbl
*.blg
*.stc*
*.bcf
*.run.xml

7
README.md
View file

@ -29,3 +29,10 @@ Oral: 15 min de présentation - 5 min de questions
trouver un argument solide sur pk on a choisi ce projet
## Compiling report
Install the necessary tools with `sudo apt-get install texlive-full`.
Navigate to `latex` directory and then run `pdflatex main.tex`. The output pdf
should be at `main.pdf`. You may want to run `biber main` first, to compile the
bibliography.

41
complexity-analysis/direct.py Normal file
View file

@ -0,0 +1,41 @@
# importing the memory tracking module
import tracemalloc
from random import random
from math import floor, sqrt
#from statistics import mean, variance
from time import perf_counter
# starting the monitoring
tracemalloc.start()
start_time = perf_counter()
# store memory consumption before
current_before, peak_before = tracemalloc.get_traced_memory()
N = 10**6
Tot = 0
Tot2 = 0
for _ in range(N):
item = random()
Tot += item
Tot2 += item ** 2
mean = Tot / N
variance = Tot2 / (N-1) - mean**2
# store memory after
current_after, peak_after = tracemalloc.get_traced_memory()
end_time = perf_counter()
print("mean :", mean)
print("variance :", variance)
# displaying the memory usage
print("Used memory before : {} B (current), {} B (peak)".format(current_before,peak_before))
print("Used memory after : {} B (current), {} B (peak)".format(current_after,peak_after))
print("Used memory : {} B".format(peak_after - current_before))
print("Time : {} ms".format((end_time - start_time) * 1000))
# stopping the library
tracemalloc.stop()

36
complexity-analysis/using_libs.py Normal file
View file

@ -0,0 +1,36 @@
# importing the memory tracking module
import tracemalloc
from random import random
from math import floor, sqrt
from statistics import mean, variance
from time import perf_counter
# starting the monitoring
tracemalloc.start()
start_time = perf_counter()
# store memory consumption before
current_before, peak_before = tracemalloc.get_traced_memory()
N = 10**6
values = [random() for _ in range(N)]
mean = mean(values)
variance = variance(values)
# store memory after
current_after, peak_after = tracemalloc.get_traced_memory()
end_time = perf_counter()
print("mean :", mean)
print("variance :", variance)
# displaying the memory usage
print("Used memory before : {} B (current), {} B (peak)".format(current_before,peak_before))
print("Used memory after : {} B (current), {} B (peak)".format(current_after,peak_after))
print("Used memory : {} B".format(peak_after - current_before))
print("Time : {} ms".format((end_time - start_time) * 1000))
# stopping the library
tracemalloc.stop()

42
latex/advanced.params/misc.commands.tex Normal file
View file

@ -0,0 +1,42 @@
\newcommand\tab[1][0.6cm]{\hspace*{#1}} %Create and define tab
\definecolor{lightgray}{gray}{0.85}
\definecolor{lightgrey}{gray}{0.85}
\definecolor{vlg}{gray}{0.85}
%Patch pour utiliser des équations dans les titres sans que hypperref nous insulte.
% Définition cyclique, compile pas. Mais c'est l"idée
%\renewcommand{\chapter}[1]{\chapter{\texorpdfstring{#1}}}
%\renewcommand{\section}[1]{\section{\texorpdfstring{#1}}}
%\renewcommand{\subsection}[1]{\subsection{\texorpdfstring{#1}}}
%\renewcommand{\subsubsection}[1]{\subsubsection{\texorpdfstring{#1}}}
%Chapter No Numbering but appears in TOC
\newcommand{\chapternn}[1]{\chapter*{#1}\addcontentsline{toc}{chapter}{#1}}
\newcommand{\sectionnn}[1]{\phantomsection\section*{#1}\addcontentsline{toc}{section}{#1}}
% phantomsection is necessary for links in TOC to function. It places the anchor
\newcommand{\subsectionnn}[1]{\subsection*{#1}\addcontentsline{toc}{subsection}{#1}}
\newcommand{\subsubsectionnn}[1]{\subsubsection*{#1}\addcontentsline{toc}{subsubsection}{#1}}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash\hspace{0pt}}p{#1}}
\newcolumntype{R}[1]{>{\raggedleft\arraybackslash\hspace{0pt}}p{#1}}
\newcolumntype{C}[1]{>{\centering\arraybackslash\hspace{0pt}}p{#1}}
\renewcommand\thesection{\arabic{section}}
\renewcommand\thesubsection{\thesection.\arabic{subsection}}
%------- Do not append new commands after :
\hypersetup{
colorlinks=false, % colorise les liens
linkbordercolor={1 1 1},
breaklinks=true, % permet le retour à la ligne dans les liens trop longs
urlcolor=blue, % couleur des hyperliens
linkcolor=black, % couleur des liens internes
citecolor=black, % couleur des références
pdftitle={}, % informations apparaissant dans
pdfauthor={}, % les informations du document
pdfsubject={} % sous Acrobat.
}

13
latex/advanced.params/tikz.conf.tex Normal file
View file

@ -0,0 +1,13 @@
% \tikzset{every picture/.style={execute at begin picture={
% \shorthandoff{:;!?};}
% }}
\tikzset{
boxnode/.style={ % requires library shapes.misc
draw,
rectangle,
text centered,
align=center,
fill=gray!5!white
},
}

57
latex/annex-performance.tex Normal file
View file

@ -0,0 +1,57 @@
For simplicity, we only include the script for the improved algorithm. For the
intuitive algorithm, simply replace the algorithm. The imports, timing and memory
usage tracking code are nearly identical.
\begin{lstlisting}[language=python]
#!/usr/bin/python3
import tracemalloc
from random import random
from math import floor, sqrt
#from statistics import mean, variance
from time import perf_counter
# starting the memory monitoring
tracemalloc.start()
start_time = perf_counter()
# store memory consumption before
current_before, peak_before = tracemalloc.get_traced_memory()
# algorithm (part to replace)
N = 10**6
Tot = 0
Tot2 = 0
for _ in range(N):
item = random()
Tot += item
Tot2 += item ** 2
mean = Tot / N
variance = Tot2 / (N-1) - mean**2
# store memory after
current_after, peak_after = tracemalloc.get_traced_memory()
end_time = perf_counter()
print("mean :", mean)
print("variance :", variance)
# displaying the memory usage
print("Used memory before : {} B (current), {} B (peak)".format(current_before,peak_before))
print("Used memory after : {} B (current), {} B (peak)".format(current_after,peak_after))
print("Used memory : {} B".format(peak_after - current_before))
print("Time : {} ms".format((end_time - start_time) * 1000))
tracemalloc.stop()
\end{lstlisting}
Example output:
\begin{lstlisting}[language=python]
mean : 0.5002592040785124
variance : 0.0833757719902084
Used memory before : 0 B (current), 0 B (peak)
Used memory after : 1308 B (current), 1336 B (peak)
Used memory : 1336 B
Time : 535.1873079998768 ms
\end{lstlisting}

5
latex/annex-probabilistic.tex Normal file
View file

@ -0,0 +1,5 @@
Script should have been provided with report. % TODO
\lstinputlisting[language=Python]{../Probas.py}
% TODO include output example

18
latex/bibliography.bib Normal file
View file

@ -0,0 +1,18 @@
@book{hofri:1987,
author = {Hofri, M.},
title = {Probabilistic Analysis of Algorithms: On Computing Methodologies for
Computer Algorithms Performance Evaluation},
year = {1987},
isbn = {0387965785},
publisher = {Springer-Verlag},
address = {Berlin, Heidelberg},
}
@misc{bin-packing-approximation:2022,
author = {{Computational Thinking}},
title = {Bin Packing Approximation},
year = {2022},
howpublished = {YouTube video},
url = {https://www.youtube.com/watch?v=R76aAh_li50},
}

3
latex/clean.sh Normal file
View file

@ -0,0 +1,3 @@
rm main.stc* main.toc main.bcf *.aux *.out main.mtc*
rm main.maf main.run.xml
rm main.log

436
latex/content.tex Normal file
View file

@ -0,0 +1,436 @@
\sectionnn{Introduction}
Bin packing is the process of packing a set of items of different sizes into
containers of a fixed capacity in a way that minimizes the number of containers
used. This has applications in many fields, such as logistics, where we want to
optimize the storage and transport of items in boxes, containers, trucks, etc.
Building mathematical models for bin packing is useful in understanding the
problem and in designing better algorithms, depending on the use case. An
algorithm optimized for packing cubes into boxes will not perform as well as
another algorithm for packing long items into trucks. Studying the mathematics
behind algorithms provides us with a better understanding of what works best.
When operating at scale, every small detail can have a huge impact on overall
efficiency and cost. Therefore, carefully developing algorithms based on solid
mathematical models is crucial. As we have seen in our Automatics class, a
small logic breach can be an issue in the long run in systems that are supposed
to run autonomously. This situation can be avoided by using mathematical models
during the design process wich will lead to better choices welding economic and
relibility concerns.
We will conduct a probabilistic analysis of multiple algorithms and compare
results to theoretical values. We will also consider the algoriths complexity
and performance, both in resource consumption and in box usage.
\clearpage
\section{Bin packing use cases}
Before studying the mathematics behind bin packing algorithms, we will have a
look at the motivations behind this project.
Bin packing has applications in many fields and allows to automize and optimize
complex systems. We will illustrate with examples focusing on two use cases:
logistics and computer science. We will consider examples of multiple dimensions
to show the versatility of bin packing algorithms.
\paragraph{} In the modern day, an effective supply chain relies on an automated production
thanks to sensors and actuators installed along conveyor belts. It is often
required to implement a packing procedure. All of this is controlled by a
computer system running continuously.
\subsection{3D : Containers}
Storing items in containers can be a prime application of bin packing. These
tree-dimensional objects of standardized size are used to transport goods.
While the dimensions of the containers are predictable, those of the transported
items are not. Storage is furthermore complicated by the fact that there can be
a void between items, allowing to move around. Multiple types of items can also
be stored in the same container.
There are many ways to optimize the storage of items in containers. For
example, by ensuring items are of an optimal standardized size or by storing a
specific item in each container, both eliminating the randomness in item size.
In these settings, it is easy to fill a container by assimilating them to
rectangular blocks. However, when items come in pseudo-random dimensions, it is
intuitive to start filling the container with larger items and then filling the
remaining gaps with smaller items. As containers must be closed, in the event
of an overflow, the remaining items must be stored in another container.
\subsection{2D : Cutting stock problem}
In industries such as woodworking bin packing algorithms are utilized to
minimize material waste when cutting large planks into smaller pieces of
desired sizes. Many tools use this two-dimensional cut process. For example, at
the Fabric'INSA Fablab, the milling machine, laser cutter and many more are
used to cut large planks of wood into smaller pieces for student projects. In
this scenario, we try to organize the desired cuts in a way that minimizes the
unusable excess wood.
\begin{figure}[ht]
\centering
\includegraphics[width=0.65\linewidth]{graphics/fraiseuse.jpg}
\caption[]{Milling machine at the Fabric'INSA Fablab \footnotemark}
\label{fig:fraiseuse}
\end{figure}
\footnotetext{Photo courtesy of Inés Bafaluy}
Managing the placement of items of complex shapes can be optimized by using
by various algorithms minimizing the waste of material.
\subsection{1D : Networking}
When managing network traffic at scale, efficiently routing packets is
necessary to avoid congestion, which leads to lower bandwidth and higher
latency. Say you're a internet service provider and your users are watching
videos on popular streaming platforms. You want to ensure that the traffic is
balanced between the different routes to minimize throttling and energy
consumption.
\paragraph{} We can consider the different routes as bins and the users'
bandwidth as the items. If a bin overflows, we can redirect the traffic to
another route. Using less bins means less energy consumption and decreased
operating costs. This is a good example of bin packing in a dynamic
environment, where the items are constantly changing. Humans are not involved
in the process, as it is fast-paced and requires a high level of automation.
\vspace{0.4cm}
\paragraph{} We have seen multiple examples of how bin packing algorithms can
be used in various technical fields. In these examples, a choice was made,
evaluating the process effectiveness and reliability, based on a probabilistic
analysis allowing the adaptation of the algorithm to the use case. We will now
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 (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 current bin if it fits, otherwise we open a new bin.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.8]
% Bins
\draw[thick] (0,0) rectangle (2,6);
\draw[thick] (3,0) rectangle (5,6);
\draw[thick] (6,0) rectangle (8,6);
% Items
\draw[fill=red] (0.5,0.5) rectangle (1.5,3.25);
\draw[fill=blue] (0.5,3.5) rectangle (1.5,5.5);
\draw[fill=green] (3.5,0.5) rectangle (4.5,1.5);
\draw[fill=orange] (3.5,1.75) rectangle (4.5,3.75);
\draw[fill=purple] (6.5,0.5) rectangle (7.5,2.75);
\draw[fill=yellow] (6.5,3) rectangle (7.5,4);
% arrow
\draw[->, thick] (8.6,3.5) -- (7.0,3.5);
\draw[->, thick] (8.6,1.725) -- (7.0,1.725);
% Labels
\node at (1,-0.75) {Bin 0};
\node at (4,-0.75) {Bin 1};
\node at (7,-0.75) {Bin 2};
\node at (10.0,3.5) {Yellow item};
\node at (10.0,1.725) {Purple item};
\end{tikzpicture}
\label{fig:nfbp}
\caption{Next Fit Bin Packing example}
\end{figure}
\paragraph{} The example in figure \ref{fig:nfbp} shows the limitations of the
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
will be of random sizes between $ 0 $ and $ 1 $.
\subsection{Variables used in models}
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^*}} $
denotes the mathematical sequence of random variables of uniform
distribution on $ [0, 1] $ representing the items' sizes.
\item $ T_i $ : the number of items in the $ i $-th bin.
\item $ V_i $ : the size of the first item in the $ i $-th bin.
\item $ H_n $ : the number of bins required to store $ n $ items.
\end{itemize}
Mathematically, the NFBP algorithm imposes the following constraint on the first box :
\begin{align*}
T_1 = k \iff & U_1 + U_2 + \ldots + U_{k} < 1 \\
\text{ and } & U_1 + U_2 + \ldots + U_{k+1} \geq 1 \qquad \text{ with } k \geq 1 \\
\end{align*}
\subsection{Implementation and results}
We implemented the NFBP algorithm in Python \footnotemark, for its ease of use
and broad recommendation. We used the \texttt{random} library to generate
random numbers between $ 0 $ and $ 1 $ and \texttt{matplotlib} to plot the
results in the form of histograms. We ran $ R = 10^6 $ simulations with
$ N = 10 $ different items each.
\footnotetext{The code is available in Annex \ref{annex:probabilistic}}
\paragraph{Distribution of $ T_i $} We first studied how many items were
present per bin.
\paragraph{Distribution of $ V_i $} We then looked at the size of the first
item in each bin.
\paragraph{Asymptotic behavior of $ H_n $} Finally, we analyzed how many bins
were needed to store $ n $ items.
% TODO histograms
% TODO analysis histograms
\cite{hofri:1987}
% TODO mettre de l'Histoire
\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.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.8]
% Bins
\draw[thick] (0,0) rectangle (2,6);
\draw[thick] (3,0) rectangle (5,6);
% Transparent Tops
\fill[white,opacity=1.0] (0,5.9) rectangle (2,6.5);
\fill[white,opacity=1.0] (3,5.9) rectangle (5,6.5);
% Items
\draw[fill=red] (0.5,0.5) rectangle (1.5,3.25);
\draw[fill=blue] (0.5,3.5) rectangle (1.5,5.5);
\draw[fill=green] (0.5,5.75) rectangle (1.5,6.75);
\draw[fill=orange] (3.5,0.5) rectangle (4.5,2.5);
\draw[fill=purple] (3.5,2.75) rectangle (4.5,5.0);
\draw[fill=yellow] (3.5,5.25) rectangle (4.5,6.25);
% Labels
\node at (1,-0.75) {Bin 0};
\node at (4,-0.75) {Bin 1};
\end{tikzpicture}
\caption{Next Fit Dual Bin Packing example}
\label{fig:nfdbp}
\end{figure}
\paragraph{} The example in figure \ref{fig:nfdbp} shows how NFDBP utilizes
less bins than NFBP, due to less stringent constraints. The top of the bin is
effectively removed, allowing for an extra item to be stored in the bin. We can
easily see how with NFDBP each bin can at least contain two items.
\paragraph{} The variables used are the same as for NFBP. Mathematically, the
new constraints on the first bin can be expressed as follows :
\begin{align*}
T_1 = k \iff & U_1 + U_2 + \ldots + U_{k-1} < 1 \\
\text{ and } & U_1 + U_2 + \ldots + U_{k} \geq 1 \qquad \text{ with } k \geq 2 \\
\end{align*}
\subsection{La giga demo}
Let $ k \in \mathbb{N} $. Let $ (U_n)_{n \in \mathbb{N}} $ be a sequence of
independent random variables with uniform distribution on $ [0, 1] $, representing
the size of the $ n $-th item.
Let $ i \in \mathbb{N} $. $ T_i $ denotes the number of items in the $ i $-th
bin. We have that
\begin{equation}
T_i = k \iff U_1 + U_2 + \ldots + U_{k-1} < 1 \text{ and } U_1 + U_2 + \ldots + U_{k} \geq 1
\end{equation}
Let $ A_k = \{ U_1 + U_2 + \ldots + U_{k-1} < 1 \}$. Hence,
\begin{align*}
% TODO = k
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*}
We will try to show that $ \forall k \geq 2 $, $ P(A_k) = \frac{1}{k!} $. To do
so, we will use induction to prove the following proposition \eqref{eq:induction},
$ \forall k \geq 2 $:
\begin{equation}
\label{eq:induction}
\tag{$ \mathcal{H}_k $}
P(U_1 + U_2 + \ldots + U_{k-1} < a) = \frac{a^k}{k!} \qquad \forall a \in [0, 1],
\end{equation}
Let us denote $ S_k = U_1 + U_2 + \ldots + U_{k-1} \qquad \forall k \geq 2 $.
\paragraph{Base cases} $ k = 2 $ : $ P(U_1 < a) = a \neq \frac{a^2}{2}$ supposedly proving $ (\mathcal{H}_2) $.
$ k = 2 $ : \[ P(U_1 + U_2 < a) = \iint_{\cal{D}} f_{U_1, U_2}(x, y) \cdot (x + y) dxdy \]
Where $ \mathcal{D} = \{ (x, y) \in [0, 1]^2 \mid x + y < a \} $.
$ U_1 $ and $ U_2 $ are independent, so
\begin{align*}
f_{U_1, U_2}(x, y) & = f_{U_1}(x) \cdot f_{U_2}(y) \\
& = \begin{cases}
1 & \text{if } x \in [0, 1] \text{ and } y \in [0, 1] \\
0 & \text{otherwise} \\
\end{cases} \\
\end{align*}
Hence,
\begin{align*}
P(U_1 + U_2 < a)
& = \iint_{\cal{D}} (x + y)dxdy \\
& = \int_{0}^{a} \int_{0}^{a - x} (x + y) dy dx \\
& = \int_{0}^{a} \left[ xy + \frac{y^2}{2} \right]_{y=0}^{y=a - x} dx \\
& = \int_{0}^{a} \left( ax - x^2 + \frac{a^2}{2} - ax + \frac{x^2}{2} \right) dx \\
& = \int_{0}^{a} \left( \frac{a^2}{2} - \frac{x^2}{2} \right) dx \\
& = \left[ \frac{a^2 x}{2} - \frac{x^3}{6} \right]_{0}^{a} \\
& = \frac{a^3}{2} - \frac{a^3}{6} \\
\end{align*}
\paragraph{Induction step} For a fixed $ k > 2 $, we assume that $
(\mathcal{H}_{k-1}) $ is true. We will try to prove $ (\mathcal{H}_{k}) $.
\[
P(S_{k-1} + U_{k-1} < a)
= \iint_{\cal{D}} f_{S_{k-1}, U_{k-1}}(x, y) \cdot (x + y) dxdy \\
\]
where $ \mathcal{D} = \{ (x, y) \in [0, 1]^2 \mid x + y < a \} $.
As $ S_{k-1} $ and $ U_{k-1} $ are independent,
\[
P(S_{k-1} + U_{k-1} < a)
= \iint_{\cal{D}} f_{S_{k-1}}(x) \cdot f_{U_{k-1}}(y) \cdot (x + y) dxdy \qquad \\
\]
$ (\mathcal{H}_{k-1}) $ gives us that $ \forall x \in [0, 1] $,
$ F_{S_{k-1}}(x) = P(S_{k-1} < x) = \frac{x^{k-1}}{(k-1)!} $.
By differentiating, we get that $ \forall x \in [0, 1] $,
\[
f_{S_{k-1}}(x) = F'_{S_{k-1}}(x) = \frac{x^{k-2}}{(k-2)!}
\]
Furthermore, $ U_{k-1} $ is uniformly distributed on $ [0, 1] $, so
$ f_{U_{k-1}}(y) = 1 $.
\begin{align*}
\text{Hence, }
P(S_{k-1} + U_{k-1} < a)
& =
& = \frac{a^{k}}{k!}
\end{align*}
\section{Complexity and implementation optimization}
Both the NFBP and NFDBP algorithms have a linear complexity $ O(n) $, as we
only need to iterate over the items once. While the algorithms themselves are
linear, calculating the statistics may not not be. In this section, we will
discuss how to optimize the implementation of the statistical analysis.
\subsection{Performance optimization}
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.
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 \\
& = \frac{1}{N-1} \sum_{i=1}^{N} X_i^2 - \frac{N}{N-1} \overline{X}^2
\end{align*}
The sum $ \frac{1}{N-1} \sum_{i=1}^{N} X_i^2 $ can be calculated iteratively
after each simulation.
\subsection{Effective resource consumption}
We set out to study the resource consumption of the algorithms. We implemented
the above formulae to calculate the mean and variance of $ N = 10^6 $ random
numbers. We wrote the following algorithms \footnotemark :
\footnotetext{The full code used to measure performance can be found in Annex \ref{annex:performance}.}
\paragraph{Intuitive algorithm} Store values first, calculate later
\begin{lstlisting}[language=python]
N = 10**6
values = [random() for _ in range(N)]
mean = mean(values)
variance = variance(values)
\end{lstlisting}
Execution time : $ 4.8 $ seconds
Memory usage : $ 32 $ MB
\paragraph{Improved algorithm} Continuous calculation
\begin{lstlisting}[language=python]
N = 10**6
Tot = 0
Tot2 = 0
for _ in range(N):
item = random()
Tot += item
Tot2 += item ** 2
mean = Tot / N
variance = Tot2 / (N-1) - mean**2
\end{lstlisting}
Execution time : $ 530 $ milliseconds
Memory usage : $ 1.3 $ kB
\paragraph{Analysis} Memory usage is, as expected, much lower when calculating
the statistics on the fly. Furthermore, something we hadn't anticipated is the
execution time. The improved algorithm is nearly 10 times faster than the
intuitive one. This can be explained by the time taken to allocate memory and
then calculate the statistics (which iterates multiple times over the array).
\footnotemark
\footnotetext{Performance was measured on a single computer and will vary
between devices. Execution time and memory usage do not include the import of
libraries.}
\subsection{NFBP vs NFDBP}
\subsection{Optimal algorithm}
\cite{bin-packing-approximation:2022}
\sectionnn{Conclusion}

BIN
latex/cover/charte_graphique.pdf Normal file
View file

Binary file not shown.

30
latex/cover/cover_in.tex Normal file
View file

@ -0,0 +1,30 @@
\input{main_variables}
\pagenumbering{gobble}
% Couverture
\thispagestyle{empty}
\definecolor{insa_blue}{RGB}{52,83,111}
\definecolor{insa_red}{RGB}{230,39,20}
\noindent\begin{tikzpicture}[remember picture, overlay, shift={(current page.south west)}]
\draw[draw=none, fill=insa_red] (12.32,1.24) -- (12.32,4.94) -- (19.7,4.94) -- (19.7,1.24) -- cycle;
\node[anchor=north west, align=left, color=white, text width=12cm] at (2.3,23.5) {\varmaintitle};
\node[anchor=north west, align=left, color=white, text width=8cm] at (2.3,22.5) {\varmainsubtitle};
% \node[anchor=north west] at (2.3,21.5) {\includegraphics[scale=1.00]{\varlogo}};
\node[anchor=north west, align=left, color=white, text width=12cm] at (2.3,16) {\varcovertext};
%\node[anchor=center, align=center, text width=12cm] at (10,6.7) {\color{black} \varcovertext}; %-3, 4.1
\node[anchor=north west, align=left, text width=12cm] at (1.2,4.7) {\varinsaaddress};
% \node[anchor=north west, align=left, color=white, text width=12cm] at (12.5,4.7) {\varcompanyaddress};
\node[anchor=north west, align=left, text width=12cm] at (0.5,4) {\includepdf[pages={1}]{cover/charte_graphique}};
\end{tikzpicture}
\newpage
% % Page de garde
% \thispagestyle{empty}
% \noindent\begin{tikzpicture}[remember picture, overlay, shift={(current page.south west)}]
% \node[anchor=north west, align=left, text width=10cm] at (2.3,23.5) {\color{black} \varmaintitle};
% \node[anchor=north west, align=left, text width=8cm] at (2.3,22.5) {\varmainsubtitle};
% \node[anchor=north west, align=left, text width=12cm] at (2.3,16) {\color{black} \varcovertext};
% \node[anchor=north west, align=left, text width=12cm] at (1.2,4.7) {\varinsaaddress};
% \node[anchor=north west, align=left, text width=12cm] at (12.5,4.7) {\color{black} \varcompanyaddress};
% \end{tikzpicture}
% \newpage

21
latex/cover/cover_out.tex Normal file
View file

@ -0,0 +1,21 @@
\newpage
\pagenumbering{gobble}
\thispagestyle{empty}
\definecolor{insa_blue}{RGB}{52,83,111}
\definecolor{insa_red}{RGB}{226,50,46}
\noindent\begin{tikzpicture}[remember picture, overlay, shift={(current page.south west)}]
% \draw[draw=none, path fading=east, left color=insa_blue, right color=insa_blue!25!white] (21,7.3) -- (4.3,16.05) -- (21,24.8) -- cycle;
% \draw[draw=none, path fading=east, left color=insa_blue, right color=insa_blue!60!white] (0,16.2) -- (7,19.75) -- (0,23.1) -- cycle;
% \draw[draw=none, path fading=east, left color=insa_blue, right color=insa_blue!25!white] (0,17.3) -- (13.7,24.5) -- (2.3,29.7) -- (0,29.7) -- cycle;
%
%
% \draw[draw=none, fill=insa_red] (6.2,0) -- (6.2,1.2) -- (11.6,0) -- cycle;
%
% \draw[draw=none, fill=white] (0,0) -- (0,10) -- (10,10) -- (10,0) -- cycle;
% \node[anchor=south west, align=left] at (1.25,3.6) {\textbf{INSA Toulouse}};
% \node[anchor=south west, align=left] at (1.25,2.20) {135, Avenue de Rangueil \\ 31077 Toulouse Cedex 4 - France \\ \href{http://www.insa-toulouse.fr}{www.insa-toulouse.fr}};
%
% \node[anchor=south west] at (11.1,2.2) {\includegraphics[height=1cm, keepaspectratio]{cover/meta/univ.png}};
%\node[anchor=south west] at (13.4,2.2) {\includegraphics[height=1cm, keepaspectratio]{cover/meta/ministere.png}};
% \node[anchor=north west, align=left, text width=12cm] at (0.5,4) {\includepdf[pages={2}]{cover/charte_graphique}};
\end{tikzpicture}

3
latex/cover/covermain.tex Normal file
View file

@ -0,0 +1,3 @@
\AtBeginDocument{\input{cover/cover_in.tex}\pagenumbering{arabic}}
%\AtEndDocument{\input{cover/cover_out.tex}}

BIN
latex/graphics/fraiseuse.jpg Normal file
View file

Binary file not shown.
Side by side

After

Width:  |  Height:  |  Size: 151 KiB

139
latex/main.tex Normal file
View file

@ -0,0 +1,139 @@
\documentclass[a4paper,11pt,twoside]{article}
%\pdfminorversion=7 % To use charte graphique (pdf 1.7)
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lscape}
\usepackage{boldline,multirow,tabularx,colortbl,diagbox,makecell,fancybox}
\usepackage{amsfonts,amssymb,amsmath,mathrsfs,array}
\usepackage{pgf,tikz,xcolor}
\usetikzlibrary{calc,positioning,shapes.geometric,shapes.symbols,shapes.misc, fit, shapes, arrows, arrows.meta,fadings,through}
\usepackage[top=2cm, bottom=2cm, left=4cm, right=4cm]{geometry}
\usepackage{hyperref}
\usepackage{titlesec}
\usepackage{eurosym}
\usepackage[english]{babel}
\usepackage{eso-pic} % for background on cover
\usepackage{listings}
\usepackage{tikz}
% Define colors for code
\definecolor{codegreen}{rgb}{0,0.4,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
\lstdefinestyle{mystyle}{
backgroundcolor=\color{backcolour},
commentstyle=\color{codegreen},
keywordstyle=\color{magenta},
numberstyle=\tiny\color{codegray},
stringstyle=\color{codepurple},
basicstyle=\ttfamily\small,
breakatwhitespace=false,
breaklines=true,
captionpos=b,
keepspaces=true,
numbers=left,
numbersep=5pt,
showspaces=false,
showstringspaces=false,
showtabs=false,
tabsize=2
}
\lstset{style=mystyle}
% table des annexes
\usepackage{minitoc}
\usepackage{pdfpages}
\usepackage[style=iso-alphabetic]{biblatex}
\addbibresource{bibliography.bib}
\input{advanced.params/tikz.conf}
\input{advanced.params/misc.commands}
\input{cover/covermain.tex}
\date{\today}
\begin{document}
\dosecttoc{} % generate TOC
% % Remerciements
% \thispagestyle{empty} % removes page number
% \subsection*{Remerciements}
% \input{remerciements}
% \clearpage
% TABLE DES MATIÈRES
\thispagestyle{empty} % removes page number
\setcounter{secnumdepth}{3}
\tableofcontents
\clearpage
% Contenu
\setcounter{page}{1}
\include{content}
% BIBLIOGRAPHIE
\addcontentsline{toc}{section}{Bibliography}
\printbibliography[title={Bibliography}]
% TABLE DES ANNEXES
\clearpage
\appendix
% \thispagestyle{empty} % removes page number
\sectionnn{Table des Annexes}
% Désactivation de la table des matières
\addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
% Personnalisation de la table des annexes
\renewcommand{\stctitle}{} % Titre (issue with previous subsection showing up)
\renewcommand\thesubsection{A\arabic{subsection}} % Numérotation
\renewcommand{\stcSSfont}{} % Police normale, pas en gras
\mtcsetrules{secttoc}{off} % Désactivation des lignes en haut et en bas de la table
% Affichage de la table des annexes
\secttoc
% ANNEXES
\clearpage
\pagenumbering{Roman}
\subsection{Performance analysis script}
\label{annex:performance}
\input{annex-performance}
\clearpage
\subsection{Probabilistic analysis script}
\label{annex:probabilistic}
\input{annex-probabilistic}
\clearpage
% \includepdf[pages={1}, scale=0.96,
% pagecommand=\subsection{Questionnaire 1 : Sensibilisation à l’Hygiène et à la Sécurité}]
% {questionnaires}
% \clearpage
%
%
% \includepdf[pages={2}, scale=0.96,
% pagecommand=\subsection{Questionnaire 2 : Sensibilisation à l’innovation en entreprise}]
% {questionnaires}
% \clearpage
%
\includepdf[pages={2}, scale=1]
{cover/charte_graphique}
\mtcsetrules{secttoc}{off}
\end{document}

34
latex/main_variables.tex Normal file
View file

@ -0,0 +1,34 @@
\def\varmaintitle{
\textbf{\Huge{One-dimensional box fitting}}
}
\def\varmainsubtitle{
\LARGE{A statistical analysis of different algorithms}
}
% \def\varlogo{logo.png} %ou 5 ou 13
\def\varcovertext{
\textbf{Clément LACAU}
\textbf{Paul ALNET}
2MIC B - Year 59
\vspace{0.33cm}
%\begin{center}
% -\hspace{0.25cm}Version du \today\hspace{0.25cm}-
%\end{center}
Defense on June Xth 2023 % TODO
}
\def\varinsaaddress{
\textbf{INSA Toulouse}
135, Avenue de Rangueil
31077 Toulouse Cedex 4 - France
\href{https://www.insa-toulouse.fr}{www.insa-toulouse.fr}
}

29
latex/remerciements.tex Normal file
View file

@ -0,0 +1,29 @@
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
tempor incididunt ut labore et dolore magna aliqua. Tortor pretium
viverra suspendisse potenti nullam ac tortor vitae. Porttitor massa id
neque aliquam vestibulum morbi blandit. Quis imperdiet massa tincidunt
nunc pulvinar sapien et ligula. Tincidunt lobortis feugiat vivamus
at. Amet justo donec enim diam. Ut tortor pretium viverra suspendisse
potenti nullam ac tortor vitae. Consectetur a erat nam at lectus urna
duis convallis convallis. Viverra nam libero justo laoreet sit amet
cursus. Non enim praesent elementum facilisis leo. Sit amet mauris
commodo quis. Lectus mauris ultrices eros in cursus turpis. Cursus euismod
quis viverra nibh cras pulvinar mattis. Duis at consectetur lorem donec
massa sapien faucibus. In hac habitasse platea dictumst quisque sagittis
purus. Ut aliquam purus sit amet. Eget egestas purus viverra accumsan
in nisl. Egestas dui id ornare arcu odio ut sem. Nunc mi ipsum faucibus
vitae. Vel pretium lectus quam id leo in vitaLorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod
tempor incididunt ut labore et dolore magna aliqua. Tortor pretium
viverra suspendisse potenti nullam ac tortor vitae. Porttitor massa id
neque aliquam vestibulum morbi blandit. Quis imperdiet massa tincidunt
nunc pulvinar sapien et ligula. Tincidunt lobortis feugiat vivamus
at. Amet justo donec enim diam. Ut tortor pretium viverra suspendisse
potenti nullam ac tortor vitae. Consectetur a erat nam at lectus urna
duis convallis convallis. Viverra nam libero justo laoreet sit amet
cursus. Non enim praesent elementum facilisis leo. Sit amet mauris
commodo quis. Lectus mauris ultrices eros in cursus turpis. Cursus euismod
quis viverra nibh cras pulvinar mattis. Duis at consectetur lorem donec
massa sapien faucibus. In hac habitasse platea dictumst quisque sagittis
purus. Ut aliquam purus sit amet. Eget egestas purus viverra accumsan
in nisl. Egestas dui id ornare arcu odio ut sem. Nunc mi ipsum faucibus
vitae. Vel pretium lectus quam id leo in vitae