tex: comment probabilistic annex

Probas.py

@@ -1,11 +1,16 @@
+#!/usr/bin/python3
 from random import random
-from math import floor, sqrt
+from math import floor, sqrt, factorial
 from statistics import mean, variance
-# from matplotlib import pyplot
+from matplotlib import pyplot as plt
+from pylab import *
+import numpy as np
+import matplotlib.pyplot as pt
+
 
 def simulate_NFBP(N):
     """
-    Tries to simulate T_i, V_i and H_n for N boxes of random size.
+    Tries to simulate T_i, V_i and H_n for N items of random size.
     """
     i = 0  # Nombre de boites
     R = [0]  # Remplissage de la i-eme boite
@@ -16,7 +21,7 @@ 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 à la boite suivante (qu'on initialise)
+            # On passe a la boite suivante (qu'on initialise)
             i += 1
             R.append(0)
             T.append(0)
@@ -28,20 +33,15 @@ def simulate_NFBP(N):
             V[i] = size
         H.append(i)
 
-    return {
+    return {"i": i, "R": R, "T": T, "V": V, "H": H}
-        "i": i,
-        "R": R,
-        "T": T,
-        "V": V,
-        "H": H
-    }
 
 
+# unused
 def stats_NFBP(R, N):
     """
-    Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
+    Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
     """
-    print("Running {} NFBP simulations with {} packages".format(R, N))
+    print("Running {} NFBP simulations with {} items".format(R, N))
     I = []
     H = [[] for _ in range(N)]  # List of empty lists
 
@@ -51,42 +51,136 @@ def stats_NFBP(R, N):
         for n in range(N):
             H[n].append(sim["H"][n])
 
-    print("Mean number of boxes : {} (variance {})".format(mean(I), variance(I)))
+    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])))
 
+
 def stats_NFBP_iter(R, N):
     """
-    Runs R runs of NFBP (for N packages) and studies distribution, variance, mean...
+    Runs R runs of NFBP (for N items) and studies distribution, variance, mean...
     Calculates stats during runtime instead of after to avoid excessive memory usage.
     """
-    print("Running {} NFBP simulations with {} packages".format(R, N))
+    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)]
+    # 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)]
+        Sum_V = [x + y for x, y in zip(Sum_V, V)]
 
-    I = ISum/R
+    Sum_T = [x / R for x in Sum_T]
-    IVariance = sqrt(IVarianceSum/(R-1) - I**2)
+    Sum_V = [round(x / R, 2) for x in Sum_V]
+    # print(Sum_V)
+    I = ISum / R
+    IVariance = sqrt(IVarianceSum / (R - 1) - I**2)
+    print("Mean number of bins : {} (variance {})".format(I, IVariance), "\n")
+    # TODO clarify line below
+    print(" {} * {} iterations of T".format(R, N), "\n")
+
+    for n in range(min(N, 10)):
+        Hn = HSum[n] / R  # moyenne
+        HVariance = sqrt(HSumVariance[n] / (R - 1) - Hn**2)  # Variance
+        print(
+            "Index of bin containing the {}th item (H_{}) : {} (variance {})".format(
+                n, n, Hn, HVariance
+            )
+        )
+    HSum = [x / R for x in HSum]
+    # print(HSum)
+    # 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), ylim=(0, 3), yticks=np.linspace(0, 3, 5)
+    )
+    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), 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), 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)
+    yb = Hn * xb / 10
+    wb = HVariance * xb / 10
+    cx.plot(xb, yb, label="Theoretical E(Hn)", color="brown")
+    cx.plot(xb, wb, label="Theoretical V(Hn)", color="purple")
+    cx.legend(loc="upper left", title="Legend")
+    plt.show()
 
-    print("Mean number of boxes : {} (variance {})".format(I, IVariance))
-    for n in range(n):
-        Hn = HSum[n]/R
-        HVariance = sqrt(HSumVariance[n]/(R-1) - Hn**2)
-        print("Index of box containing the {}th package (H_{}) : {} (variance {})".format(n, n, Hn, HVariance))
 
 def simulate_NFDBP(N):
     """
-    Tries to simulate T_i, V_i and H_n for N boxes of random size.
+    Tries to simulate T_i, V_i and H_n for N items of random size.
+    Next Fit Dual Bin Packing : bins should overflow
     """
     i = 0  # Nombre de boites
     R = [0]  # Remplissage de la i-eme boite
@@ -95,83 +189,185 @@ def simulate_NFDBP(N):
     H = []  # Rang de la boite contenant le n-ieme paquet
     for n in range(N):
         size = random()
-        R[i] += size
+        if R[i] >= 1:
-        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 à la boite suivante (qu'on initialise)
+            # On passe a 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 {
+    return {"i": i, "R": R, "T": T, "V": V, "H": H}
-        "i": i,
-        "R": R,
-        "T": T,
-        "V": V,
-        "H": H
-    }
 
 
-def stats_NFDBP(R, N):
+def stats_NFDBP(R, N, t_i):
     """
-    Runs R runs of NFDBP (for N packages) and studies distribution, variance, mean...
+    Runs R runs of NFDBP (for N items) and studies distribution, variance, mean...
     """
-    print("Running {} NFDBP simulations with {} packages".format(R, N))
+    print("## Running {} NFDBP simulations with {} items".format(R, N))
+    # TODO comment this function
+    P = N * R  # Total number of items
     I = []
     H = [[] for _ in range(N)]  # List of empty lists
-    Tmean=[]
+    T = []
+    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"]
         for n in range(N):
             H[n].append(sim["H"][n])
+            Tk[n].append(sim["T"][n])
+        Ti.append(sim["T"])
+        Sum_T = [x + y for x, y in zip(Sum_T, T)]
+    Sum_T = [x / R for x in Sum_T]  # Experimental [Ti=k]
+    Sum_T = [
+        x * 100 / (sum(Sum_T)) for x in Sum_T
+    ]  # Pourcentage de la repartition des items
 
-        for k in range(sim["i"]):
+    print("Mean number of bins : {} (variance {})".format(mean(I), variance(I)))
-            # for o in range(sim["i"]):
-                Tmean+=sim["T"]
-    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])))
-    for k in range(int(mean(I))+1):
+    # TODO variance for T_k doesn't see right
-        print(Tmean[7])
+    print("Mean T_{} : {} (variance {})".format(k, mean(Sum_T), variance(Sum_T)))
-       # print("Mean T_{} : {} (variance {})".format(k, mean(Tmean[k]), variance(Tmean[k])))
+    # Loi math
+    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)
+    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")
 
-N = 10 ** 1
+    # TODO fix the graph below
-sim = simulate_NFBP(N)
+    # Mathematical P(Ti=k) plot. It shows the Ti(t_i) law with the probability of each number of items.
+    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")
 
-print("Simulation NFBP pour {} packaets. Contenu des boites :".format(N))
+    # Loi mathematique
-for j in range(sim["i"] + 1):
+    print(T_maths)
-    remplissage = floor(sim["R"][j] * 100)
+    cx = fig.add_subplot(224)
-    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")
+    plt.show()
 
-print()
-stats_NFBP(10 ** 4, 10)
 
-N = 10 ** 1
+# unused
-sim = simulate_NFDBP(N)
+def basic_demo():
-print("Simulation NFDBP pour {} packaets. Contenu des boites :".format(N))
+    N = 10**1
-for j in range(sim["i"] + 1):
+    sim = simulate_NFBP(N)
-    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()
+    print("Simulation NFBP pour {} packaets. Contenu des boites :".format(N))
-stats_NFDBP(10 ** 4, 10)
+    for j in range(sim["i"] + 1):
-stats_NFBP_iter(10**6, 10)
+        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()
-# pyplot.plot([1, 2, 4, 4, 2, 1], color = 'red', linestyle = 'dashed', linewidth = 2,
+    stats_NFBP(10**3, 10)
-#   markerfacecolor = 'blue', markersize = 5)
-# pyplot.ylim(0, 5)
-# pyplot.title('Un exemple')
 
+    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**3, 10)
+print("\n\n")
+stats_NFDBP(10**3, 10, 1)

latex/annex-performance.tex

@@ -1,5 +1,5 @@
 For simplicity, we only include the script for the improved algorithm. For the
-intuitive algorithm, simply replace the algorithm. The imports timing and memory
+intuitive algorithm, simply replace the algorithm. The imports, timing and memory
 usage tracking code are nearly identical.
 
 \begin{lstlisting}[language=python]

latex/annex-probabilistic.tex

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

latex/main.tex

@@ -113,6 +113,12 @@
 
 \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}]