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No commits in common. "master" and "iterative" have entirely different histories.
27 changed files with 75 additions and 1792 deletions
21
.gitignore
vendored
21
.gitignore
vendored
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*.aux
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*.log
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*.swp
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*.out
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*.pdf
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*.toc
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*.maf
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*.mtc
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*.mtc0
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*.stc
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*.stc0
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*.stc1
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*.stc2
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*.stc3
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*.bbl
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*.blg
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*.stc*
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*.bcf
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*.run.xml
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350
Boxes.ipynb
350
Boxes.ipynb
File diff suppressed because one or more lines are too long
400
Probas.py
Executable file → Normal file
400
Probas.py
Executable file → Normal file
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@ -1,15 +1,11 @@
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#!/usr/bin/python3
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from random import random
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from random import random
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from math import floor, sqrt, factorial,exp
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from math import floor, sqrt
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from statistics import mean, variance
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from statistics import mean, variance
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from matplotlib import pyplot as plt
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# from matplotlib import pyplot
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from pylab import *
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import numpy as np
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def simulate_NFBP(N):
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def simulate_NFBP(N):
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"""
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"""
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Tries to simulate T_i, V_i and H_n for N items of random size.
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Tries to simulate T_i, V_i and H_n for N boxes of random size.
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"""
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"""
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i = 0 # Nombre de boites
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i = 0 # Nombre de boites
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R = [0] # Remplissage de la i-eme boite
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R = [0] # Remplissage de la i-eme boite
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@ -20,7 +16,7 @@ def simulate_NFBP(N):
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size = random()
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size = random()
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if R[i] + size >= 1:
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if R[i] + size >= 1:
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# Il y n'y a plus de la place dans la boite pour le paquet.
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# Il y n'y a plus de la place dans la boite pour le paquet.
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# On passe a la boite suivante (qu'on initialise)
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# On passe à la boite suivante (qu'on initialise)
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i += 1
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i += 1
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R.append(0)
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R.append(0)
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T.append(0)
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T.append(0)
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@ -32,15 +28,20 @@ def simulate_NFBP(N):
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V[i] = size
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V[i] = size
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H.append(i)
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H.append(i)
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return {"i": i, "R": R, "T": T, "V": V, "H": H}
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return {
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"i": i,
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"R": R,
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"T": T,
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"V": V,
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"H": H
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}
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# unused
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def stats_NFBP(R, N):
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def stats_NFBP(R, N):
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"""
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"""
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Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
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Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
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"""
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"""
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print("Running {} NFBP simulations with {} items".format(R, N))
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print("Running {} NFBP simulations with {} packages".format(R, N))
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I = []
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I = []
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H = [[] for _ in range(N)] # List of empty lists
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H = [[] for _ in range(N)] # List of empty lists
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@ -50,161 +51,42 @@ def stats_NFBP(R, N):
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for n in range(N):
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for n in range(N):
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H[n].append(sim["H"][n])
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H[n].append(sim["H"][n])
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print("Mean number of bins : {} (variance {})".format(mean(I), variance(I)))
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print("Mean number of boxes : {} (variance {})".format(mean(I), variance(I)))
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for n in range(N):
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for n in range(N):
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print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
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print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
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def stats_NFBP_iter(R, N):
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def stats_NFBP_iter(R, N):
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"""
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"""
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Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
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Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
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Calculates stats during runtime instead of after to avoid excessive memory usage.
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Calculates stats during runtime instead of after to avoid excessive memory usage.
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"""
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"""
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Hmean=0
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print("Running {} NFBP simulations with {} packages".format(R, N))
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Var=[]
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H=[]
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Exp=0
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P = R * N # Total number of items
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print("## Running {} NFBP simulations with {} items".format(R, N))
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# number of bins
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ISum = 0
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ISum = 0
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IVarianceSum = 0
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IVarianceSum = 0
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# index of the bin containing the n-th item
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HSum = [0 for _ in range(N)]
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HSum = [0 for _ in range(N)]
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HSumVariance = [0 for _ in range(N)]
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HSumVariance = [0 for _ in range(N)]
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# number of items in the i-th bin
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Sum_T = [0 for _ in range(N)]
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TSumVariance = [0 for _ in range(N)]
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# size of the first item in the i-th bin
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Sum_V = [0 for _ in range(N)]
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for i in range(R):
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for i in range(R):
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sim = simulate_NFBP(N)
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sim = simulate_NFBP(N)
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ISum += sim["i"]
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ISum += sim["i"]
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IVarianceSum += sim["i"] ** 2
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IVarianceSum += sim["i"]**2
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for n in range(N):
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for n in range(N):
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HSum[n] += sim["H"][n]
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HSum[n] += sim["H"][n]
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HSumVariance[n] += sim["H"][n] ** 2
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HSumVariance[n] += sim["H"][n]**2
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T = sim["T"]
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V = sim["V"]
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# ensure that T, V have the same length as Sum_T, Sum_V
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for i in range(N - sim["i"]):
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T.append(0)
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V.append(0)
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Sum_T = [x + y for x, y in zip(Sum_T, T)]
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TSumVariance = [x + y**2 for x, y in zip(TSumVariance, T)]
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Sum_V = [x + y for x, y in zip(Sum_V, V)]
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Sum_T = [x / R for x in Sum_T]
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I = ISum/R
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print(min(Sum_T[0:20]))
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IVariance = sqrt(IVarianceSum/(R-1) - I**2)
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print(mean(Sum_T[0:35]))
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print(Sum_T[0])
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TVariance = sqrt(TSumVariance[0] / (R - 1) - Sum_T[0]**2) # Variance
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print(TVariance)
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Sum_V = [round(x / R, 2) for x in Sum_V]
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print("Mean number of boxes : {} (variance {})".format(I, IVariance))
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# print(Sum_V)
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for n in range(n):
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I = ISum / R
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Hn = HSum[n]/R
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IVariance = sqrt(IVarianceSum / (R - 1) - I**2)
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HVariance = sqrt(HSumVariance[n]/(R-1) - Hn**2)
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print("Mean number of bins : {} (variance {})".format(I, IVariance), "\n")
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print("Index of box containing the {}th package (H_{}) : {} (variance {})".format(n, n, Hn, HVariance))
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# TODO clarify line below
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print(" {} * {} iterations of T".format(R, N), "\n")
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for n in range(N):
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Hn = HSum[n] / R # moyenne
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HVariance = sqrt(HSumVariance[n] / (R - 1) - Hn**2) # Variance
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Var.append(HVariance)
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H.append(Hn)
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print(
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"Index of bin containing the {}th item (H_{}) : {} (variance {})".format(
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n, n, Hn, HVariance
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)
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)
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print(HSum)
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print(len(HSum))
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for x in range(len(HSum)):
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Hmean+=HSum[x]
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Hmean=Hmean/P
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print("Hmean is : {}".format(Hmean))
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Exp=np.exp(1)
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HSum = [x / R for x in HSum]
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HSumVariance = [x / R for x in HSumVariance]
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print(HSumVariance)
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# Plotting
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fig = plt.figure()
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# T plot
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x = np.arange(N)
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# print(x)
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ax = fig.add_subplot(221)
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ax.bar(
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x,
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Sum_T,
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width=1,
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label="Empirical values",
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edgecolor="blue",
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linewidth=0.7,
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color="red",
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)
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ax.set(
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xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 3), yticks=np.linspace(0, 3, 4)
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)
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ax.set_ylabel("Items")
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ax.set_xlabel("Bins (1-{})".format(N))
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ax.set_title("T histogram for {} items (Number of items in each bin)".format(P))
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ax.legend(loc="upper left", title="Legend")
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# V plot
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bx = fig.add_subplot(222)
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bx.bar(
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x,
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Sum_V,
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width=1,
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label="Empirical values",
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edgecolor="blue",
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linewidth=0.7,
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color="orange",
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)
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bx.set(
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xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 1), yticks=np.linspace(0, 1, 10)
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)
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bx.set_ylabel("First item size")
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bx.set_xlabel("Bins (1-{})".format(N))
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bx.set_title("V histogram for {} items (first item size of each bin)".format(P))
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bx.legend(loc="upper left", title="Legend")
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# H plot
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# We will simulate this part for a asymptotic study
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cx = fig.add_subplot(223)
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cx.bar(
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x,
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HSum,
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width=1,
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label="Empirical values",
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edgecolor="blue",
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linewidth=0.7,
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color="green",
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)
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cx.set(
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xlim=(0, N), xticks=np.arange(0, N,N/10), ylim=(0, 10), yticks=np.linspace(0, N, 5)
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)
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cx.set_ylabel("Bin ranking of n-item")
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cx.set_xlabel("n-item (1-{})".format(N))
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cx.set_title("H histogram for {} items".format(P))
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xb = linspace(0, N, 10)
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xc=linspace(0,N,50)
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yb = [Hmean for n in range(N)]
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db =(( HSum[30] - HSum[1])/30)*xc
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wb =(( HSumVariance[30] - HSumVariance[1])/30)*xc
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cx.plot(xc, yb, label="Experimental Hn_Mean", color="brown")
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cx.plot(xc, H, label="Experimental E(Hn)", color="red")
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cx.plot(xc, Var, label="Experimental V(Hn)", color="purple")
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cx.legend(loc="upper left", title="Legend")
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plt.show()
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def simulate_NFDBP(N):
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def simulate_NFDBP(N):
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"""
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"""
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Tries to simulate T_i, V_i and H_n for N items of random size.
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Tries to simulate T_i, V_i and H_n for N boxes of random size.
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Next Fit Dual Bin Packing : bins should overflow
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"""
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"""
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i = 0 # Nombre de boites
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i = 0 # Nombre de boites
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R = [0] # Remplissage de la i-eme boite
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R = [0] # Remplissage de la i-eme boite
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@ -213,215 +95,83 @@ def simulate_NFDBP(N):
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H = [] # Rang de la boite contenant le n-ieme paquet
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H = [] # Rang de la boite contenant le n-ieme paquet
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for n in range(N):
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for n in range(N):
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size = random()
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size = random()
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if R[i] >= 1:
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R[i] += size
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T[i] += 1
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if R[i] + size >= 1:
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# Il y n'y a plus de la place dans la boite pour le paquet.
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# Il y n'y a plus de la place dans la boite pour le paquet.
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# On passe a la boite suivante (qu'on initialise).
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# On passe à la boite suivante (qu'on initialise)
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i += 1
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i += 1
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R.append(0)
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R.append(0)
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T.append(0)
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T.append(0)
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V.append(0)
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V.append(0)
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if V[i] == 0:
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if V[i] == 0:
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# C'est le premier paquet de la boite
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# C'est le premier paquet de la boite
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V[i] = size
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V[i] = size
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H.append(i)
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H.append(i)
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R[i] += size
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T[i] += 1
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return {"i": i, "R": R, "T": T, "V": V, "H": H}
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return {
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"i": i,
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"R": R,
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"T": T,
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"V": V,
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"H": H
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}
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def stats_NFDBP(R, N, t_i):
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def stats_NFDBP(R, N):
|
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"""
|
"""
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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...
|
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"""
|
"""
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print("## Running {} NFDBP simulations with {} items".format(R, N))
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print("Running {} NFDBP simulations with {} packages".format(R, N))
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# TODO comment this function
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T1=[]
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P = N * R # Total number of items
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I = []
|
I = []
|
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H = [[] for _ in range(N)] # List of empty lists
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H = [[] for _ in range(N)] # List of empty lists
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T = []
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Tmean=[]
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Tk = [[] for _ in range(N)]
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Ti = []
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T_maths = []
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# First iteration to use zip after
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sim = simulate_NFDBP(N)
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Sum_T = [0 for _ in range(N)]
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for i in range(R):
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for i in range(R):
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sim = simulate_NFDBP(N)
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sim = simulate_NFDBP(N)
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I.append(sim["i"])
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I.append(sim["i"])
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for k in range(N):
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T.append(0)
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T = sim["T"]
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T1.append(sim["T"][0])
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for n in range(N):
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for n in range(N):
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H[n].append(sim["H"][n])
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H[n].append(sim["H"][n])
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Tk[n].append(sim["T"][n])
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Ti.append(sim["T"])
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for k in range(sim["i"]):
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Sum_T = [x + y for x, y in zip(Sum_T, T)]
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# for o in range(sim["i"]):
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Sum_T = [x / R for x in Sum_T] # Experimental [Ti=k]
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Tmean+=sim["T"]
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Sum_T = [
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print("Mean number of boxes : {} (variance {})".format(mean(I), variance(I)))
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x * 100 / (sum(Sum_T)) for x in Sum_T
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] # Pourcentage de la repartition des items
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T1=[x/100 for x in T1]
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print("Mean number of bins : {} (variance {})".format(mean(I), variance(I)))
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for n in range(N):
|
for n in range(N):
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print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
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print("Mean H_{} : {} (variance {})".format(n, mean(H[n]), variance(H[n])))
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# TODO variance for T_k doesn't see right
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for k in range(int(mean(I))+1):
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print("Mean T_{} : {} (variance {})".format(k, mean(Sum_T), variance(Sum_T)))
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print(Tmean[7])
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||||||
# Loi math
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# print("Mean T_{} : {} (variance {})".format(k, mean(Tmean[k]), variance(Tmean[k])))
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for u in range(N):
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|
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u = u + 2
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|
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T_maths.append(1 / (factorial(u - 1)) - 1 / factorial(u))
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||||||
E = 0
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|
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sigma2 = 0
|
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# print(T_maths)
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|
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T_maths = [x * 100 for x in T_maths]
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for p in range(len(T_maths)):
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|
||||||
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)
|
print()
|
||||||
dx.hist(
|
stats_NFBP(10 ** 4, 10)
|
||||||
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()
|
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]))
|
||||||
|
|
||||||
|
print()
|
||||||
|
stats_NFDBP(10 ** 4, 10)
|
||||||
|
stats_NFBP_iter(10**6, 10)
|
||||||
|
|
||||||
# unused
|
#
|
||||||
def basic_demo():
|
# pyplot.plot([1, 2, 4, 4, 2, 1], color = 'red', linestyle = 'dashed', linewidth = 2,
|
||||||
N = 10**1
|
# markerfacecolor = 'blue', markersize = 5)
|
||||||
sim = simulate_NFBP(N)
|
# pyplot.ylim(0, 5)
|
||||||
|
# pyplot.title('Un exemple')
|
||||||
|
|
||||||
print("Simulation NFBP 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]
|
|
||||||
)
|
|
||||||
)
|
|
||||||
|
|
||||||
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")
|
|
||||||
|
|
|
||||||
36
README.md
36
README.md
|
|
@ -1,38 +1,2 @@
|
||||||
# Projet_Boites
|
# 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.
|
|
||||||
|
|
|
||||||
|
|
@ -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()
|
|
||||||
|
|
@ -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()
|
|
||||||
|
|
@ -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.
|
|
||||||
}
|
|
||||||
|
|
@ -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
|
|
||||||
},
|
|
||||||
}
|
|
||||||
|
|
@ -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}
|
|
||||||
|
|
@ -1,5 +0,0 @@
|
||||||
Script should have been provided with report. % TODO
|
|
||||||
|
|
||||||
\lstinputlisting[language=Python]{../Probas.py}
|
|
||||||
|
|
||||||
% TODO include output example
|
|
||||||
|
|
@ -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},
|
|
||||||
}
|
|
||||||
|
|
||||||
|
|
@ -1,3 +0,0 @@
|
||||||
rm main.stc* main.toc main.bcf *.aux *.out main.mtc*
|
|
||||||
rm main.maf main.run.xml
|
|
||||||
rm main.log
|
|
||||||
|
|
@ -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}
|
|
||||||
|
|
||||||
Binary file not shown.
|
|
@ -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
|
|
||||||
|
|
@ -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}
|
|
||||||
|
|
@ -1,3 +0,0 @@
|
||||||
\AtBeginDocument{\input{cover/cover_in.tex}\pagenumbering{arabic}}
|
|
||||||
|
|
||||||
%\AtEndDocument{\input{cover/cover_out.tex}}
|
|
||||||
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139
latex/main.tex
139
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
|
|
||||||
\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}
|
|
||||||
|
|
@ -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}
|
|
||||||
}
|
|
||||||
|
|
@ -1,29 +0,0 @@
|
||||||
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
|
|
||||||
Loading…
Reference in a new issue