.gitignore

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

Probas.py

@@ -1,15 +1,11 @@
-#!/usr/bin/python3
 from random import random
-from math import floor, sqrt, factorial,exp
+from math import floor, sqrt
 from statistics import mean, variance
-from matplotlib import pyplot as plt
+# from matplotlib import pyplot
-from pylab import *
-import numpy as np
-
 
 def simulate_NFBP(N):
     """
-    Tries to simulate T_i, V_i and H_n for N items of random size.
+    Tries to simulate T_i, V_i and H_n for N boxes of random size.
     """
     i = 0  # Nombre de boites
     R = [0]  # Remplissage de la i-eme boite
@@ -20,7 +16,7 @@ def simulate_NFBP(N):
         size = random()
         if R[i] + size >= 1:
             # Il y n'y a plus de la place dans la boite pour le paquet.
-            # On passe a la boite suivante (qu'on initialise)
+            # On passe à la boite suivante (qu'on initialise)
             i += 1
             R.append(0)
             T.append(0)
@@ -32,15 +28,20 @@ def simulate_NFBP(N):
             V[i] = size
         H.append(i)
 
-    return {"i": i, "R": R, "T": T, "V": V, "H": H}
+    return {
+        "i": i,
+        "R": R,
+        "T": T,
+        "V": V,
+        "H": H
+    }
 
 
-# unused
 def stats_NFBP(R, N):
     """
-    Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
+    Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
     """
-    print("Running {} NFBP simulations with {} items".format(R, N))
+    print("Running {} NFBP simulations with {} packages".format(R, N))
     I = []
     H = [[] for _ in range(N)]  # List of empty lists
 
@@ -50,161 +51,42 @@ def stats_NFBP(R, N):
         for n in range(N):
             H[n].append(sim["H"][n])
 
-    print("Mean number of bins : {} (variance {})".format(mean(I), variance(I)))
+    print("Mean number of boxes : {} (variance {})".format(mean(I), variance(I)))
 
     for n in range(N):
         print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
 
-
 def stats_NFBP_iter(R, N):
     """
-    Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
+    Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
     Calculates stats during runtime instead of after to avoid excessive memory usage.
     """
-    Hmean=0
+    print("Running {} NFBP simulations with {} packages".format(R, N))
-    Var=[]
-    H=[]
-    Exp=0
-    P = R * N  # Total number of items
-    print("## Running {} NFBP simulations with {} items".format(R, N))
-    # number of bins
     ISum = 0
     IVarianceSum = 0
-    # index of the bin containing the n-th item
     HSum = [0 for _ in range(N)]
     HSumVariance = [0 for _ in range(N)]
-    # number of items in the i-th bin
-    Sum_T = [0 for _ in range(N)]
-    TSumVariance = [0 for _ in range(N)]
-    # size of the first item in the i-th bin
-    Sum_V = [0 for _ in range(N)]
 
     for i in range(R):
         sim = simulate_NFBP(N)
         ISum += sim["i"]
-        IVarianceSum += sim["i"] ** 2
+        IVarianceSum += sim["i"]**2
         for n in range(N):
             HSum[n] += sim["H"][n]
-            HSumVariance[n] += sim["H"][n] ** 2
+            HSumVariance[n] += sim["H"][n]**2
-        T = sim["T"]
-        V = sim["V"]
-        # ensure that T, V have the same length as Sum_T, Sum_V
-        for i in range(N - sim["i"]):
-            T.append(0)
-            V.append(0)
-        Sum_T = [x + y for x, y in zip(Sum_T, T)]
-        TSumVariance = [x + y**2 for x, y in zip(TSumVariance, T)]
-        Sum_V = [x + y for x, y in zip(Sum_V, V)]
 
-    Sum_T = [x / R for x in Sum_T]
+    I = ISum/R
-    print(min(Sum_T[0:20]))
+    IVariance = sqrt(IVarianceSum/(R-1) - I**2)
-    print(mean(Sum_T[0:35]))
-    print(Sum_T[0])
-    TVariance = sqrt(TSumVariance[0] / (R - 1) - Sum_T[0]**2)  # Variance
-    print(TVariance)
 
-    Sum_V = [round(x / R, 2) for x in Sum_V]
+    print("Mean number of boxes : {} (variance {})".format(I, IVariance))
-    # print(Sum_V)
+    for n in range(n):
-    I = ISum / R
+        Hn = HSum[n]/R
-    IVariance = sqrt(IVarianceSum / (R - 1) - I**2)
+        HVariance = sqrt(HSumVariance[n]/(R-1) - Hn**2)
-    print("Mean number of bins : {} (variance {})".format(I, IVariance), "\n")
+        print("Index of box containing the {}th package (H_{}) : {} (variance {})".format(n, n, Hn, HVariance))
-    # TODO clarify line below
-    print(" {} * {} iterations of T".format(R, N), "\n")
-    for n in range(N):
-        Hn = HSum[n] / R  # moyenne
-        HVariance = sqrt(HSumVariance[n] / (R - 1) - Hn**2)  # Variance
-        Var.append(HVariance)
-        H.append(Hn)
-        print(
-            "Index of bin containing the {}th item (H_{}) : {} (variance {})".format(
-                n, n, Hn, HVariance
-            )
-        )
-    print(HSum)
-    print(len(HSum))
-    for x in range(len(HSum)):
-        Hmean+=HSum[x]
-    Hmean=Hmean/P
-    print("Hmean is : {}".format(Hmean))
-    Exp=np.exp(1)
-    HSum = [x / R for x in HSum]
-    HSumVariance = [x / R for x in HSumVariance]
-    print(HSumVariance)
-    # Plotting
-    fig = plt.figure()
-    # T plot
-    x = np.arange(N)
-    # print(x)
-    ax = fig.add_subplot(221)
-    ax.bar(
-        x,
-        Sum_T,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="red",
-    )
-    ax.set(
-            xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 3), yticks=np.linspace(0, 3, 4)
-    )
-    ax.set_ylabel("Items")
-    ax.set_xlabel("Bins (1-{})".format(N))
-    ax.set_title("T histogram for {} items (Number of items in each bin)".format(P))
-    ax.legend(loc="upper left", title="Legend")
-    # V plot
-    bx = fig.add_subplot(222)
-    bx.bar(
-        x,
-        Sum_V,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="orange",
-    )
-    bx.set(
-        xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 1), yticks=np.linspace(0, 1, 10)
-    )
-    bx.set_ylabel("First item size")
-    bx.set_xlabel("Bins (1-{})".format(N))
-    bx.set_title("V histogram for {} items (first item size of each bin)".format(P))
-    bx.legend(loc="upper left", title="Legend")
-    # H plot
-    # We will simulate this part for a asymptotic study
-    cx = fig.add_subplot(223)
-    cx.bar(
-        x,
-        HSum,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="green",
-    )
-    cx.set(
-        xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 10), yticks=np.linspace(0, N, 5)
-    )
-    cx.set_ylabel("Bin ranking of n-item")
-    cx.set_xlabel("n-item (1-{})".format(N))
-    cx.set_title("H histogram for {} items".format(P))
-    xb = linspace(0, N, 10)
-    xc=linspace(0,N,50)
-    yb =  [Hmean for n in range(N)]
-    db =(( HSum[30] - HSum[1])/30)*xc
-    wb =(( HSumVariance[30] - HSumVariance[1])/30)*xc
-    cx.plot(xc, yb, label="Experimental Hn_Mean", color="brown")
-    cx.plot(xc, H, label="Experimental E(Hn)", color="red")
-    cx.plot(xc, Var, label="Experimental V(Hn)", color="purple")
-    cx.legend(loc="upper left", title="Legend")
-   
-
-    plt.show()
 
 def simulate_NFDBP(N):
     """
-    Tries to simulate T_i, V_i and H_n for N items of random size.
+    Tries to simulate T_i, V_i and H_n for N boxes of random size.
-    Next Fit Dual Bin Packing : bins should overflow
     """
     i = 0  # Nombre de boites
     R = [0]  # Remplissage de la i-eme boite
@@ -213,215 +95,83 @@ def simulate_NFDBP(N):
     H = []  # Rang de la boite contenant le n-ieme paquet
     for n in range(N):
         size = random()
-        if R[i] >= 1:
+        R[i] += size
+        T[i] += 1
+        if R[i] + size >= 1:
             # Il y n'y a plus de la place dans la boite pour le paquet.
-            # On passe a la boite suivante (qu'on initialise).
+            # On passe à la boite suivante (qu'on initialise)
             i += 1
             R.append(0)
             T.append(0)
             V.append(0)
+
         if V[i] == 0:
             # C'est le premier paquet de la boite
             V[i] = size
         H.append(i)
-        R[i] += size
-        T[i] += 1
 
-    return {"i": i, "R": R, "T": T, "V": V, "H": H}
+    return {
+        "i": i,
+        "R": R,
+        "T": T,
+        "V": V,
+        "H": H
+    }
 
 
-def stats_NFDBP(R, N, t_i):
+def stats_NFDBP(R, N):
     """
-    Runs R runs of NFDBP (for N items) and studies distribution, variance, mean...
+    Runs R runs of NFDBP (for N packages) and studies distribution, variance, mean...
     """
-    print("## Running {} NFDBP simulations with {} items".format(R, N))
+    print("Running {} NFDBP simulations with {} packages".format(R, N))
-    # TODO comment this function
-    T1=[]
-    P = N * R  # Total number of items
     I = []
     H = [[] for _ in range(N)]  # List of empty lists
-    T = []
+    Tmean=[]
-    Tk = [[] for _ in range(N)]
-    Ti = []
-    T_maths = []
-    # First iteration to use zip after
-    sim = simulate_NFDBP(N)
-    Sum_T = [0 for _ in range(N)]
     for i in range(R):
         sim = simulate_NFDBP(N)
         I.append(sim["i"])
-        for k in range(N):
-            T.append(0)
-            T = sim["T"]
-            T1.append(sim["T"][0])
         for n in range(N):
             H[n].append(sim["H"][n])
-            Tk[n].append(sim["T"][n])
+
-        Ti.append(sim["T"])
+        for k in range(sim["i"]):
-        Sum_T = [x + y for x, y in zip(Sum_T, T)]
+            # for o in range(sim["i"]):
-    Sum_T = [x / R for x in Sum_T]  # Experimental [Ti=k]
+                Tmean+=sim["T"]
-    Sum_T = [
+    print("Mean number of boxes : {} (variance {})".format(mean(I), variance(I)))
-        x * 100 / (sum(Sum_T)) for x in Sum_T
-    ]  # Pourcentage de la repartition des items
-    T1=[x/100 for x in T1]
-    print("Mean number of bins : {} (variance {})".format(mean(I), variance(I)))
 
     for n in range(N):
         print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
-    # TODO variance for T_k doesn't see right
+    for k in range(int(mean(I))+1):
-    print("Mean T_{} : {} (variance {})".format(k, mean(Sum_T), variance(Sum_T)))
+        print(Tmean[7])
-    # Loi math
+       # print("Mean T_{} : {} (variance {})".format(k, mean(Tmean[k]), variance(Tmean[k])))
-    for u in range(N):
-        u = u + 2
-        T_maths.append(1 / (factorial(u - 1)) - 1 / factorial(u))
-    E = 0
-    sigma2 = 0
-    # print(T_maths)
-    T_maths = [x * 100 for x in T_maths]
-    for p in range(len(T_maths)):
-        E = E + (p + 1) * T_maths[p]
-        sigma2 = ((T_maths[p] - E) ** 2) / (len(T_maths) - 1)
-    print(
-        "Mathematical values : Empiric mean T_{} : {} Variance {})".format(
-            t_i, E, sqrt(sigma2)
-        )
-    )
-   # T_maths = [x * 100 for x in T_maths]
-    # Plotting
-    fig = plt.figure()
-    # T plot
-    x = np.arange(N)
-    print(x)
-    print(Sum_T)
-    ax = fig.add_subplot(221)
-    ax.bar(
-        x,
-        Sum_T,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="red",
-    )
-    ax.set(
-        xlim=(0, N), xticks=np.arange(0, N), ylim=(0, 20), yticks=np.linspace(0, 20, 2)
-    )
-    ax.set_ylabel("Items(n) in %")
-    ax.set_xlabel("Bins (1-{})".format(N))
-    ax.set_title(
-        "Items percentage for each bin and {} items (Number of items in each bin)".format(
-            P
-        )
-    )
-    ax.legend(loc="upper right", title="Legend")
 
-    # TODO fix the graph below
+N = 10 ** 1
-    # Mathematical P(Ti=k) plot. It shows the Ti(t_i) law with the probability of each number of items.
+sim = simulate_NFBP(N)
-    print(len(Tk[t_i]))
-    bx = fig.add_subplot(222)
-    bx.hist(
-        Tk[t_i],
-        bins=10,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="red",
-    )
-    bx.set(
-        xlim=(0, N),
-        xticks=np.arange(0, N),
-        ylim=(0, len(Tk[t_i])),
-        yticks=np.linspace(0, 1, 1),
-    )
-    bx.set_ylabel("P(T{}=i)".format(t_i))
-    bx.set_xlabel("Bins i=(1-{}) in %".format(N))
-    bx.set_title(
-        "T{} histogram for {} items (Number of items in each bin)".format(t_i, P)
-    )
-    bx.legend(loc="upper right", title="Legend")
 
-    # Loi mathematique
+print("Simulation NFBP pour {} packaets. Contenu des boites :".format(N))
-    print("ici")
+for j in range(sim["i"] + 1):
-    print(T_maths)
+    remplissage = floor(sim["R"][j] * 100)
-    cx = fig.add_subplot(223)
+    print("Boite {} : Rempli à {} % avec {} paquets. Taille du premier paquet : {}".format(j, remplissage, sim["T"][j],
-    cx.bar(
+                                                                                           sim["V"][j]))
-        x,
-        T_maths,
-        width=1,
-        label="Theoretical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="red",
-    )
-    cx.set(
-        xlim=(0, N),
-        xticks=np.arange(0, N),
-        ylim=(0, 100),
-        yticks=np.linspace(0, 100, 10),
-    )
-    cx.set_ylabel("P(T{}=i)".format(t_i))
-    cx.set_xlabel("Bins i=(1-{})".format(N))
-    cx.set_title("Theoretical T{} values in %".format(t_i))
-    cx.legend(loc="upper right", title="Legend")
-   
-    dx = fig.add_subplot(224)
-    dx.hist(
-        T1,
-        bins=10,
-        width=1,
-        label="Empirical values",
-        edgecolor="blue",
-        linewidth=0.7,
-        color="black",
-    )
-    dx.set(
-        xlim=(0, 10),
-        xticks=np.arange(0, 10,1),
-        ylim=(0, 100),
-        yticks=np.linspace(0, 100, 10),
-    )
-    dx.set_ylabel("Number of items in T1 for {} iterations")
-    dx.set_xlabel("{} iterations for T{}".format(R,1))
-    dx.set_title(
-        "T{} items repartition {} items (Number of items in each bin)".format(1, P)
-    )
-    dx.legend(loc="upper right", title="Legend")
 
-    plt.show()
+print()
+stats_NFBP(10 ** 4, 10)
 
+N = 10 ** 1
+sim = simulate_NFDBP(N)
+print("Simulation NFDBP pour {} packaets. Contenu des boites :".format(N))
+for j in range(sim["i"] + 1):
+    remplissage = floor(sim["R"][j] * 100)
+    print("Boite {} : Rempli à {} % avec {} paquets. Taille du premier paquet : {}".format(j, remplissage,
+                                                                                               sim["T"][j],
+                                                                                               sim["V"][j]))
 
-# unused
+print()
-def basic_demo():
+stats_NFDBP(10 ** 4, 10)
-    N = 10**1
+stats_NFBP_iter(10**6, 10)
-    sim = simulate_NFBP(N)
 
-    print("Simulation NFBP pour {} packaets. Contenu des boites :".format(N))
+#
-    for j in range(sim["i"] + 1):
+# pyplot.plot([1, 2, 4, 4, 2, 1], color = 'red', linestyle = 'dashed', linewidth = 2,
-        remplissage = floor(sim["R"][j] * 100)
+#   markerfacecolor = 'blue', markersize = 5)
-        print(
+# pyplot.ylim(0, 5)
-            "Boite {} : Rempli a {} % avec {} paquets. Taille du premier paquet : {}".format(
+# pyplot.title('Un exemple')
-                j, remplissage, sim["T"][j], sim["V"][j]
-            )
-        )
 
-    print()
-    stats_NFBP(10**3, 10)
-
-    N = 10**1
-    sim = simulate_NFDBP(N)
-    print("Simulation NFDBP pour {} packaets. Contenu des boites :".format(N))
-    for j in range(sim["i"] + 1):
-        remplissage = floor(sim["R"][j] * 100)
-        print(
-            "Boite {} : Rempli a {} % avec {} paquets. Taille du premier paquet : {}".format(
-                j, remplissage, sim["T"][j], sim["V"][j]
-            )
-        )
-
-
-stats_NFBP_iter(10**5, 50)
-print("\n\n")
-stats_NFDBP(10**3, 10, 1)
-
-print("Don't run code you don't understand or trust without a sandbox")

README.md

@@ -1,38 +1,2 @@
 # Projet_Boites
 
-TODO LIST:
-
-
-Rapport: 15 pages max
-
-Explication du projet et des 2 modeles 
-
-Exemples d'applications(cf Internet ou description du cours)
-
-Analyse des liens entre experiences et lois 
-
-
-Programme
-
-Pour les differentes variable, faire plusieurs essais avec des experiences variables
-
--NFBP : Lois mathematiques + representations pour T, V et H
-
-Pour Hn comparaison historigramme et densite de la loi normale pour plusieurs valeurs de n 
-
--NFDBP: Terminer programme sur T et recreer les lois maths (mettre le calcul pour obtenir explicitement la loi des Ti. Comparer simulations et theorie 
-
-
-Oral: 15 min de présentation - 5 min de questions
-
--Reflechir aux 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.

complexity-analysis/direct.py

@@ -1,41 +0,0 @@
-# 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()

complexity-analysis/using_libs.py

@@ -1,36 +0,0 @@
-# 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()

latex/advanced.params/misc.commands.tex

@@ -1,42 +0,0 @@
-\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. 
-}

latex/advanced.params/tikz.conf.tex

@@ -1,13 +0,0 @@
-% \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
-		},
-}

latex/annex-performance.tex

@@ -1,57 +0,0 @@
-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}

latex/annex-probabilistic.tex

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

latex/bibliography.bib

@@ -1,18 +0,0 @@
-@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},
-}
-

latex/clean.sh

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

latex/content.tex

@@ -1,589 +0,0 @@
-\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.
-
-\footnotetext{The code is available in Annex \ref{annex:probabilistic}}
-
-We will try to approximate $ \mathbb{E}[R] $ and $ \mathbb{E}[V] $ with $
-	\overline{X_N} $ using $ {S_n}^2 $. This operation will be done for both $ R =
-	2 $ and $ R = 10^6 $ simulations.
-
-\[
-	\overline{X_N} = \frac{1}{N} \sum_{i=1}^{N} X_i
-\]
-
-As the variance value is unknown, we will use $ {S_n}^2 $ to estimate the
-variance and further determine the Confidence Interval (95 \% certainty).
-
-\begin{align*}
-	{S_N}^2      & = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X_N})^2                                      \\
-	IC_{95\%}(m) & = \left[ \overline{X_N} \pm \frac{S_N}{\sqrt{N}} \cdot t_{1 - \frac{\alpha}{2}, N-1} \right] \\
-\end{align*}
-
-
-
-
-\paragraph{2 simulations} We first ran $ R = 2 $ simulations to observe the
-behavior of the algorithm and the low precision of the results.
-
-\begin{figure}[h]
-	\centering
-	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Ti-2-sim}
-	\caption{Histogram of $ T_i $ for $ R = 2 $ simulations and $ N = 50 $ items (number of items per bin)}
-	\label{fig:graphic-NFBP-Ti-2-sim}
-\end{figure}
-
-On this graph (figure \ref{fig:graphic-NFBP-Ti-2-sim}), we can see each value
-of $ T_i $. Our calculations have yielded that $ \overline{T_1} = 1.0 $ and $
-	{S_N}^2 = 2.7 $. Our Student coefficient is $ t_{0.95, 2} = 4.303 $.
-
-We can now calculate the Confidence Interval for $ T_1 $ for $ R = 2 $ simulations :
-
-\begin{align*}
-	IC_{95\%}(T_1) & = \left[ 1.0 \pm 1.96 \frac{\sqrt{2.7}}{\sqrt{2}} \cdot 4.303 \right] \\
-	               & = \left[ 1 \pm 9.8 \right]                                            \\
-\end{align*}
-
-We can see that the Confidence Interval is very large, which is due to the low
-number of simulations. Looking at figure \ref{fig:graphic-NFBP-Ti-2-sim}, we
-easily notice the high variance.
-
-\begin{figure}[h]
-	\centering
-	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Vi-2-sim}
-	\caption{Histogram of $ V_i $ for $ R = 2 $ simulations and $ N = 50 $ items (size of the first item in a bin)}
-	\label{fig:graphic-NFBP-Vi-2-sim}
-\end{figure}
-
-On the graph of $ V_i $ (figure \ref{fig:graphic-NFBP-Vi-2-sim}), we can see
-that the sizes are scattered pseudo-randomly between $ 0 $ and $ 1 $, which is
-unsuprising given the low number of simulations. The process determinig the statistics
-is the same as for $ T_i $, yielding $ \overline{V_1} = 0.897 $, $ {S_N}^2 =
-	0.2 $ and $ IC_{95\%}(V_1) = \left[ 0.897 \pm 1.3 \right] $. In this particular run,
-the two values for $ V_1 $ are high (being bouded between $ 0 $ and $ 1 $).
-
-
-\paragraph{100 000 simulations} In order to ensure better precision, we then
-ran $ R = 10^5 $ simulations with $ N	= 50 $ different items each.
-
-\begin{figure}[h]
-	\centering
-	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Ti-105-sim}
-	\caption{Histogram of $ T_i $ for $ R = 10^5 $ simulations and $ N = 50 $ items (number of items per bin)}
-	\label{fig:graphic-NFBP-Ti-105-sim}
-\end{figure}
-
-On this graph (figure \ref{fig:graphic-NFBP-Ti-2-sim}), we can see each value
-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}.
-
-\begin{figure}[h]
-	\centering
-	\includegraphics[width=0.8\textwidth]{graphics/graphic-NFBP-Vi-105-sim}
-	\caption{Histogram of $ V_i $ for $ R = 10^5 $ simulations and $ N = 50 $ items (size of the first item in a bin)}
-	\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)}
-
-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{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
-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 \}$. Hence,
-
-\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*}
-
-We will try to show that $ \forall k \geq 1 $, $ P(A_k) = \frac{1}{k!} $. To do
-so, we will use induction to prove the following proposition \eqref{eq:induction},
-$ \forall k \geq 1 $:
-
-\begin{equation}
-	\label{eq:induction}
-	\tag{$ \mathcal{H}_k $}
-	P(U_1 + U_2 + \ldots + U_{k} < 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} \qquad \forall k \geq 1 $.
-
-\paragraph{Base case} $ k = 1 $ : $ P(U_1 < a) = a = \frac{a^1}{1!}$, proving $ (\mathcal{H}_1) $.
-
-\paragraph{Induction step} Let $ k \geq 2 $. We assume $ (\mathcal{H}_{k-1}) $ is
-true. We will show that $ (\mathcal{H}_{k}) $ is true.
-
-\begin{align*}
-	P(S_k < a)                & = P(S_{k-1} + U_k < a)                                                             \\
-	                          & = \iint_{\cal{D}} f_{S_{k-1}, U_k}(x, y)  dxdy                                     \\
-	\text{Where } \mathcal{D} & = \{ (x, y) \in [0, 1]^2 \mid x + y < a \}                                         \\
-	                          & = \{ (x, y) \in [0, 1]^2 \mid 0 < x < a \text{ and } 0 < y < a - x \}              \\
-	P(S_k < a)                & = \iint_{\cal{D}} f_{S_{k-1}}(x) \cdot f_{U_k}(y) dxdy \qquad
-	\text{because $ S_{k-1} $ and $ U_k $ are independent}                                                         \\
-	                          & = \int_{0}^{a} f_{S_{k-1}}(x) \cdot \left( \int_{0}^{a-x} f_{U_k}(y) dy \right) dx \\
-\end{align*}
-
-$ (\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 $. We can then integrate by parts :
-
-\begin{align*}
-	P(S_k < a)
-	 & = \int_{0}^{a} f_{S_{k-1}}(x) \cdot \left( \int_{0}^{a-x} 1 dy \right) dx        \\
-	 & = \int_{0}^{a} f_{S_{k-1}}(x) \cdot (a - x) dx                                   \\
-	 & = a \int_{0}^{a} f_{S_{k-1}}(x) dx - \int_{0}^{a} x f_{S_{k-1}}(x) dx            \\
-	 & = a \int_0^a F'_{S_{k-1}}(x) dx - \left[ x F_{S_{k-1}}(x) \right]_0^a
-	+ \int_{0}^{a} F_{S_{k-1}}(x) dx \qquad \text{(IPP : }x, F_{S_{k-1}} \in C^1([0,1]) \\
-	 & = a \left[ F_{S_{k-1}}(x) \right]_0^a - \left[ x F_{S_{k-1}}(x) \right]_0^a
-	+ \int_{0}^{a} \frac{x^{k-1}}{(k-1)!} dx                                            \\
-	 & = \left[ \frac{x^k}{k!} \right]_0^a                                              \\
-	 & = \frac{a^k}{k!}                                                                 \\
-\end{align*}
-
-\paragraph{Conclusion} We have shown that $ (\mathcal{H}_{k}) $ is true, so by induction, $ \forall k \geq 1 $,
-$ \forall a \in [0, 1] $, $ P(U_1 + U_2 + \ldots + U_{k} < a) = \frac{a^k}{k!} $. Take
-$ a = 1 $ to get
-
-\[ P(U_1 + U_2 + \ldots + U_{k} < 1) = \frac{1}{k!} \]
-
-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 $.
-
-\begin{align*}
-	\mu = E(T_i) & = \sum_{k=2}^{\infty} k \cdot P(T_i = k)                    \\
-	             & = \sum_{k=2}^{\infty} (\frac{k}{(k-1)!} - \frac{1}{(k-1)!}) \\
-	             & = \sum_{k=2}^{\infty} \frac{k-1}{(k-1)!}                    \\
-	             & = \sum_{k=0}^{\infty} \frac{1}{k!}                          \\
-	             & = e                                                         \\
-\end{align*}
-
-\begin{align*}
-	E({T_i}^2) & = \sum_{k=2}^{\infty} k^2 \cdot P(T_i = k)                    \\
-	           & = \sum_{k=2}^{\infty} (\frac{k^2}{(k-1)!} - \frac{k}{(k-1)!}) \\
-	           & = \sum_{k=2}^{\infty} \frac{(k-1)k}{(k-1)!}                   \\
-	           & = \sum_{k=2}^{\infty} \frac{k}{(k-2)!}                        \\
-	           & = \sum_{k=0}^{\infty} \frac{k+2}{k!}                          \\
-	           & = \sum_{k=0}^{\infty} (\frac{1}{(k-1)!} + \frac{2}{(k)!})     \\
-	           & = \sum_{k=0}^{\infty} \frac{1}{(k)!} - 1 + 2e                 \\
-	           & = 3e - 1
-\end{align*}
-
-\begin{align*}
-	\sigma^2 = E({T_i}^2) - E(T_i)^2 = 3e - 1 - e^2
-\end{align*}
-
-$ H_n $ is asymptotically normal, following a $ \mathcal{N}(\frac{N}{\mu}, \frac{N \sigma^2}{\mu^3}) $
-
-
-\section{Complexity and implementation optimization}
-
-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{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}
-

latex/cover/cover_in.tex

@@ -1,30 +0,0 @@
-\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

latex/cover/cover_out.tex

@@ -1,21 +0,0 @@
-\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}

latex/cover/covermain.tex

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

latex/main.tex

@@ -1,139 +0,0 @@
-\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
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-\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}

latex/main_variables.tex

@@ -1,34 +0,0 @@
-\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 7th 2023
-}
-
-\def\varinsaaddress{
-	\textbf{INSA Toulouse}
-
-	135, Avenue de Rangueil
-
-	31077 Toulouse Cedex 4 - France
-
-	\href{https://www.insa-toulouse.fr}{www.insa-toulouse.fr}
-}

latex/remerciements.tex

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