correlation_clustering.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Correlation Clustering\n",
    "Correlation clustering techniques provide the ability to cluster similar data into groups *(Bansal, Blum, and Chawla 2004)*. Unlike other clustering algorithms, such as k-means, correlation clustering does not require an initial guess for the number of clusters. Instead, it takes as input a similarity matrix, which provides a similarity measure between each pair of data points. The algorithm produces the optimal clustering based on the similarity matrix. The correlation clustering problem is known to be NP-hard.\n",
    "\n",
    "This notebook will compare various implementations of correlation clustering. Specifically, Integer Linear Programming (ILP) formulations will be compared against weighted Max-SAT formulations *(Berg and Järvisalo 2017)*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "from sklearn.datasets import make_blobs\n",
    "from sklearn.cluster import KMeans\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's generate some clustered data points in two-dimensional space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "X, y = make_blobs(n_samples=100, centers=5, n_features=2, cluster_std=0.4, random_state=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], c=y)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "K-means clustering is fast and effective if we know the number of clusters ahead of time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kmeans = KMeans(n_clusters=5)\n",
    "kmeans.fit(X)\n",
    "y_kmeans = kmeans.predict(X)\n",
    "plt.scatter(X[:, 0], X[:, 1], c=y_kmeans)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But what if we have no good guess for the number of clusters? This is where correlation clustering can be of help. Correlation  clustering takes an $N$ x $N$ similarity matrix $W$ as an input, where $N$ is the number of points in the data set. Each entry $w_{ij}$ is a weight representing the similarity between points $v_i$ and $v_j$. Similar points will have a positive weight. Dissimilar points will have a negative weight. The numerical value of the weight indicates the magnitude of similarity or dissimilarity. The way to calculate $W$ depends on the data. There, are several possible  methods, and examining these is beyond the scope of this project. One nice thing about correlation clustering is that it doesn't care how $W$ was created (though the quality of the results will be directly tied to how well $W$ measures similarity in the data). We will use a very simple measure of similarity based on Euclidean distance, along with a threshold value. If the distance between points is larger than the threshold, they will receive negative similarity values. Distances smaller than the threshold get positive values. The Max-SAT solvers we use later will require integer weights, so we multiply each value by some factor and round."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dist(a, b):\n",
    "    return np.linalg.norm(a-b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "mult_factor = 100\n",
    "dim = X.shape[0]\n",
    "alpha = 2\n",
    "W = np.zeros((dim, dim))\n",
    "for i in range(dim):\n",
    "    for j in range(dim):\n",
    "        W[i,j] = round((alpha - dist(X[i], X[j])) * mult_factor)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Correlation Clustering Using Integer Linear Programming\n",
    "The ILP formulation is described in *(Berg and Järvisalo 2017)*. We use binary variables $x_{ij} \\in \\{0,1\\}$ for $1 \\leq i < j \\leq N$. Two points $v_i$ and $v_j$ are co-clustered if $x_{ij} = 1$. We set up the objective function to minimize the weight of dissimilar pairs of co-clustered points. We define a transitivity constraint $x_{ij} + x_{jk} - x_{ik} \\leq 1$. This ensures that if $v_i$ and $v_j$ are co-clustered, and $v_j$ and $v_k$ are co-clustered, then $v_i$ and $v_k$ must also be co-clustered. We formulate the ILP as follows:\n",
    "\n",
    "$$\\text{minimize}~\\sum_{\\substack{-\\infty<w_{ij}<0\\\\i<j}} x_{ij}|w_{ij}| - \\sum_{\\substack{\\infty>w_{ij}>0\\\\i<j}} x_{ij}w_{ij}\\\\ \\text{subject to}~x_{ij} + x_{jk} - x_{ik} \\leq 1~\\text{for all distinct}~i, j, k\\\\ x_{ij} \\in \\{0,1\\}~\\text{for all}~i, j$$\n",
    "\n",
    "This produces $O(N^2)$ variables and $O(N^3)$ constraints.\n",
    "\n",
    "To optimize this ILP, we will utilize two state-of-the-art LP solvers: [CPLEX](https://www.ibm.com/products/ilog-cplex-optimization-studio) and [Gurobi](https://www.gurobi.com/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Write LP  File\n",
    "We will write the ILP to file using a [format](https://www.gurobi.com/documentation/9.0/refman/lp_format.html) that is readable by both CPLEX and Gurobi."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.zeros(W.shape, dtype=np.int32)\n",
    "bin_vars = []\n",
    "var_num = 0\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        var_num += 1\n",
    "        x[i,j] = var_num\n",
    "        bin_vars.append('x{}'.format(var_num))\n",
    "\n",
    "with open('files/correlation_clustering.lp', 'w') as f:\n",
    "    f.write('Minimize\\n')\n",
    "    \n",
    "    # Objective\n",
    "    obj = ''\n",
    "    for i in range(dim-1):\n",
    "        for j in range(i+1, dim):\n",
    "            if W[i,j] > 0:\n",
    "                obj += ' - {} x{}'.format(int(W[i,j]), x[i,j])\n",
    "            elif W[i,j] < 0:\n",
    "                obj += ' + {} x{}'.format(int(-W[i,j]), x[i,j])\n",
    "    f.write('  {}\\n'.format(obj))\n",
    "    obj = None\n",
    "    \n",
    "\n",
    "    # Constraints\n",
    "    f.write('Subject To\\n')\n",
    "    constraint_i = 0\n",
    "    for i in range(dim-2):\n",
    "        for j in range(i+1, dim-1):\n",
    "            for k in range(j+1, dim):\n",
    "                constraint_i += 1\n",
    "                f.write('  x{} + x{} - x{} <= 1\\n'.format(x[i,j], x[j,k], x[i,k]))\n",
    "\n",
    "    # Binary\n",
    "    f.write('Binary\\n')\n",
    "    for bin_var in bin_vars:\n",
    "        f.write('{}\\n'.format(bin_var))\n",
    "    bin_vars=None\n",
    "    \n",
    "    f.write('End\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### CPLEX\n",
    "We optimize the ILP using CPLEX:\n",
    "\n",
    "`\n",
    "cplex -c \"read files/correlation_clustering.lp\" \"optimize\" \"display solution variables -\" > files\\cplex.sol\n",
    "`\n",
    "\n",
    "The folowing code reads in the resulting solution and sorts the points into their assigned clusters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solution time =    0.20 sec.  Iterations = 54  Nodes = 0\n",
      "\n",
      "Found 5 clusters\n"
     ]
    }
   ],
   "source": [
    "cplex_out = np.zeros(var_num, dtype=np.int32)\n",
    "with open('files/cplex.sol', 'r') as f:\n",
    "    while True: \n",
    "        line = f.readline()\n",
    "        if not line:\n",
    "            break\n",
    "        if line.startswith('Solution time'):\n",
    "            print(line)\n",
    "        if line.startswith('Variable Name'):\n",
    "            break\n",
    "        \n",
    "    for i in range(dim-1):\n",
    "        for j in range(i+1, dim):\n",
    "            line = f.readline()\n",
    "            if line.startswith('x'):\n",
    "                var_sol = line.split()\n",
    "                xi = int(var_sol[0][1:])\n",
    "                xi_val = int(var_sol[1][0])\n",
    "                cplex_out[xi-1] = xi_val\n",
    "\n",
    "x = np.zeros(W.shape, dtype=np.int32)\n",
    "var_i = 0\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        x[i,j] = cplex_out[var_i]\n",
    "        var_i += 1\n",
    "\n",
    "\n",
    "# Sort into clusters\n",
    "assigned = []\n",
    "for i in range(dim):\n",
    "    assigned.append(False)\n",
    "\n",
    "clusters = []\n",
    "for i in range(dim-1):\n",
    "    cluster = []\n",
    "    for j in range(i+1, dim):\n",
    "        if x[i,j] == 1:\n",
    "            if not assigned[i]:\n",
    "                cluster.append(X[i])\n",
    "                assigned[i] = True\n",
    "            if not assigned[j]:\n",
    "                cluster.append(X[j])\n",
    "                assigned[j] = True\n",
    "    if len(cluster) > 0:\n",
    "        clusters.append(cluster)\n",
    "\n",
    "print('Found {} clusters'.format(len(clusters)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for cluster in clusters:\n",
    "    cluster_np = np.array(cluster)\n",
    "    plt.scatter(cluster_np[:,0], cluster_np[:,1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Memory consumption increases rapidly as the number of data points increase. I was unable to solve problems with more than 600 points without surpassing my machine's available memory. Note that neither CPLEX nor Gurobi report their memory usage. Memory usage was estimated by manually monitoring system resources. The solvers execute very quickly on small data sets, so I was unable to obtain estimates for sizes 50 and 100.\n",
    "\n",
    "CPLEX results for test data sets:\n",
    "\n",
    "| Samples | Time (s) | Mem(MB) |\n",
    "|---------|----------|---------|\n",
    "| 50      | 0.01     | -       |\n",
    "| 100     | 0.20     | -       |\n",
    "| 200     | 2.39     | 302     |\n",
    "| 300     | 9.36     | 1038    |\n",
    "| 400     | 27.98    | 2444    |\n",
    "| 500     | 65.56    | 5593    |\n",
    "| 600     | 140.08   | 10699   |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gurobi\n",
    "We optimize the ILP using Gurobi:\n",
    "\n",
    "`\n",
    "gurobi_cl ResultFile=\"files\\gurobi.sol\" \"files\\correlation_clustering.lp\"\n",
    "`\n",
    "\n",
    "The folowing code reads in the resulting solution and sorts the points into their assigned clusters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 5 clusters\n"
     ]
    }
   ],
   "source": [
    "gurobi_out = np.zeros(var_num, dtype=np.int32)\n",
    "with open('files/gurobi.sol', 'r') as f:\n",
    "    f.readline()\n",
    "        \n",
    "    for i in range(dim-1):\n",
    "        for j in range(i+1, dim):\n",
    "            line = f.readline()\n",
    "            if line.startswith('x'):\n",
    "                var_sol = line.split()\n",
    "                xi = int(var_sol[0][1:])\n",
    "                xi_val = int(var_sol[1][0])\n",
    "                cplex_out[xi-1] = xi_val\n",
    "\n",
    "x = np.zeros(W.shape, dtype=np.int32)\n",
    "var_i = 0\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        x[i,j] = cplex_out[var_i]\n",
    "        var_i += 1\n",
    "\n",
    "# Sort into clusters\n",
    "assigned = []\n",
    "for i in range(dim):\n",
    "    assigned.append(False)\n",
    "\n",
    "clusters = []\n",
    "for i in range(dim-1):\n",
    "    cluster = []\n",
    "    for j in range(i+1, dim):\n",
    "        if x[i,j] == 1:\n",
    "            if not assigned[i]:\n",
    "                cluster.append(X[i])\n",
    "                assigned[i] = True\n",
    "            if not assigned[j]:\n",
    "                cluster.append(X[j])\n",
    "                assigned[j] = True\n",
    "    if len(cluster) > 0:\n",
    "        clusters.append(cluster)\n",
    "\n",
    "print('Found {} clusters'.format(len(clusters)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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k5VmpXUOkWSjIZZr//Od9PPXowMTX+RuRQGphnj+PZq2IJKcglyme2zU0JcTzanEj8uxL5yu4RVKgHrlM8fgDP4t9TTciRRqTKvImV9gLLxbWUTcitahHJHuqyJtY1KKcYhZd8HslP//QN5/hP/95X20GLCKRVJE3sahFOcXs2zlE91ldU25URn3+qUcHJt6nil2k9hTkTaxYBW4t4AUZnb/hCSdmm8TJv2/yFMNazH4RETB3r/tF+/r6vL+/v+7Xlanu+fR/R4ZxqV5520ktZVXycefJ/5BQhS5SGTPb7e59hcfVI29iS1aeRdtJU/8J5BflWJF/GUlCHE5U+tooSyQdCvImdval83nbn507MRtlzhtm8bY/O5ezL50/ra1SqUUX/F5Zy+0nt2tEpDrqkTe5uEU5pdorpezbOcS5l81n386hkhW85qeLJKOKXCItWXkWLa3V7048evQ4T//XAOdeNn+iMo9r12ijLJFkFOQS6exL59PeUfqfR7EQ9uO5ynzJyrP4yD9cyVU3nh/bkxeR6inIJdZrh8eKvm4tcOMXlxYN88k98GI9eRGpnoJcYpVqefQu7gKiZ79Mph64SG0pyCVWqYAe+vlveG7X0ESlXaoHrud0itSGglxiFbZCChW2TaJ64HBij5Ziz+kUkeopyKWosy+dz41fXBr7+uS2ydmXzufcy6b3u/ftHOK5XUN6TqdIjWgeuRSV3/QqTmG1/r9P/Wrae/JVd9zcdE0/FElGFbnEKuxpF4qaOlis6i62JYCIVE8VucQqts1t3IZXxapuPadTpDZSCXIzuwbYALQCd7v7ujTOK9kq1ruO65svWXnWlK1rYWrVred0iqQvcWvFzFqBrwPvAM4HPmBm5yc9r2QvrnddrKetRT8i9ZdGRf4m4Kfu/gKAmf0rsBJ4JoVzS4ZKVddxVHWL1FcaQd4LvDTp6/3ApYVvMrNVwCqAhQsXpnBZqTX1tEXCkEaQR22RN+2xQ+6+EdgIuScEpXBdqQNV1yKNL43ph/uB0yd9vQAYSOG8IiJShjSC/H+AxWZ2hpmdBNwAfDeF84qISBkSt1bcfdTMPgo8SG764Tfc/enEIxMRkbKkMo/c3b8PfD+Nc4mISGW0RF9EJHAKchGRwCnIRUQCpyAXEQmcglxEJHAKchGRwCnIRUQCpyAXEQmcglxEJHAKchGRwCnIRUQCpyAXEQmcglxEJHAKchGRwCnIRUQCpyAXEQmcglxEJHAKchGRwCnIRUQCpyAXkaZyaOtWnr9yGXvPO5/nr1zGoa1bsx5SYqk8fFlEJASHtm5l8LOfw48cAWB0YIDBz34OgLkrVmQ5tERUkYtI03j5zrsmQjzPjxzh5Tvvqul1a/1bgCpyEWkao4ODFR1PQz1+C1BFLiJNo627u6LjaajHbwGJgtzM1pvZPjN70sy+Y2ZdaQ1MRCRtp918E9bRMeWYdXRw2s03lfX5alok9fgtIGlF/hBwgbtfCDwH3Jp8SCIi6coH8MAnP0VLRwfW1QVmtPX00P2F28tqceRbJKMDA+DO6MAAA7d8kr3nnlc01OvxW0CiIHf37e4+Ov7lTmBB8iGJiKSnMIDHhofhyBF6vvwlFj+8o+w+dVSLJC/f944K86S/BZQjzR75XwI/iHvRzFaZWb+Z9R88eDDFy4qIxEurR12qFRJ3zrkrVtD9hdtp6+mp+LeAcpWctWJm/w7Mj3jpM+7+wPh7PgOMApvizuPuG4GNAH19fV7VaEVEynRo61ZevvOuXCUeodIedVt3d+y5Sp1z7ooVNZ2nXjLI3f2qYq+b2Y3AtcAyd1dAi0jmCqf8RbG5c3n+ymWMDg7S1t3NaTffVDRsT7v5JgZu/TSMjsa+p5azX4pJOmvlGuBTwLvc/XfpDElEJJli/WwAWlrg8OEpNy4Le9yFM1QAWufMiT1l2n3vSliSItrMfgrMAn41fminu3+41Of6+vq8v7+/6uuKiBSz97zzoVi2mUW+bl1dnLvz8bIq+kI9679c82X+Zrbb3fsKjyda2enuf5jk8yIitVCynx0T8j48PNFbryTE23p6Mt2rRSs7RWTGiZryN0Vra+xLL995V0U3QrNsqeQpyEVkxslP+WvtillsPjYW+9n8zc+ytLYy90+vy3znRAW5iMxIc1es4Oydj9Oz/su5OdyQ642X0tJScprhhLExDn3n/sz3NFeQi8iMNnfFChY/vCMX5uVM7ihSrUepxza4pSjIRaQpFO17m8X2zVu7uiZWZVZ17jpQkItIU4jdvKqnh/P2PhNbiY8ND7P44R2ct/eZEy2aMs9dLwpyEWkKcZtXzXnLFRMLfiJNqtTrsQFWNfSEIBFpCvmZJfnphW3d3cx5yxUc+s79xeeMj40x+PnPM7z527mq3Qzr7MSPHClraX89JFrZWS2t7BSRRvD8lctKz1Dp7ISRkWmHuz5wA9233VajkUWLW9mp1oqINK1SNymtowNiqvXhzd+uxZCqoiAXkaZV7CZlft/w2CmLFU5TrCUFuYg0nfzOhqMDA9OmFVpHBz3rv3zi6UFxy/mLLPOvNwW5iDSVKY9+gykVd9TTe7quf1/keeKOZ0GzVkSkqcTtbNjW08Pih3dMO56/oTkxa6W1la7r31f3G53FKMhFpKnE3eAsduOz+7bbGiq4C6m1IiJNJXaFZ8arM5NQkItIU2nU1ZlJqLUiIk0laoVnI6zOTEJBLiJNZ+6KFUEHdyG1VkREAqcgFxEJnIJcRCRwCnIRkcApyEVEAqcgFxEJnIJcRCRwmTwhyMwOAi8mOMWpwC9TGk6W9H00Fn0fjWMmfA+Q/vfxB+4+r/BgJkGelJn1Rz3uKDT6PhqLvo/GMRO+B6jf96HWiohI4BTkIiKBCzXIN2Y9gJTo+2gs+j4ax0z4HqBO30eQPXIRETkh1IpcRETGKchFRAIXbJCb2Xoz22dmT5rZd8ysK+sxVcLMrjGzZ83sp2a2JuvxVMPMTjezR8xsr5k9bWarsx5Ttcys1cz2mNn3sh5Ltcysy8y2jP//Yq+ZLcl6TNUws5vH/z09ZWb/YmYdpT+VPTP7hpm9bGZPTTr2BjN7yMyeH//vKbW4drBBDjwEXODuFwLPAbdmPJ6ymVkr8HXgHcD5wAfM7PxsR1WVUeAT7n4ecBnwkUC/D4DVwN6sB5HQBuDf3P1c4I8J8Psxs17g40Cfu18AtAI3ZDuqsv0TcE3BsTXADndfDOwY/zp1wQa5u29399HxL3cCC7IcT4XeBPzU3V9w96PAvwIrMx5Txdx90N2fGP/zb8kFR2+2o6qcmS0AlgN3Zz2WapnZ64ErgH8EcPej7j6c7aiq1gZ0mlkbcDIwkPF4yuLujwKvFBxeCdwz/ud7gOtqce1gg7zAXwI/yHoQFegFXpr09X4CDMDJzGwRcBGwK9uRVOUu4JPA8awHksCZwEHgm+MtorvNbHbWg6qUux8AvgL8AhgEDrn79mxHlcjvu/sg5Aof4LRaXKShg9zM/n28T1b4v5WT3vMZcr/ib8pupBWziGPBzgM1sznAvcBN7v6brMdTCTO7FnjZ3XdnPZaE2oCLgb9394uAw9To1/haGu8hrwTOAHqA2Wb2wWxH1fga+uHL7n5VsdfN7EbgWmCZhzUhfj9w+qSvFxDIr4+FzKydXIhvcvf7sh5PFZYC7zKzdwIdwOvN7FvuHlp47Af2u3v+N6ItBBjkwFXAz939IICZ3QdcDnwr01FV7//MrNvdB82sG3i5Fhdp6Iq8GDO7BvgU8C53/13W46nQ/wCLzewMMzuJ3M2c72Y8poqZmZHrye51969mPZ5quPut7r7A3ReR+3t4OMAQx92HgJfM7JzxQ8uAZzIcUrV+AVxmZieP//taRoA3bSf5LnDj+J9vBB6oxUUauiIv4e+AWcBDub9vdrr7h7MdUnncfdTMPgo8SO6u/Dfc/emMh1WNpcCfAz8xsx+NH/u0u38/wzE1s48Bm8aLgxeAD2U8noq5+y4z2wI8Qa5luodAluub2b8AbwVONbP9wG3AOmCzmf0VuR9S76vJtcPqSIiISKFgWysiIpKjIBcRCZyCXEQkcApyEZHAKchFRAKnIBcRCZyCXEQkcP8PZ8T0sdlWBCMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for cluster in clusters:\n",
    "    cluster_np = np.array(cluster)\n",
    "    plt.scatter(cluster_np[:,0], cluster_np[:,1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gurobi requires even more memory than CPLEX, and solve time is much higher. I was unable to solve problems with more than 500 points without surpassing my machine's available memory.\n",
    "\n",
    "Gurobi results for test data sets:\n",
    "\n",
    "| Samples | Time (s) | Mem (MB) |\n",
    "|---------|----------|----------|\n",
    "| 50      | 0.02     | -        |\n",
    "| 100     | 0.42     | -        |\n",
    "| 200     | 6.91     | 425      |\n",
    "| 300     | 33.31    | 1480     |\n",
    "| 400     | 113.38   | 3643     |\n",
    "| 500     | 251.72   | 7241     |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Correllation Clustering Using MaxSAT Encoding\n",
    "The [Maximum Satisfiability](https://en.wikipedia.org/wiki/Maximum_satisfiability_problem) (MaxSAT) problem is defined as follows: given a formula written in [conjunctive normal form](https://en.wikipedia.org/wiki/Conjunctive_normal_form), find some assignment of variables that results in the maximum number of clauses evaluating to true.\n",
    "\n",
    "An extension of this is *weighted* MaxSat, where each clause is assigned a non-negative weight. We seek to maximize the sum of the weights of satisfied clauses.\n",
    "\n",
    "To implement correlation clustering, we utilize *partial weighted* MaxSAT. In the partial weighted MaxSAT problem, we have a set of *hard* clauses, and a set of *soft* clauses. A solution must satisfy *all* of the hard clauses. We seek to maximize the sum of the weights of all satisfied soft clauses *(Berg and Järvisalo 2017)*.\n",
    "\n",
    "We will encode our problem as a weighted partial MaxSAT and utilize [UWrMaxSat](https://github.com/marekpiotrow/UWrMaxSat) to optimize. UWrMaxSat was a top contender in the [2019 MaxSat Evaluation](https://maxsat-evaluations.github.io/2019/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Helper Methods\n",
    "Most MaxSat solvers take an input file in the [DIMACS](https://maxsat-evaluations.github.io/2019/rules.html#input) format. Solution output is also in this format. The folowing  method reads the output file and returns a dictionary that maps variables to their solution values (true or  false)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def read_solution(filename):\n",
    "    \"\"\"\n",
    "    Read MaxSAT solution file.\n",
    "\n",
    "    Returns:\n",
    "        dict: dictionary indicating which variables are True or False.\n",
    "    \"\"\"\n",
    "    solution = None\n",
    "    with open(filename, 'r') as f:\n",
    "        while True: \n",
    "            line = f.readline()  \n",
    "            if not line: \n",
    "                break\n",
    "            if line.startswith('c ') or line.startswith('s '):\n",
    "                print(line.strip())\n",
    "            if line.startswith('v '):\n",
    "                solution = line.strip().split(' ', 1)[1]\n",
    "\n",
    "    sol_arr = map(int, solution.split(' '))\n",
    "    inc_dict = dict()\n",
    "    for sol in sol_arr:\n",
    "        if sol > 0:\n",
    "            inc_dict[sol] = True\n",
    "        else:\n",
    "            inc_dict[-sol] = False\n",
    "    \n",
    "    return inc_dict"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For weighted MaxSAT, solvers require that hard clauses be assigned a weight that is greater than the sum of all soft clause weights. We calculate hard clause weight below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_hard_clause_weight(W):\n",
    "    dim = W.shape[0]\n",
    "    hard_cluase_weight = 0\n",
    "    W_abs = np.abs(W)\n",
    "    for i in range(dim-1):\n",
    "        for j in range(i+1, dim):\n",
    "            hard_cluase_weight += W_abs[i,j]\n",
    "    return int(hard_cluase_weight + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transitive Encoding\n",
    "We must encode our clustering problem into conjunctive normal form. We will first try the *transitive* encoding *(Berg and Järvisalo 2017)*. This encoding has hard clauses analogous to the ILP transitive constraints. The soft clauses are analogous to the ILP objective function.\n",
    "\n",
    "**Hard Clauses:**  \n",
    "$(\\neg x_{ij} \\vee \\neg x_{jk} \\vee x_{ik}) \\text{ for all } (v_i,v_j,v_k) \\in V^3 \\text{ where  } i,j,k \\text{ are distinct}$\n",
    "\n",
    "**Soft Clauses:**  \n",
    "$(x_{ij}) \\text{ for all similar } v_i, v_j \\text{ s.t. } i<j$  \n",
    "$(\\neg x_{ij}) \\text{ for all dissimilar } v_i, v_j \\text{ s.t. } i<j$\n",
    "\n",
    "**Soft Clause Weight:**  \n",
    "$w_{ij} \\text{ for all similar } v_i, v_j \\text{ s.t. } i<j$  \n",
    "$|w_{ij}| \\text{ for all dissimilar } v_i, v_j \\text{ s.t. } i<j$\n",
    "\n",
    "This encoding produces $O(N^2)$ variables and $O(N^3)$ clauses."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "hard_clause_weight = get_hard_clause_weight(W)\n",
    "\n",
    "# Map variables to integers for WDIMACS format\n",
    "int_map = np.zeros((dim, dim), dtype=np.int32)\n",
    "var_num = 0\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        var_num += 1\n",
    "        int_map[i,j] = var_num\n",
    "\n",
    "# Hard clauses\n",
    "hard_clauses = []\n",
    "for i in range(dim-2):\n",
    "    for j in range(i+1, dim-1):\n",
    "        for k in range(j+1, dim):\n",
    "            hard_clauses.append('{} -{} -{} {} 0\\n'.format(hard_clause_weight, int_map[i,j], int_map[j,k], int_map[i,k]))\n",
    "\n",
    "# Soft clauses\n",
    "soft_clauses= []\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        if W[i,j] > 0:\n",
    "            soft_clauses.append('{} {} 0\\n'.format(int(W[i,j]), int_map[i,j]))\n",
    "        elif W[i,j] < 0:\n",
    "            soft_clauses.append('{} -{} 0\\n'.format(int(-W[i,j]), int_map[i,j]))\n",
    "\n",
    "with open('files/maxsat_transitive.wcnf', 'w') as f:\n",
    "    f.write('c Correlation Clustering Max-SAT Transitive\\n')\n",
    "    f.write('c\\n')\n",
    "    f.write('p wcnf {} {} {}\\n'.format(var_num, len(hard_clauses) + len(soft_clauses), hard_clause_weight))\n",
    "\n",
    "    # Hard clauses\n",
    "    for clause in hard_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "    # Soft clauses\n",
    "    for clause in soft_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "hard_clauses = None\n",
    "soft_clauses = None"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We optimize the transitive encoding using UWrMaxSat:\n",
    "\n",
    "`./uwrmaxsat <input.wcnf> -m > <outfile>`\n",
    "\n",
    "The solution file is read."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c Parsing MaxSAT file...\n",
      "c ============================[  Problem Statistics ]============================\n",
      "c |  Number of variables:          4950                                         |\n",
      "c |  Number of clauses:          166647 (incl.         4947 soft in queue)      |\n",
      "c ===============================================================================\n",
      "c Using COMiniSatPS SAT solver by Chanseok Oh (2016)\n",
      "c \u001b[1mOptimal solution: 784\u001b[0m\n",
      "c _______________________________________________________________________________\n",
      "c\n",
      "c restarts               : 698\n",
      "c conflicts              : 2              (4 /sec)\n",
      "c decisions              : 305135         (0.00 % random) (557961 /sec)\n",
      "c propagations           : 4002367        (7318614 /sec)\n",
      "c conflict literals      : 9              (0.00 % deleted)\n",
      "c Memory used            : 21.80 MB\n",
      "c CPU time               : 0.546875 s\n",
      "c OptExp Enc: Srt/BDD/Add: 23 0 0\n",
      "c _______________________________________________________________________________\n",
      "s OPTIMUM FOUND\n"
     ]
    }
   ],
   "source": [
    "inc_dict = read_solution('files/maxsat_transitive.sol')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We sort the points into clusters in the same way we did for the ILP solutions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 5 clusters\n"
     ]
    }
   ],
   "source": [
    "assigned = []\n",
    "for i in range(dim):\n",
    "    assigned.append(False)\n",
    "\n",
    "clusters = []\n",
    "for i in range(dim-1):\n",
    "    cluster = []\n",
    "    for j in range(i+1, dim):\n",
    "        if inc_dict[int_map[i,j]]:\n",
    "            if not assigned[i]:\n",
    "                cluster.append(X[i])\n",
    "                assigned[i] = True\n",
    "            if not assigned[j]:\n",
    "                cluster.append(X[j])\n",
    "                assigned[j] = True\n",
    "    if len(cluster) > 0:\n",
    "        clusters.append(cluster)\n",
    "\n",
    "print('Found {} clusters'.format(len(clusters)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for cluster in clusters:\n",
    "    cluster_np = np.array(cluster)\n",
    "    plt.scatter(cluster_np[:,0], cluster_np[:,1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are the results for the test data sets using the transitive encoding:\n",
    "\n",
    "| Samples | Time (s) | Memory (MB) |\n",
    "|---------|----------|-------------|\n",
    "| 50      | 0.04     | 13.66       |\n",
    "| 100     | 0.54     | 21.80       |\n",
    "| 200     | 6.81     | 91.75       |\n",
    "| 300     | 22.85    | 336.73      |\n",
    "| 400     | 80.53    | 543.36      |\n",
    "| 500     | 200.60   | 1059.09     |\n",
    "| 600     | 379.14   | 2227.91     |\n",
    "| 700     | 648.51   | 3830.15     |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unary Encoding\n",
    "We can improve the efficiency of the algorithm by setting an upper limit on the number of clusters. We define this maximum as $K \\leq N$. With this, we can formulate the *unary* encoding *(Berg and Järvisalo 2017)*. We use $N \\cdot K$ variables $y_i^k$. We encode logic to ensure that $\\sum_{k=1}^Ky_i^k=1$ for all $i = 1,\\ldots,N$. If $y_i^k=1$ then $v_i$ is assigned to cluster $k$. The encoding  makes use of a few additional auxiliary variables. For details, refer to the referenced paper.\n",
    "\n",
    "This encoding produces $O(E \\cdot K + N \\cdot K)$ variables and $O(E \\cdot K)$ clauses where $E$ is the number of nonzero values in $W$. This makes it more compact than the transitive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = 8\n",
    "hard_clause_weight = get_hard_clause_weight(W)\n",
    "\n",
    "# Map variables to integers for WDIMACS format\n",
    "var_num = 0\n",
    "\n",
    "# Auxilary variables for unary encoding\n",
    "y = []\n",
    "for i in range(dim):\n",
    "    yi =[]\n",
    "    for k in range(K):\n",
    "        var_num += 1\n",
    "        yi.append(var_num)\n",
    "    y.append(yi)\n",
    "\n",
    "s = []\n",
    "for i in range(dim):\n",
    "    si = []\n",
    "    for k in range(K-1):\n",
    "        var_num += 1\n",
    "        si.append(var_num)\n",
    "    s.append(si)\n",
    "\n",
    "A = np.zeros((dim, dim, K), dtype=np.int32)\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        for k in range(K):\n",
    "            var_num += 1\n",
    "            A[i,j,k] = var_num\n",
    "\n",
    "D = np.zeros((dim, dim), dtype=np.int32)\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        var_num += 1\n",
    "        D[i,j] = var_num\n",
    "\n",
    "# Hard clauses\n",
    "hard_clauses = []\n",
    "\n",
    "# ExactlyOne(i)\n",
    "for i in range(dim):\n",
    "    hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, y[i][0], s[i][0]))\n",
    "    hard_clauses.append('{} -{} -{} 0\\n'.format(hard_clause_weight, y[i][K-1], s[i][K-2]))\n",
    "    for k in range(1, K-1):\n",
    "        hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, y[i][k], s[i][k]))\n",
    "        hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, s[i][k-1], s[i][k]))\n",
    "        hard_clauses.append('{} -{} -{} 0\\n'.format(hard_clause_weight, y[i][k], s[i][k-1]))\n",
    "\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        if W[i,j] > 0:\n",
    "            for k in range(K):\n",
    "                # HardSimilar(i,j,k)\n",
    "                hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, A[i,j,k], y[i][k]))\n",
    "                hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, A[i,j,k], y[j][k]))\n",
    "                hard_clauses.append('{} {} -{} -{} 0\\n'.format(hard_clause_weight, A[i,j,k], y[i][k], y[j][k]))\n",
    "        elif W[i,j] < 0:\n",
    "            for k in range(K):\n",
    "                # HardDissimilar(i,j,k)\n",
    "                hard_clauses.append('{} {} -{} -{} 0\\n'.format(hard_clause_weight, D[i,j], y[i][k], y[j][k]))\n",
    "\n",
    "# Soft clauses\n",
    "soft_clauses= []\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        if W[i,j] > 0:\n",
    "            # SoftSimilar(i,j)\n",
    "            soft_sim = '{} '.format(int(W[i,j]))\n",
    "            for k in range(K):\n",
    "                soft_sim += '{} '.format(A[i,j,k])\n",
    "            soft_sim += '0\\n'\n",
    "            soft_clauses.append(soft_sim)\n",
    "        elif W[i,j] < 0:\n",
    "            # SoftDissimilar(i,j)\n",
    "            soft_clauses.append('{} -{} 0\\n'.format(int(-W[i,j]), D[i,j]))\n",
    "\n",
    "with open('files/maxsat_unary.wcnf', 'w') as f:\n",
    "    f.write('c Correlation Clustering Max-SAT Unary\\n')\n",
    "    f.write('c\\n')\n",
    "    f.write('p wcnf {} {} {}\\n'.format(var_num, len(hard_clauses) + len(soft_clauses), hard_clause_weight))\n",
    "\n",
    "    # Hard clauses\n",
    "    for clause in hard_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "    # Soft clauses\n",
    "    for clause in soft_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "hard_clauses = None\n",
    "soft_clauses = None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c Parsing MaxSAT file...\n",
      "c ============================[  Problem Statistics ]============================\n",
      "c |  Number of variables:         14361                                         |\n",
      "c |  Number of clauses:           62363 (incl.         4947 soft in queue)      |\n",
      "c ===============================================================================\n",
      "c Using COMiniSatPS SAT solver by Chanseok Oh (2016)\n",
      "c \u001b[1mOptimal solution: 955\u001b[0m\n",
      "c _______________________________________________________________________________\n",
      "c\n",
      "c restarts               : 1028\n",
      "c conflicts              : 1465           (1803 /sec)\n",
      "c decisions              : 4613417        (0.00 % random) (5678052 /sec)\n",
      "c propagations           : 12187663       (15000201 /sec)\n",
      "c conflict literals      : 13927          (9.68 % deleted)\n",
      "c Memory used            : 32.61 MB\n",
      "c CPU time               : 0.8125 s\n",
      "c OptExp Enc: Srt/BDD/Add: 32 0 0\n",
      "c _______________________________________________________________________________\n",
      "s OPTIMUM FOUND\n"
     ]
    }
   ],
   "source": [
    "inc_dict = read_solution('files/maxsat_unary.sol')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 5 clusters\n"
     ]
    }
   ],
   "source": [
    "y_unary = np.zeros(dim)\n",
    "var_i=0\n",
    "for i in range(dim):\n",
    "    count = 0\n",
    "    for k in range(K):\n",
    "        var_i += 1\n",
    "        if inc_dict[var_i] == True:\n",
    "            y_unary[i] = k\n",
    "            count += 1\n",
    "    if count != 1:\n",
    "        print('y_{} assigned {} clusters'.format(i, count))\n",
    "\n",
    "print('Found {} clusters'.format(len(set(y_unary))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], c=y_unary)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are the results for the test data sets using the unary encoding:\n",
    "\n",
    "| Samples | Time (s)  | Memory (MB) |\n",
    "|---------|-----------|-------------|\n",
    "| 50      | 0.07      | 17.12       |\n",
    "| 100     | 0.81      | 32.61       |\n",
    "| 200     | 6.32      | 92.16       |\n",
    "| 300     | 14.12     | 187.04      |\n",
    "| 400     | 69.23     | 342.77      |\n",
    "| 500     | 170.45    | 513.50      |\n",
    "| 600     | 265.98    | 735.01      |\n",
    "| 700     | 417.42    | 924.57      |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binary Encoding\n",
    "The *binary* encoding *(Berg and Järvisalo 2017)* is even more compact than the unary. We choose a maximum number of clusters $K = 2^a \\leq N$. We define variables $b_i^k$ for $i=1,\\ldots,N$ and $k=1,\\ldots,a$. These variables can be interpreted as the bits in a binary integer. Combining the bits $b_i^a,\\ldots,b_i^1$ gives a binary value for the cluster number to which we will assign $v_i$. Like the unary encoding, the binary encoding employs several additional auxiliary variables. See the referenced paper for a detailed description.\n",
    "\n",
    "This encoding produces $O(E + N \\cdot log_2K)$ variables and $O(E \\cdot log_2K)$ clauses where $E$ is the number of nonzero values in $W$. We have managed to reduce the size of our encoding even more."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "hard_clause_weight = get_hard_clause_weight(W)\n",
    "a = 3\n",
    "K = 2**a\n",
    "assert(K <= dim)\n",
    "\n",
    "var_num = 0\n",
    "\n",
    "# Binary variables (b)\n",
    "b = []\n",
    "for i in range(dim):\n",
    "    bi = []\n",
    "    for k in range(a):\n",
    "        var_num += 1\n",
    "        bi.append(var_num)\n",
    "    b.append(bi)\n",
    "\n",
    "# Auxilary variables (EQ, S)\n",
    "EQ = np.zeros((dim, dim, a), dtype=np.int32)\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        for k in range(a):\n",
    "            var_num += 1\n",
    "            EQ[i,j,k] = var_num\n",
    "\n",
    "S = np.zeros((dim, dim), dtype=np.int32)\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        var_num += 1\n",
    "        S[i,j] = var_num\n",
    "\n",
    "# Hard clauses\n",
    "hard_clauses = []\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        same_cluster_hard = '{} {} '.format(hard_clause_weight, S[i,j])\n",
    "        for k in range(a):\n",
    "            # Equality(i,j,k)\n",
    "            hard_clauses.append('{} {} {} {} 0\\n'.format(hard_clause_weight, EQ[i,j,k], b[i][k], b[j][k]))\n",
    "            hard_clauses.append('{} {} -{} -{} 0\\n'.format(hard_clause_weight, EQ[i,j,k], b[i][k], b[j][k]))\n",
    "            hard_clauses.append('{} -{} -{} {} 0\\n'.format(hard_clause_weight, EQ[i,j,k], b[i][k], b[j][k]))\n",
    "            hard_clauses.append('{} -{} {} -{} 0\\n'.format(hard_clause_weight, EQ[i,j,k], b[i][k], b[j][k]))\n",
    "\n",
    "            # SameCluster(i,j)\n",
    "            hard_clauses.append('{} -{} {} 0\\n'.format(hard_clause_weight, S[i,j], EQ[i,j,k]))\n",
    "            same_cluster_hard += '-{} '.format(EQ[i,j,k])\n",
    "        same_cluster_hard += '0\\n'\n",
    "        hard_clauses.append(same_cluster_hard)\n",
    "\n",
    "# Soft clauses\n",
    "soft_clauses= []\n",
    "for i in range(dim-1):\n",
    "    for j in range(i+1, dim):\n",
    "        if W[i,j] > 0:\n",
    "            soft_clauses.append('{} {} 0\\n'.format(int(W[i,j]), S[i,j]))\n",
    "        elif W[i,j] < 0:\n",
    "            soft_clauses.append('{} -{} 0\\n'.format(int(-W[i,j]), S[i,j]))\n",
    "\n",
    "# Write file\n",
    "with open('files/maxsat_binary.wcnf', 'w') as f:\n",
    "    f.write('c Correlation Clustering Max-SAT Binary\\n')\n",
    "    f.write('c\\n')\n",
    "    f.write('p wcnf {} {} {}\\n'.format(var_num, len(hard_clauses) + len(soft_clauses), hard_clause_weight))\n",
    "\n",
    "    # Hard clauses\n",
    "    for clause in hard_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "    # Soft clauses\n",
    "    for clause in soft_clauses:\n",
    "        f.write(clause)\n",
    "\n",
    "hard_clauses = None\n",
    "soft_clauses = None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c Parsing MaxSAT file...\n",
      "c ============================[  Problem Statistics ]============================\n",
      "c |  Number of variables:         20100                                         |\n",
      "c |  Number of clauses:           84147 (incl.         4947 soft in queue)      |\n",
      "c ===============================================================================\n",
      "c Using COMiniSatPS SAT solver by Chanseok Oh (2016)\n",
      "c \u001b[1mOptimal solution: 955\u001b[0m\n",
      "c _______________________________________________________________________________\n",
      "c\n",
      "c restarts               : 1030\n",
      "c conflicts              : 2072           (2600 /sec)\n",
      "c decisions              : 273513         (0.00 % random) (343232 /sec)\n",
      "c propagations           : 4580508        (5748088 /sec)\n",
      "c conflict literals      : 17128          (7.28 % deleted)\n",
      "c Memory used            : 33.40 MB\n",
      "c CPU time               : 0.796875 s\n",
      "c OptExp Enc: Srt/BDD/Add: 28 0 0\n",
      "c _______________________________________________________________________________\n",
      "s OPTIMUM FOUND\n"
     ]
    }
   ],
   "source": [
    "inc_dict = read_solution('files/maxsat_binary.sol')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found 5 clusters\n"
     ]
    }
   ],
   "source": [
    "y_bin = []\n",
    "var_i = 0\n",
    "for i in range(dim):\n",
    "    c = 0\n",
    "    for k in range(a):\n",
    "        var_i += 1\n",
    "        if inc_dict[var_i] == True:\n",
    "            c += 2**k\n",
    "    y_bin.append(c + 1)\n",
    "print('Found {} clusters'.format(len(set(y_bin))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], c=y_bin)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are the results for the test data sets using the binary encoding:\n",
    "\n",
    "| Samples | Time (s) | Memory (MB) |\n",
    "|---------|----------|-------------|\n",
    "| 50      | 0.12     | 16.82       |\n",
    "| 100     | 0.79     | 33.40       |\n",
    "| 200     | 6.06     | 95.45       |\n",
    "| 300     | 12.79    | 210.08      |\n",
    "| 400     | 56.67    | 318.43      |\n",
    "| 500     | 121.76   | 491.46      |\n",
    "| 600     | 229.31   | 729.07      |\n",
    "| 700     | 375.54   | 891.93      |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algorithm Comparison\n",
    "We compare results for each algorithm tested. The X's in the plots indicate the data set size after which I was unable to test further due to memory constraints."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hhyeK0xzvvfceFy5cwGAwMHPmfyPOExMTadq0qcWOK0R28P3Z77kReYMx9cZgZ5NNh4lprXPso06dOvphZ8+efWRZVsuXL5/28PDQUVFRWmutt23bpt3d3fXLL7/8RPsbMGCAnj9/fvL7EydOPLC+Xr16ukmTJnr16tWPtG3evLk+fPhwmvu/evWqrl69+mPXx8fHZyxgM8THx2snJ6dM36/W2ePfgBAPC4kK0Q3WNdDDdgyzdigaOKIf870qZwoW8tJLL/Hbb78BsH79+uTCcwCHDh2iUaNGeHp60qhRIy5cuADA3Llz6du3LwCnTp2iRo0aREVFcevWrQdGRdeqVSv59eXLl4mMjGT69OmZOiK5SZMmTJw4kWbNmrFo0SK2bt1KgwYN8PT0pF27dgQHGyueT5o0iX79+tG8eXPKly/P4sWLAYiIiOCll17C3d2dGjVqJJ99NGnSBB8fH8aNG0dERETyGUxCQgIFCxYE4LXXXuOvv/5KjuWdd95h69atJCQkMGrUKOrXr0+tWrVYuXJlpn1eISxt0fFFxCbEZrsuqA/LpucvmWPWoVmcv3M+U/dZpXAVPq7/cbrb9ezZk2nTptGpUydOnjxJ37592bdvn3EfVaqwd+9e7Ozs2LFjBxMmTODHH3/kww8/pEWLFmzZsoUZM2awfPly8ubNy9ChQ3njjTdYtGgRbdq04b333qNkyZLAfwmnadOmXLhwgeDgYIoWLZpWaGa7d+9eckXVsLAwXnnlFZRSLFu2jDlz5iSX3Lh48SLe3t7cvXuXqlWrMmjQILZt24abmxu///47QPKlrSQzZ85k5cqV+Pj4AMZLbCn/dhs3bqRdu3bExMSwZ88eVq1axYoVKyhatCiHDh0iNjaWF198kXbt2lGmTJlM+bxCWMr5O+f536X/8U61d3BzcrN2OGl6ppOCNdWqVQs/Pz/Wr1//yD2A8PBw3n33XS5duoRSivj4eABsbGz49ttvqVWrFgMHDqRx48YAtG/fnitXrvDHH3/w+++/4+npyenTp3FxcWHDhg1s2bIFGxsbunXrxubNmxk6NHOKzfbs2TP59fXr13n99dcJDAwkNjY2uRQ3GCf4yZUrF0WLFqVw4cKEhIRQq1Ytxo0bx7hx4+jcuXPyZzHHyy+/zOjRo4mPj+e3336jVatW5M6dm7/++otz586xYcMGwPh3vHTpkiQFka1prZl1aBZOuZ0YWOvRWRazm2c6KZjzi96SXnnlFT766CN2795NaGho8vJPPvmEli1bsmXLFvz8/GjRokXyukuXLuHo6MjNmw9OOle4cGHeeust3nrrLTp16sTevXupWLEily5dom3btgDExcVRvnz5NJPCe++9x/HjxylZsiTbtm1LM/58+fIlvx46dCgTJkygY8eO7Nix44GbxLlz505+bWtrS0JCAlWrVuXIkSNs27aNMWPG0KlTJyZMmJD2H8wkb968NG7cmO3bt7Nx40bee+89wPg/15IlS2jd+pHK60JkWzuv7+RI0BEmNpiIU24na4eTLrmnYEF9+/Zl8uTJj8ysFh4eTqlSpQDjtJopl3/wwQfs3buX0NDQ5OvwO3fuTC6bHRERweXLlylTpgzr169n6tSp+Pn54efnx82bN7lx4wbXrl17bEyrV6/Gx8cn3YTwsKSYtdasWbMm3e1v3LiBo6MjvXr1YtSoURw7duyB9XZ2xt8jKS8bpdSzZ09WrVrFP//8Q5s2bQDjGdOSJUuS21y4cIHo6OgMfQ4hslJcYhyzj8ymglMFulfqnn6DbECSggW5urrywQcfPLJ87NixjB8/nsaNG5OYmJi8fOTIkQwZMoRKlSqxatUqxo0bR3BwMEePHqVu3brUqlWLhg0b8v7771OvXj02bNhA165dH9h3165dky+vZKapU6fStWtXmjdvTrFixdLd/sSJE9SrVw8PDw++/PLLVM8S+vXrR61atR7pKgvQoUMHvL296dChA/b29gAMHDiQihUr4uHhQY0aNRg8ePBjk4oQ2cG6c+sIiAxgbL2x2bcL6kOkdLZ45si/AZEd3I6+TactnahTrA6LWy+2djgPkNLZQgiRxRb7LM4RXVAfJklBCCEy2YU7F/jfpf/Rs0pPyjuVt3Y4GSJJQQghMpE2VUHNnys/g9wHpd8gm7FoUlBK+SmlTimlfJRSR0zLCiultiulLpmeC5mWK6XUQqWUr1LqpFIq9UmHhRAiG9vlv4tDgYcY4j4kR3RBfVhWnCm01Fp7pLipMQ7w1lpXBLxN7wFeAiqaHgOApVkQmxBCZJqkLqjlncrTo3IPa4fzRKxx+agLkNTRfQ3waorla031mv4FCiqlSlghPiGEeCLrz6/HP8KfMfXGYG9jb+1wnoilk4IG/lJKHVVKJU0xVExrfQvA9JxUqKcU4J+ibYBpWY6TU0pnA7Ro0YKU3Xr9/PyoUaPGE8UpxPPsTswdlp1YRpNSTWhSqom1w3lilh5N0VhrfVMpVRTYrpRKqzqdSmXZI4MoTMllAJBta97ky5eP06dPEx0dTZ48edi+fXvyCOYnMXnyZNq2bZs8EO7kyZMPrF+/fj316tVjy5Yt9OnTh/bt29O+fXvA+KU/e/Zs6tZNtUuyRSUmJmJra5vlxxXCGhYfX0x0QjRj6o6xdihPxaJnClrrm6bnYGALUB8ISrosZHoONm0eAJRO0dwVeLAAkHFfK7TWdbXWdV1cXCwZ/lPJ6aWzwViCo1u3bnTo0IGKFSsyduzY5HWDBw+mbt26VK9enSlTpiQvd3NzY9q0aTRp0oSZM2dSu/Z//QUuXbpEnTp1MjVGIbKDi2EX+eHSD7xR+Q3KF8xZXVAfZrEzBaVUPsBGax1het0OmAb8DLwLzDQ9bzU1+RkYppTaADQAwpMuMz2pwM8/J/Zc5pbOzl21CsXNKOz2LJTOBvDx8eH48ePkzp2bypUrM3z4cEqXLs2MGTMoXLgwiYmJtG7dmpMnTyYnKwcHB/bv3w/Ajh078PHxwcPDg9WrV9OnT59Mi02I7CCpC6qjvSOD3QdbO5ynZskzhWLAfqXUCeAQ8JvW+g+MyaCtUuoS0Nb0HmAbcAXwBb4GhlgwNotLr3R2jx49qFGjBiNHjuTMmTPAf6Wze/XqRfPmzR8pnd2/f3/Onz+Pp6cnISEhAGzYsIGePXs+UDo7I5R69KpdymWtW7fGyckJBwcHqlWrllxsb9OmTdSuXRtPT0/OnDnD2bNnk9u88cYbya/ff/99Vq9eTWJiIhs3buStt97KUHxCZHd7AvZw8NZBhngMoaBDQWuH89Qsdqagtb4CuKeyPBR4pPaxaYq4zJkIwMScX/SWlBNKZzs7OxMWFpa8/s6dOxQpUiT5fWplsa9evcrs2bM5fPgwhQoVok+fPsTExCRvl7Lk9muvvcann35Kq1atqFOnDs7Ozhn4CwqRvcUnxjP7yGzcCrjxeuXXrR1OppARzRaUE0pnt2jRgu+//56kwohr1qyhZcuWaX6ue/fukS9fPpycnAgKCkqeXS01Dg4OtG/fnsGDByfPiyDEs2L9+fVcu3ctR3dBfZgkBQvKCaWzBwwYQP78+XF3d8fd3Z3IyEg++uijNNu4u7vj6elJ9erV6du3b7qzqr399tsopWjXrp3ZcQmR3YXFhLHsxDIal2xM01JNrR1OppHS2cLiZs+eTXh4OJ999lmWHE/+DYisMP3f6fxw8Qd+6PwDLxR6wdrhZEhapbNzxqwPIsfq2rUrly9fZufOndYORYhMcynsEpsvbub1Sq/nuISQHkkKwqK2bNli7RCEyFRaa7wOe5HPPh9DPHJ0J8lUyT0FIYTIgH039vHPrX8Y7D6YQg6FrB1OppOkIIQQZoo3xON12Au3Am70rNzT2uFYhFw+EkIIM208vxG/e34sarUIe9tnowvqw+RMQQghzHA35i5LTiyhYYmGNHNtZu1wLEaSggXkpNLZffr0oVy5cnh4eFClShU+/fTT5HXvv//+A+UrhHieLTmxhPvx9xlTb0yq5WGeFXL5yAJyWulsLy8vunfvTkxMDNWqVaN3796UK1eOlStXPnHMKSUkJGBnJ//URM51+e5lNl3YRI9KPahYqKK1w7EoOVOwkJxYOjupflFS7aKUE/A4OjoyceJE3N3defHFFwkKCgLgl19+oUGDBnh6etKmTZvk5VOnTmXAgAG0a9eO3r1707RpU3x8fJKP1bhx40eSmxDZldcRL/La5X0mu6A+7Jn++bZv00Vu+0dm6j6LlHak6euV0t0uJ5XOHjNmDNOnT8fX15cRI0ak2v7+/fu8+OKLzJgxg7Fjx/L1118zadIkmjRpwr///otSipUrV/Lll18yZ84cAI4ePcr+/fvJkycPa9as4dtvv2X+/PlcvHiR2NjYB5KbENnVvoB9/H3jbz6q+xGFHQpbOxyLkzMFC8kppbPBePnIx8eHwMBAvL29OXDgwCPb5MqVK/meSJ06dfDz8wMgICCA9u3bU7NmTby8vJI/CxirxObJkweAHj168OuvvxIfH88333wj8yqIHCHeEI/XES/K5C/DW1Wej7Lvz/SZgjm/6C0pJ5TOTsnR0ZEWLVqwf/9+GjVq9MA6e3v75JtrSSW0AYYPH86oUaN45ZVX2L17N1OnTk1uk7KEdt68eWnbti1bt25l06ZNPFyzSojsaNOFTVwNv8rClguf2S6oD5MzBQvKCaWzU0pISODgwYNUqFDB7M+Y8rOsWbMmzW3ff/99RowYQb169Shc+Nk/DRc5W3hsOEt8ltCgRANalG5h7XCyjCQFC8oJpbPBeE/Bw8ODWrVqUbNmTbp162Z226lTp9KjRw+aNm36wOQ8qalTpw4FChSQeRVEjrD0xFIi4yMZU/fZ7oL6MCmdLbLMzZs3adGiBefPn8fGxnK/R+TfgHhaV+5eodvP3ehWsRuTG062djiZLq3S2XKmILLE2rVradCgATNmzLBoQhAiM8w+Mps8dnkY6pGpMwRnmjg/Pyz1g17+7xRZonfv3vj7+9OjRw9rhyJEmvbf2M++G/sYWGsgznmy35zikfv/5kqXVwlbu9Yi+38mk0JOviQmno78txdPI8GQgNdhL0rnL81bVbNfF9TIffsIGDKEXG5uFOjc2SLHeOaSgoODA6GhofLl8BzSWhMaGoqDg4O1QxE51OaLm7kSfoXRdUeTyzaXtcN5QOSePQQMGUquChUo8+1q7CzUg8/i4xSUUrbAEeCG1rqTUqocsAEoDBwDemmt45RSuYG1QB0gFHhDa+2X0eO5uroSEBCQPLhLPF8cHBweKAkihLnCY8NZ7LOY+sXr06p0K2uH84CInbsI+OADHCpWpMw3q7AtWNBix8qKwWsfAOeAAqb3s4B5WusNSqllQD9gqek5TGv9glKqp2m7NzJ6MHt7e8qVK5c5kQshnhvLTizjXuw9xtYbm626oEbs2EHAyFE4VKlCmZVfY+vkZNHjWfTykVLKFXgZWGl6r4BWwA+mTdYAr5pedzG9x7S+tcpO/2WEEM+sq+FX2XB+A90qdqNy4crWDifZvT//IuDDkThUq0qZVSstnhDA8vcU5gNjAYPpvTNwV2udYHofACTVlC4F+AOY1oebtn+AUmqAUuqIUuqIXCISQmSGOUfmkNsuN8M8h1k7lGT3fv+dG6NGkadmTcqsWoVtgQLpN8oEFksKSqlOQLDW+mjKxalsqs1Y998CrVdoretqreu6uLhkQqRCiOfZgRsH2BOwhwG1BlAkT9qj8rNK+K+/cWP0R+Tx8KD0119j6+iYZce25D2FxsArSqmOgAPGewrzgYJKKTvT2YArkFT5LQAoDQQopewAJ+COBeMTQjznEgwJeB3xwtXRlXeqvmPtcAAI//lnbo4bT97atSm9fBk2KQpLZgWLnSlorcdrrV211m5AT2Cn1vptYBfQ3bTZu8BW0+ufTe8xrd+ppV+pEMKCfrz4I753fbNNF9S7W37i5sfjyFuvHqVXLM/yhADWGafwMTBKKeWL8Z7BKtPyVYCzafkoYJwVYhNCPCfuxd1jkc8i6harS+syra0dDnd//JFbEyaQr+GLlF62FJu8ea0SR5bMp6C13g3sNr2+AtRPZZsYQGogCCGyxPITywmPDc8WXZabTJkAACAASURBVFDDNm4icMoU8jVujOviRdhYcQBmuknB1K20J9AUKAlEA6eB34DftdaGNJoLIUS2s+v6Lr47+x3dKnajqrN1K+qGrV9P4KfTyNesKa5ffYVN7tzptrl0JAi3WkWwz2Wb6fGkeflIKbUa+AaIwziY7E1gCLAD6ADsV0o1y/SohBDCQs6EnuHjfR9T3bk6H9f/2Kqx3PnuewI/nYZjixa4LlqUbkLQWnNk21X+WnmGkzv9LRJTemcKc7TWp1NZfhr4n1IqF1Am88MSQojMF3g/kOHewymYuyBftf6KPHZ5rBbLnTVrCPpiJo6tW+M6by4qV9o3urXW/LPlMsf/uk7lF4vj2dYyX71pJoXUEoJSqhBQWmt9UmsdB/haJDIhhMhEkXGRDPEeQnRCNN+99J1VxySEfrOa4C+/JH/btpSaMzv9hGDQ7N14kdN7blCjWSma9ayEsrHMfRCzbjQrpXYDr5i29wFClFJ7tNajLBKVEEJkogRDAh/t+Yird6+ypM0SXij0gtViuf3114TMmUv+Dh0o5fUlyt4+ze0NiQZ2fXee8/8G4tm2DA27VbDojXFzu6Q6aa3vAd2A1VrrOkAbi0UlhBCZRGvN5wc/5++bfzPpxUk0LNnQarHcXraMkDlzKdCxI6Vme6WbEBITDPy16gzn/w2kfudyFk8IYH5SsFNKlQBeB361YDxCCJGp1pxZw+aLm+lXox+vVXrNanGELF5MyPwFFOjcmZJfzkLZpX2hJiEukd+XneLysRAad3+Bei+Xy5Kus+aOU5gG/Ans11ofVkqVBy5ZLiwhhHh6269tZ87RObR3a8+I2iOsEoPWmttffcXtJUtx6tKFEp/PQNmm3ZU0LiaBbUtPcuPiXVq8XZnqTUuluX1mMispaK03A5tTvL8CWC/lCiFEOk6GnGT8vvG4u7gzvfF0bFTWF3DQWhMyfwGhy5fj1K0bJT6blm5CiLkfz6+LThB8LYI2fapRuUHxLIrWKL1xCpOUUo+d800p1cpUDVUIIbKNG5E3GL5zOC55XFjYaiEOdlk/QlhrTcjcuYQuX07BHt0pMf2zdBNCdEQcW+cfJ8Q/gg79a2R5QoD0zxROAb8opWIwTp0ZgrHiaUXAA+Mgts8tGqEQQmTAvbh7DNkxhHhDPIvbLKawg2XmMk6L1prgL724s3o1BXu+QfHJk1E2aZ+pRIbF8vOC40SExvDy4FqUqf7IdDJZIr1xCluBrUqpihhLYZcA7gHfAwO01tGWD1EIIcwTnxjPqF2juB5xnRVtV1DeqXyWx6C1JuiLLwhb+x2F3nqLYp9MSvcG8b3b0Wydf5zoyHg6j3CnZMVCWRTto8y9p3AJubEshMjGtNZ89u9nHAw8yIwmM6hXvJ5VYgiaPoOwdeso1LsXxcaPTzchhAXeZ+t8HxLiEunyoSfF3LJmhrXHyZIqqUIIYWmrTq9ii+8WBrkP4pUKr2T58bXBQOBnn3F3/QYK9+lD0Y/Tr756OyCCnxf4APDqqNoUcc26GdYeR5KCECLH+/3q7yw4toCO5ToyxH1Ilh9fGwwETpnK3c2bcX6/Hy6jR6ebEAKvhvPrVyewz21Llw89KVjMOvMnPEySghAiRzsefJxJ+ydRu2htPmv8WZbPjaANBm598gnhP/4P5wEDcBn5Ybox3LgQxm9LTpKnQC66fOBBgSLWK8z3MLM67iqlKimlvJVSp03vaymlJlk2NCGESNv1e9cZsXMEJRxLsKDlgiyfUlMnJnJrwkTCf/wfRYYMNishXDsTyi+LTuBY2IFuo2tnq4QA5pe5+BoYD8QDaK1PYpx4RwghrOJuzF2Geg8FYEnrJRR0KJilx9cJCdwcN57wn36iyLBhuIwYkW5CuHw8mG1LTlKoeF66jvYkX8H0J9TJauZePsqrtT700AdOsEA8QgiRrrjEOD7c/SE3Im+wst1KyhTI2mlddEICN8d+zL1t23D5YARFBg9Ot82Fg4F4rzlHMbf8dBrmTu68aRfDsxZzk8JtpVQFQAMopboDtywWlRBCPIbWmikHpnA06Cizms6idrHaWXv8+HhujBlLxB9/4DJqFEUG9E+3zem9N9iz/gKlKhWi4+Ca5HLIvrdzzY1sKLACqKKUugFcBd6xWFRCCPEYy04s49crvzLcczgdy3fM0mPruDhujP6IiO3bKTpmDM79+qbb5vj26xz40ZeyNZ3pMKAGdvaZP69yZjJ38NoVoI1SKh9go7WOSK+NUsoB2AvkNh3nB631FKVUOWADUBhj6YxeWus4pVRuYC1QBwgF3tBa+z3BZxJCPKN+ufwLS04soUuFLvSvmf4v9Myk4+IIGDmKSG9vio77GOc+fdLeXmsO/+bH4V+vUqF2Udr2rYatXdYX5csoc2deKwj0Btwwzq0AgNY6rVq0sUArrXWkUsoe2K+U+h0YBczTWm9QSi0D+gFLTc9hWusXlFI9gVnAG0/2sYQQz5rDgYeZfGAyDYo3YErDKVna9dQQF8eNDz4kctcuik2cSOFeaV8o0Vpz4EdffHb4U6VhcVr2qoqNhabPzGzmpq1tGBPCKeBoisdjaaNI01t700MDrYAfTMvXAK+aXncxvce0vrXK6g7HQohs6Wr4VT7c9SFl8pdhbsu52Ntm3U1aQ2wsAcOHGxPC5E/STwgGzZ71F/HZ4U/NFq60ykEJAcy/p+DwJPMxK6VsMSaPF4DFwGXgrtY6qedSAJA0e0QpwB9Aa52glAoHnIHbD+1zADAAoEyZrO1xIITIendi7jBkxxDsbOxY3HoxBXJlXW0gQ0wMAcOGc3//fop/+imF3ng97e0TDexce54LBwOp3b4sL75aPssH0z0tc88UvlNK9VdKlVBKFU56pNdIa52otfYAXIH6QNXUNjM9p/aX048s0HqF1rqu1rqui4uLmeELIXKi2MRYRuwcQUh0CAtbLcQ1v2uWHdsQHU3AkKHc//tvSkz/LN2EkBhv4M+VZ7hwMJAGXcrTsKvl51O2BHPPFOIAL2Ai/31Ra8CsurRa67tKqd3Ai0BBpZSd6WzBFbhp2iwAKA0EKKXsACfgjpnxCSGeMQZtYNL+SZwIOcGc5nNwd3HPumNHReE/ZChRBw9SYsYMCnbrmub28XGJ/LH8FNfP3KFJj4q4ty6dRZFmPnPPFEYBL2it3bTW5UyPNBOCUsrFdIMapVQeoA1wDtgFdDdt9i6w1fT6Z9N7TOt3aq0fOVMQQjwfFh1fxB9+fzCqzijaubXLsuMa7t/Hf+Agog4douTML9JNCHExCfz61Qmun71Dy15VcnRCAPPPFM4AURncdwlgjem+gg2wSWv9q1LqLLBBKTUdOA6sMm2/CuNlKl+MZwhSRkOI59SWS1v4+tTXdK/UnT7V+2TZcRMj7+M/aCDRx45TctYsnDqnPdtwzP14fvnqBCHXI2jbtxqV6mX99JmZzdykkAj4KKV2YexqCqTdJdVUH8kzleVXMN5feHh5DNDDzHiEEM+of27+w7R/ptGoZCMmNJiQZdflEyMj8e8/gOiTJyk124sCHdMeGBd1L46fF/gQFnSfDgNqUN7j2bjHaW5S+Mn0EEIIi/EN82XU7lG4Obkxu/ls7G2yputpYkQE199/n5gzZyk1Zw4FOrRPc/vIsBi2zvchMiyGTkPcKV0t6+eBthRzRzSvSX8rIYR4crejbzPUeygOdg4sab2E/LnyZ8lxE+/d43q/94k5d45S8+ZSoG3bNLcPD4li63wfYu/H03mEByVfyNrqrJaWZlJQSm3SWr+ulDpF6t1Da1ksMiHEcyM6IZrh3sMJiw1jdYfVlHAskSXHTbx715gQLl7EdeEC8rdqleb2d27d5+f5x0lIMNBlpCdFy1p3PmVLSO9M4QPTc9p3W4QQ4gkZtIHx+8ZzJvQM81vOp7pz9Sw5bkJYGNf79iPO1xfXrxaSv0WLNLcP8TfOp2xjo+g6qjbOpaw/n7IlpNklVWudVB57iNb6WsoHkPUToQohnjnzjs7D+7o3Y+qNoVWZtH+pZ5ao48e59vY7xF2+jOviRekmhMAr4Wyddxw7exu6jn52EwKYP04htYtsL2VmIEKI58+mC5v49sy3vFnlTd6pavlq/Inh4dyaPIVrb76FISqK0iuW49isWZptAi6EsXWBDw757Ok2pg4Fi+W1eJzWlN49hcEYzwjKK6VOpliVH/jbkoEJIbKx/fOhVG0ol+IL9epeuHEMmnxo1i72BexjxsEZNHNtxth6Yy3a9VRrzb1ffyNo5kwSw8Io/O67FBk+HFvHfGm28zt1mz9WnMbJJQ+vfOBBPqfsN31mZkvvnsL/Ab8DXwDjUiyP0FpLCQohnlelasPmPtDjW2NiuLr3v/dmuHDnAh/t+YhKhSrh1cwLOxvLzUQWd+0agZ9O4/6BAzjUrEmZr1fgUK1auu18jwaz/ZszOJdy5JURHjg4Zs/pMzNbmv8ltNbhQDjwZtaEI4TIEco1MyaA9W9C2cZw48h/CSIdwVHBDPUeimMuRxa1WkRee8tcjjHExRG6ciWhy5ajcuWi2CeTKNSzJ8o2/ZnPzv9zi51rz1G8ghMvD3Und57sO31mZnt+PqkQIvNoDTeOQlwkXPoTmo4xKyFExUcxzHsYEXERrH1pLcXyFbNIePcPHSJw6qfEXblC/g4dKDZ+PPbFiprV9tTuAPZuuIhrlUJ0HFwL+9zZe/rMzCZJQQiRMQYD/DkBDi4F21zQcDgc/QbKN0szMSQaEhm7dywXwi7wVauvqFy4cqaHlhAWRvCXXoRv2YJ9qVJm3UhO6dif1/hny2XcahWhff/q2X4+ZUuQpCCEMF9CLGwZCGe2gJ0DvLkJKjSHCi0evMeQCq8jXuwJ2MPEBhNp5mr+F7U5tNaE/28LwV5eJEZG4ty/P0WGDMYmTx6z2x/65SpHtvlRsW5RWr9XDVvb7D+fsiVIUhBCmCfmHmx823hTuWI7aDgMyjc3rku6x3DjWKpJYd25daw7t45e1XrRs0rmFkCOvXKFwClTiTp8mDyenhT/dCoOlSqZ3V5rzd8/+HLC25+qjUvQ4u0qOWr6zMwmSUEIkb6IIFj3GgSfg67LwT2VL/ZyqV8+2nV9F7MOzaJV6VaMrjM600IyxMRwe/lyQleuwiZvXopP+5SC3bujbMz/hW8waPb83wXO7r9JrZauNOlREfUcJwSQpCCESM9tX/i+G9y/DW9uhIptzG56JvQMH+/7mGrO1fii6RfY2mTONfrIv/8m8NNpxF+/ToFXOlPs44+xc3bO0D4SEw14f3uOS4eDqPNSWRq8kvPmU7YESQpCiMe7cRTWmaY56fMLlKpjdtPA+4EM9x5OwdwFWdQ6c7qeJty+TdDMWdz79VdylS1LmW9Wka9RowzvJ+Z+PN7fnsXvVCgvvlqeOh3cnjq2Z4UkBSFE6i7tgE29IJ8L9NoCzhXMbhoZF8kQ7yFEJ0Sz9qW1FMlT5KlC0QYDdzdtJnjuXHR0NEWGDMF54ABscmd8hLH/2Tt4rzlLdGQ8zd+sRI3mrk8V27NGkoIQ4lEnNsDWoVC0Krz9I+Q3fzxBgiGBj/Z8xJW7V1jSZgkVC1V8qlBiLlwkcMoUon18yFu/PsWnTiV3+XIZ3k9CXCL//HSZkzsDKFQ8Ly8PdcelTNbM2ZCTSFIQQvxHa/h7AeyYYrxp/MY6cDB/zgCtNZ8f/Jy/b/7NlIZTaFQy45d2khiiori9ZAmh367BNn9+Ssz8AqcuXZ7oun+IfwTbvzlL2K371GrpSsOuFbDL9fyNQTCHJAUhhJHBAH9NhH+XQPVu0HUZ2GXs8szas2vZfHEzfWv0pXul7k8cSsTu3QRN+4z4mzdxeq0bRT/6CLtChTK8H4NB47P9Ogd/voKDoz2dR7hTplrGbkg/byQpCCGMg9J+Ggynf4QGg6H955CBrp0A269tZ86RObQr244Pan+QfoNUxAcFE/T550T8+Se5KlSg7Pffkbdu3Sfa173b0ez49iy3fMOpUNuFFm9VeW6K2j0NSQpCPO9i7sHGd+DqHmjzKTT+ADJ4ieZkyEnG7xtPTZeazGgyAxuVsYSiExMJ+7/1hMyfj05IwOXDD3Hu+x4qV64M7QeMl7AuHgxk74aLaKB1n6pUblBcupuayWJJQSlVGlgLFAcMwAqt9QKlVGFgI+AG+AGva63DlPG/2AKgIxAF9NFaH7NUfEIITIPSukPQGXh1GXhkvCDyjcgbDN85nCJ5irCw5UIc7Bwy1D76zBkCp0wl5vRp8jVuTPEpk8lVpkyG4wBjV9Pd6y5w+VgwJV5wok2fahQoYl6pC2FkyTOFBGC01vqYUio/cFQptR3oA3hrrWcqpcZhnKfhY4wzuVU0PRoAS03PQghLCL0M33WF+yHw1kaomNoEi2m7F3ePITuGEG+IZ3Xr1TjnMf96fWLkfW5/tZA7332PbeHClJwzmwIdOz7xL/qUXU1ffLU8nu3KPtflKp6UxZKCaX7nW6bXEUqpc0ApoAvQwrTZGmA3xqTQBVirtdbAv0qpgkqpEinmiRZCZJYbR2Hd64CGd38FV/MHpSUJjgpm5K6RXI+4zoq2KyhfsLxZ7bTWROzYQdCMz0kICqJgzzcoOnIktgXM7+WUknQ1zVxZck9BKeUGeAIHgWJJX/Ra61tKqaQi56UA/xTNAkzLHkgKSqkBwACAMk94iinEc813B2zsDfmc4Z0tUOSFDO/icOBhxuwZQ1RCFF7NvKhXvJ5Z7eJv3iTws+lE7tpF7sqVcZ0/jzweHhk+fpKUXU1rtnSlkXQ1fWoWTwpKKUfgR+BDrfW9NE4NU1uhH1mg9QpgBUDdunUfWS+ESMOJjbB1CLhUhXd+gPzFM9Rca83as2uZd3QepfOXZmW7lbxQKP2kohMSuLP2O0IWLQKtKTpmDIV790LZP1lvoEe6mg53p0x16WqaGSyaFJRS9hgTwjqt9f9Mi4OSLgsppUoAwablAUDpFM1dgZuWjE+I58rfC2H7J6ZBad+Dg1OGmkfFRzH5wGT+9PuT1mVaM73xdBxzOabbLvrECW5NmUrs+fM4tmxJ8UkTsS9V6kk/BfdCo/H+9hw3L92VrqYWYMneRwpYBZzTWs9Nsepn4F1gpul5a4rlw5RSGzDeYA6X+wlCZAKDAf6aBP8uhupdjaWvMzgo7Ur4FUbuGonfPT9G1hnJe9XfS/eGcOK9ewTPm8fdDRuxK1qUUl8tJH+bNk98I1m6mmYNS54pNAZ6AaeUUj6mZRMwJoNNSql+wHXAVIKRbRi7o/pi7JL6ngVjE+L5kBBnGpT2AzQYBO2/yPCgtB3XdjDp70nkssnFirYraFAi7U6BWmsifv+dwC++IDH0DoV6vYPLiA+wdcz3xB9DuppmHUv2PtpP6vcJAFqnsr0GhloqHiGeO7ERxkFpV3ZD6ynQZGSGBqUlGBJYeHwhq0+vpmaRmsxtMZfi+dK+BxHn70/gp9O4v38/DtWrU3rpMvLUqP5UH8P/3B28v5WupllFRjQL8SyKDDYOSgs8DV2WgOfbGWoeGh3K2L1jORR4iNcrvc7H9T8ml+3jRxfruDhCv1nN7aVLUXZ2FJs4kUJvvYmyffKeQNLV1DokKQjxrAm9bJwpLTL4iQalnQg5wajdowiPDeezxp/x6guvprl91NGj3JoyhTjfy+Rv145iEydgX8z8Utupka6m1iNJQYhnyY1jxpnStAHe/QVczS8mp7Vm04VNzDw8k2J5i/HdS99R1bnqY7dPjIgg+Esv7m7ejH3JkrguW0r+Fi2eKnzpamp9khSEeFb4esPGXpDXGXr9D4qYP7lNdEI00/+dzs+Xf6ZJqSbMbDoTp9yP77IauWcPtyZPISEkhMJ9++IybCg2eZ9uus0Hupp6utDibelqag2SFIR4FpzcZOxl5FIV3t4MBUqY3dQ/wp+Ru0ZyMewiQ9yHMNB94GOrnCbevUvQFzMJ37qV3BVfwHXRV+SpWfOpQn+kq+m7Van8onQ1tRZJCkLkdAe+Mo5DcGsKPddlaFDa3oC9jNs3DoBFrRfRzLXZY7eN2LGDW59+SuKdMJwHD6LI4MHYPEFp65Skq2n2I0lBiJzKYDCOUP5nEVR7FbqtMHtQmkEbWHZiGUtPLKVK4SrMbTGX0vlLp7ptQlgYQZ9N5962beSuWpUyy5fjUK3aU4ef3NU0QrqaZieSFITIiRLijDWMTm2G+gOgw0ywMa93TnhsOOP2jWP/jf28UuEVJr04iTx2j/4611oT8ccfBH42ncSICIqMGE6R/v2fuF7Rf6FLV9PsTJKCEDlNbITxhvKVXdB6MjQZZfagtHOh5xi5eyRBUUF88uIn9KjUI9Vr9wm3bxP46TQitm/HoUYNynw+A4dKlZ46dOlqmv1JUhAiJ4kMNnY5DTyV4UFpP/n+xPR/p1Mwd0HWdFhDLZdaj2yjtebeL78QNONzDNHRuIwehfN776Hsnu6rIjt1NV225zK1XJ1oVKFI8rIDl29zMiCcQc0rWCWm7ESSghA5xZ0r8F03iAiEN9dDpfZmNYtLjGPmoZlsvriZ+sXr82WzL1OdIS0+KIjAKVOJ3L2bPB4elPh8BrnLmzdxTlqyW1fTWq5ODPu/4yx6y5NGFYpw4PLt5PdCkoIQOcPN48YzBEOCcVBaafMmtQm8H8io3aM4dfsUfWv0ZbjncOxsHvzfXmtN+P+2EDRzJjo+nmLjx1HonXeeqkRF0n6zY1fTRhWKMLtHLfqtOULbqsXY73s7OUEISQpCZH+XdxrvIeQpDO/8CC7mXdv/99a/jN0zljhDHPNazKNN2TaPbBN/8ya3Jk/h/v795K1blxIzppOrbNmnDvmBrqYVnGjzXvboahoeHc/aA36s+vsq0XGJ/HziJiNavSAJIQVJCkJkZyc3GwelFalkTAhmDErTWvPN6W9YeHwh5QqUY17LeZRzKvfgNgYDdzdtJtjLC601xSZ/QqGePVEZLKudmuzY1fTO/Ti+2X+VNQf8iIhNoHaZgvgGR9KnkRvfH7zOixWcJTGYSFIQIrs6sAj+mghlmxgHpeUpmG6TiLgIPvn7E7yve9PerT3TGk0jr/2D5Sfi/P25NekTog4eJF+jhhSf9hm5XJ98JrQk2bGraXBEDCv3XeX7f68RHZ9IxxolaPyCM7P/usiyXnVoVKEIL1ZwfuAew/NOkoIQ2Y3BADsmG0cqV+sCXVeAvUO6zXzDfBm5eyT+Ef6MqTuGXtV6PXD9XhsMhK37P4LnzkXZ2lL8s2kU7N49U67xZ7eupjfvRrN8z2XWH/YnIdFAF49SDGlRgYrF8rNsz+UHEkCjCkVY9JYnJwPCJSkAyji3Tc5Ut25dfeTIEWuHIUTmSYiDrUPh1Cao1x9emmXWoLTfr/7OlANTyGuXl9nNZ1O3+IPVUeP8/Lg5cRLRR4+Sr1lTSnz6KfYlzK+P9DgPdzVt3buqVauaXg+NYsluX348FgDAa7VdGdS8Am5FnnzWt2eRUuqo1jrVErpypiBEdhEbCZt6GW8st/oEmo5Od1BavCGeuUfm8v257/Fw8WBOizkUzVs0eb1OTOTOmrWELFiAyp2bEl98gdOrXZ767CAhPpFLh4M5ucuf2/6RVu9q6hscwZJdl9l64ia2Noq36pdhQPMKlCpo/ZvbOY0kBSGyg8gQ+L8ecOskvLIIavdKt0lIVAgf7fmIY8HHeLvq24yuMxp72/++lGN9fbk5cSIxJ07i2Lo1xadMxr5o0TT2mL6IOzGc3nODs/tvEnM/nkIl8tHmvWpUql/MKl1Nz968x+Jdvmw7fQsHO1v6Nnajf9PyFC2Q/uU2kTpJCkJYW8gFWN8T7t2Cnv8HlTuk2+RY0DFG7xnN/fj7zGw6k5fLv5y8TickELrqG24vWoRNvnyUnD2bAi93fOIvba01Ny6EcWr3Da6eCAGgnLsLNVuUolTlQlZJBj7+d1m08xI7zgWTP7cdQ1u8QN8m5Sic7+mqtgpJCkJYz93rsGcW+Pyfsdz1uz9D6fppNtFas+7cOuYcmUNJx5Isb7ucSoX+G7cQc+ECt8ZPIObsWfJ36EDxTyZh5/xk1/jjYhK4eDCQk7tvEHbrPg757PFsV5bqzUpSwNk6l2UOXb3DVzsvse/SbQrmtWdU20q828gNpzwyGU9msVhSUEp9A3QCgrXWNUzLCgMbATfAD3hdax2mjD81FgAdgSigj9b6mKViE8KqIoJg3xw4uhpQ0GCQsaido0uazaLio5j6z1R+v/o7LUq3YEaTGRTIVQAAHRfH7eUruL18ObZOTpRasIAC7ds9UXh3g6I4tSeA8wduEReTiEuZ/LTqXZWKdYtapUeR1pq/fUNZuPMSh67eoYhjLsa/VIW3XyyLY275XZvZLPkX/RZYBKxNsWwc4K21nqmUGmd6/zHwElDR9GgALDU9C/HsiLoDBxbCweWQEAue70DzseDkmm5Tv3A/Ru4eyeW7lxnhOYJ+Nfslz44WffoMtyZOJPbCBQp07kyxCeOxK1QoQ6Fpg+bamVBO7Q7g+pk72NgqKtQuSq2WrhQrV8Aql4i01uw8H8xXO33x8b9L8QIOTO1cjZ71y+BgL5VVLcViSUFrvVcp5fbQ4i5AC9PrNcBujEmhC7BWG/vH/quUKqiUKqG1vmWp+ITIMrER8O9S47iD2Aio2R1ajAdn8ypy7ry+k4n7J2JnY8eyNstoVKoRAIa4OG4vWkzoqlXYFS6M65LF5G/VKmOhRcVz7sAtTu25wb2QaPI65aJ+53JUa1KSfE7mTdiT2QwGzR9nAvlqpy/nbt3DtVAePu9ak9fqlCK3nSQDS8vqc69iSV/0WutbSqmkrhClAP8U2wWYlj2SFJRSA4ABAGXKlLFstEI8jfhoOLwK9s+FqFCo0glaToBi1c1qnmhIZJHPIlaeWkl15+rMbTGXko4lAYg+cYKbEyYSd/kyTt26UWzcx9gWKGB2aKE39xl6jQAAGhdJREFUIjm1O4ALBwNJiDNQooITL3YpT3kPF2ztnr7UxZNISDTwy8mbLN51Gd/gSMq75GNOD3de8SiJva11YnoeZZcLcqmdm6Y6qk5rvQJYAcbBa5YMSognkhgPx7+DPV4QcRPKtzSOO3CtY/Yu7sTc4eO9H/PvrX95reJrjG8wnty2uTHExBCyYCF31qzBrlgxSn+9AsemTc3apyHRwNUTtzm1O4AbF+9ia29DpXrFqNnC1arlKOISDPx/e3ceHVd1H3D8+5tFo82WNJLlRfIiWfK+gvGCsWPA6QEHMElY07RAk3JOYtokNM1SaBLaNIRTTsmeQmloGhpIQhJkOwRIsB3AKwYcrxhrsyVZ1kga7TOjWd7tH+9pLG/YGEsj2b/POe+89+7cN/pZHt3fvPveu/e3b9fzo01VHG4NMW3MCH7wiflcP2ssbp2ec9ANdlJo6usWEpGxQMAprwf6TxBbDBwd5NiU+mCsBOx5DjZ9C9pqYfwie97kknNrtPvsad7D/X+6n2A4yENXPsTHyj8GQOjNN2n8pweIHj5M7u23U/iPX8SdnX3W9wt1Rtm/+Sj7Xm2gu62XEf50lnx0MjOWjkvpvAaRWIJf7qzjPzdVcbQjwpziHJ74q8tZOX10ygfQu5QNdlJYC9wFfNtZV/Qrv09EnsW+wNyh1xPUsGEMvLMeNvwbNB+AMbPhE7+C8g+f8zSZ9tsYnjv0HA9vf5hRGaP431X/y8z8mVihEIHHvkPb00/jHTeOCU/9hKwlS876fk21nezZVM+hnU1YcUPxtDyW3T6FSXMKUtrohqJxfr79CI+/Wk1zVy8LJubx8MfnsLy8IOVzLaiBvSX1GeyLygUiUg98HTsZ/FJEPgUcAW51qr+AfTtqJfYtqfcMVFxKXTDG2ENSbPhXexKc/HK49X9g+mp4n0NQR+IRvrX9W/y28rdcOe5KHln2CLnpufRs207jgw8Sq68n75OfpPALn8eVdeZxfBIxi8q3AuzZVE9TTSden5uZS8cxa0Ux/rGpHf+nMxLjZ1sP8+Rr1bSFYiwty+d7d8xncalfk8EQMpB3H915hpeuPU1dA6wZqFiUuuAOb7WTweHNkDPBni95zu3gfv9/UgeDB/nnzf/MgeAB7p1zL5+d+1kIRWj8xjdof/YXeCdOYOLTPyNzwWnHLwOgu62Xfa81sO+1BsJdMXJHZ7Ls9nKmLR5LWkZqLx229UR5anMNT22ppSsS55pphay5uozLJ76/22bV4BgqF5qVGh6O7oIN34TKP0D2aFj1KFz21+B5f7dvtkXaeKHmBSoqKzgQPMAI7wi+f833WTF+Bd2vvU7j179GvPEY/nvuYdTf/x2ujFOfIDbG0FjZwe6N9VTvasYYw6TZBcxeUcT4aX4kxf3yzV29PPl6NU9vPUxPNMF1M8dw3zVlzCrKSWlc6r1pUlDqXDQfhI3/BvsrICMPVj4EC++FtMyzH+uIWTG2NGyhoqqCjXUbiVtxpvun85WFX2FVySpGRt0cfeABOn79G9JKS5n0zM/JmDfv1PeJJji0o4ndm+ppre/Gl+lh7rXjmbW8iJxRqR8VtLEjzON/quaZHUeIJSxunDuONVeXMWV0aifcUedGk4JS76WtFjY9ArufBW8mfOjLsGSNPVbROTrUdoiKygrWV6+nNdKKP93PndPuZPXk1Uz1TwWga+NGqr/+DeKtreT/7d9ScN8aXL4Tzz46msPsfbWBA5uP0huKk1+UzdWfnEb5wtF4UzihTZ+6YIgf/6mK53bWYxnDxy4r4jMryijRuQyGFU0KSp1OZyO89ii8+VN7kpsla2DpFyDr3AaXa4+0291DVRXsb92PRzx8aPyHWD15NVcVX4XX5SXR1UX7r39Dx9q1hLZvxzdlCsU//CEZs2cl38dYhrp3guzZWE/t3lZEhMnzRzF7RTFjy3KGxAXaquZufrSxiud3NeAW4bYr7IltivPO/SxKDR2aFJTqLxSE1x+DHU+AFYfL7oLlX4SR4856aNyKs+XoFp6vfP6U7qHrS67Hn+7HRKN0b3qNprXr6N6wARON4p04gVH/cD/5d92FpNlDP0fDcd7Z1sieTQ20N4XIGOFlwfWTmLmsiOy81Aw/0Scat9jT0M626iDbqlt5vbIFn8fF3VdO4t7lpYzWuQyGNU0KSgFEOmHbj2DLDyDaDXPvsLuK/CVnPbSyrZKKqgrWVa2jNdJKni+PO6bewc1lNzPVPxVjDOG3d9G47vt0vfB7Eh0duPPyyL31VnJuupH0OXOS3/iDjT3s3VTPO9uOEetNMLpkJCvvmUHZZYW4vakZ6iESS7Crrp3t1UG217Ty1pE2IjELgPLCbD67YjL3LC2hIDu1yUpdGJoU1KUtGoI3/gte/w6EgzD9Jrj6ASic9p6HdfR2JO8e2te6D494WF68nNVlq1lWtAyv20tvdTWBn32XznXridXXI+npjLj2WkbeeAPZS5ciXvtpYithUbPHHn6i/p02XB6hfIE9/MToSec+ntGFEorGefNwG9urg+yoCbKrrp1owkIEpo8ZyZ0LJ7CoxM8Vk/zkayK46GhSUJemeBTe+im8+ih0H4OylXDNgzBu/pkP6dc9tKluEzErxtS8qXz5ii+zqnQV/nQ/8ZYWOv/vGTrWriOydy+4XGQtXkzBfWsYsfLDuLPti66hzihHdjZyeG8rdQeC9IbiZOf5WLS6lJlXjSNjxODNINYZifFmbRvbalrZXh1kb0MHccvgdgmzxo3k7qWTWFTiZ8FEPzmZOpnNxU6Tgrq0WAnY/QvY9LA989mEK+HWp2DilWc8pKq9iorKCtZVr6Ml3EKeL4/bp97O6rLVTPNPw+rpoevlVziybj09W7ZAIkH6jBkUfuXLjFy1Cm9hIVbCoqmmk8P7qjiyL0jzkS4AMkemUTK3gJK5o5g0Ox/XIIwG2tYTZUdt0D4TqG1l/9FOLANetzC3OJd7l5eyqDSfyyfm6SQ2lyD9H1eXBsuCA2vtZw1a3oWx8+CGx2Dytacdn6ijt4Pf1/yeisoK9rbuxSMelhUvY3XZapYXLcdjhJ6tW2lY+xO6/vhHTDiMd9w48j/9aXJuvAFfWRmhzihV+1o5XLE3eTYgLmFM6UgWrS5l4sx8CoqzB/whs+auXnbU2NcDtlcHOdhkJySfx8X8Cbn83TXlLCr1M398HhlD4NZWlVqaFNTFzRio/CO88i9wbDcUTIXbfgbTbzwlGcStOFuPbqWiqoINRzYQs2JMyZvCl674EqtK7O6hyN69tD7973T+7gUSra24cnLIuekmOxHMnUfgcDdv72vlyC/fOOVsYOKsAoqn5ZGeNbBdMI0d4eRF4e01Qaqbe+w40txcPjGPG+eOZVFpPnOKc3TSGnUKTQrq4lW72U4GddsgdyJ89HGYfav93EE/Ve1VVFRVsL5qPc3hZnJ9udw29TZuLruZaf5pROvq6HjqF1SvXUe0thbxesm++mpybroRmbeY+kNd7H67lbqfbxn0swFjDHXBcDIBbK9ppS4YBmBEuocrJvm5fcF4Fpb4mVWUo5PVqLPSpKAuPg1v2YPVVW2AEWPhI/8B8/8KPMcv3nb0dvBizYtUVFWwp2UPbnGzrHgZN0++meXFy5HObrpefJHatf9C+O23AchcuJDce/6G8PQrqa+JsGVHkObndtivjUyjZN4oJs7MH9CzAWMM1S09yTOBHTVBGjsiAORlellY4ufuK0tYVOJn+tiROkmNet/EHqB0eFqwYIHZuXNnqsNQQ0XggD1Y3TvrIcMPy+6HKz4NXns8oISVYGvjVioq7e6hqBWlPK+cmyffbN89JNl0b9xIx9p1dL/2GsTj+MrLSLv+o3SULKG+PnHKtYEJM/Pts4Hx2QPydLFlGd4NdNnXBKqDbK8J0tLdC0BBto9FpX4Wl/hZWJJPeWG2Tk6jzomIvGmMOe2wu3qmoIafRAxaqyCw304EfetgNfhGwIp/gsWfgXT7Hv/q9upk91AgHCDHl8MtU26xHy7LKSe8cycd//oYh156CaunB1fhaKzbPkNb8QKONkLz7m7Y3XjC2cD46Xn4BuD2zIRlONDYybZq+yzgjdogbaEYAGNz0rmqLJ9FpfksKvFTUpA1JIa5UBcXTQpq6LIsaD98YsMfOGDfPWTZDSXiAv9kGD0T5v8lXH4PZPrp6O3gpYO/pKKygt0tu+3uoaJlfLXsqywrWoaprKHjv9dR9bvfEW9qIpYzmp4VdxMcNZvGgNB7NI4c6znx2sAFPhsIReM0tIWpbw9z8FgX26tb2VnbRldvHIAJ/kxWTh+dTALFeRmaBNSA06SgUs8Y6Dp2YsMf2A/N70AsdLxezgQonG5Pc1k4w94umAJee6ydhJVgW+M2nn/j+WT3UFluGV9c8EU+UvoRctpjdKxfT8Pa7xGurKIzdzJdc2+ldfFUgh0u6IFMt5uSefkf+GzAGEN7KEZDe5j6tjAN7WEa2sI0tIeS231nAH0mj8rihrnjWFzqZ2GJn7E5qR8GW116NCmowRUKnvrNP7AfIu3H62QV2g3+ZXfZ68IZMGoqli+btkgbgVDAXroOEmh6jUAoQFOoiXeD79IcbibHl8PHp3yc1WWrmeoppuvll+n89v3U//kgQf8M2ktvomXiJGJxl31tYNRIFq3IZ+Is506hc/g2blmG5u7eZINf3xZyGn27wT/aHqYnmjjhmAyvm6K8DIpyM5hTnEtRbgbFzv6kgiwdO0gNCXqhWQ2M3m57YpqT+/27jx2v48txGv3phArKCYwcTSAzh4DpPd7w91uaw83ErfgJP0YQ8jPyKcwspDi7mOtKrmN54RKim7fRVrGeo2/V0jKynLYx8+n0jQEgMycteYH4TGcD0bjFsY4I9e0nNvYN7fbS2B4hmrBOOCY300tRrt3I9zX+dqOfSVFeBnmZXu3+UUPCe11o1qSgPph4L7QcOunb/377WgAQB1p9WTQXlNKUO45Alp+AL4OAQCDebTf2oWa6Yl2nvHWWN4vCzEIKMwrt9WmW/PR83JEoiZ4eYocP01TxMrVvNtCSUUIwfwZxdwYiMKY0h4mz85ngPDcQjiWS/fmnNPptYZq6IvT/0xCBwhE+p8HPTDb8xc56XG6GDgmhhg1NCuqDsxIQrEk2/KZpH93NBwh0HqbJJQQ8bpo9Xpqy8wmkZ9HsdhMwMVpi3Vj0+0ZtDOmWmyKXnyJXPmMkh0JG4Ley8CeyyO5NJyOaRlrUg4kkiIZixCIx4pE4sd4E8ahFLGaIxyGegLhxYbnSSLh9RHx5dI8YD0B6OuROycOMy6J9pIujPdH37M/3uoWxOSd+y+/f6I/JSdenf9VFQ29JVefOGGJtNTQ37CDQ9GcCre8SaD9CsLOV9jh0J3yE45lE4xl4YiPI6L2M9JiP9LgPX9xLrpXJWJOFz8rAY3y4TRpieTHGg2U8JIybhCsNy51G3O3DcvtIuNIIuNMIyFmetk1zFgBj4SIBYoFYWGKIuwxVIxLswuKIFYcjYThiV89Mcycb+rnFuad074wa4dMHvZRiiCUFEbkO+C7gBp40xnw7xSENWYl4lFCohXDbMXram+kJthLp7CTS1UlvT5hoKEI0FCXemyAeTZDotScSs+IurLhgLDfG8oDlAcsLeMB4AS92y+sDmY5xzcVypZHm9lHgTiNfzvBtWZxDgYizYAwuE8NFHLCABAa7AU+IISbQS4yIxAjTQzdCt7jocXvocbvpdbmICcSw6/bfTgAuF2SmechM85Ll85Dt8zAuN52VTh9+/wu5udqfr9Q5GTJJQUTcwA+BDwP1wBsistYYsz+1kZ3KsixMwsIkEvY6nsCKWxjLwsTjJOIJIt3d9ASbCXW0Ee5sp7e7h95QiGgoQiwSJRaJE49aJGIGKy5YCcEkXFh9jbVxY5yG2ogXgxfE3rbEi3F5sVxpWC4PiBcY4yznwO0sxkJMHJcVQ0wUMXHExBDiGBJYEiMhcRISJoqLXgvCQA9CD0LY7SbkdtPt9tDj8RJyuYm5JNlwxwEEXCJkpfnISHOT5fOQ4XWT5XM7Dbq9zvK5yUtzk3VSmV33eFlm2vF9n8elDb1SF9iQSQrAQqDSGFMNICLPAquBC54Ufvq5B+jtnGs/+IQAgjlhWwCXsz7+enL/bN0cJ/ABo89eTSxcrhhCDJfEoK+RJgYmBoSAGMYkgLidkKwERiwSYhF3vklHxUWvuAiLl5ArjW5Jp1sy6JRMwm4vIbeHiNtNXIQE9j/H7XLjdmXiFsHjkmTjbTfEpzbeo9LcTNLGW6mL0lBKCkVAXb/9emDRyZVE5F7gXoAJEyac1w/y5npItFeDMRgxCAYDdv80xl7EJLdN375TZqTv4rxxjsF+HwHTt+8yiAeMB/C4wOvB+NLsidkzMsGXDVk5uLL9SHYB7vQReLzeZMPs7rd4XILLdWq5WwSPW3C7XLjFqesWXCe9x6nv58Llwl4L2mArpZKGUlI4Xct0yq1RxpgngCfAvvvofH7QJx566HwOU0qpi95QGly9Hhjfb78YOJqiWJRS6pI0lJLCG0C5iJSISBpwB7A2xTEppdQlZch0Hxlj4iJyH/AS9r0xPzHG7EtxWEopdUkZMkkBwBjzAvBCquNQSqlL1VDqPlJKKZVimhSUUkolaVJQSimVpElBKaVU0rAeOltEmoHDqY7jDAqAllQHcZ6Ga+zDNW7Q2FPlUo19ojFm1OleGNZJYSgTkZ1nGq98qBuusQ/XuEFjTxWN/VTafaSUUipJk4JSSqkkTQoD54lUB/ABDNfYh2vcoLGnisZ+Er2moJRSKknPFJRSSiVpUlBKKZWkSeE8iMhPRCQgInv7lflF5A8icshZ5znlIiLfE5FKEdktIpelLnIQkfEislFEDojIPhH53HCJX0TSRWSHiPzZif0hp7xERLY7sf/CGXodEfE5+5XO65NSFbsTj1tE3haR9cMs7loR2SMiu0Rkp1M25D8vTjy5IvKciLzjfOaXDIfYRWSq8/vuWzpF5PODEbsmhfPzP8B1J5V9BXjFGFMOvOLsA1wPlDvLvcCPBynGM4kD/2CMmQ4sBtaIyAyGR/y9wDXGmLnAPOA6EVkMPAI85sTeBnzKqf8poM0YUwY85tRLpc8BB/rtD5e4Aa42xszrd1/8cPi8AHwXeNEYMw2Yi/37H/KxG2MOOr/vecDl2JO0/5bBiN0Yo8t5LMAkYG+//YPAWGd7LHDQ2X4cuPN09YbCAlQAHx5u8QOZwFvY83i3AB6nfAnwkrP9ErDE2fY49SRF8RY7f8TXAOuxp58d8nE7MdQCBSeVDfnPCzASqDn5dzccYj8p3r8ANg9W7HqmcOGMNsY0AjjrQqe8CKjrV6/eKUs5p1tiPrCdYRK/0wWzCwgAfwCqgHZjTPw08SVjd17vAPIHN+Kk7wBfAixnP5/hETfYc6W/LCJvisi9Ttlw+LyUAs3AU0633ZMiksXwiL2/O4BnnO0Bj12TwsCT05Sl/D5gEckGfg183hjT+V5VT1OWsviNMQljn1IXAwuB6aer5qyHROwicgMQMMa82b/4NFWHVNz9LDXGXIbdRbFGRJa/R92hFLsHuAz4sTFmPtDD8e6W0xlKsQPgXGe6CfjV2aqepuy8YtekcOE0ichYAGcdcMrrgfH96hUDRwc5thOIiBc7IfyfMeY3TvGwiR/AGNMObMK+LpIrIn2zCPaPLxm783oOEBzcSAFYCtwkIrXAs9hdSN9h6McNgDHmqLMOYPdrL2R4fF7qgXpjzHZn/znsJDEcYu9zPfCWMabJ2R/w2DUpXDhrgbuc7buw++r7yv/auTtgMdDRd/qXCiIiwH8DB4wx/9HvpSEfv4iMEpFcZzsDWIl94XAjcItT7eTY+/5NtwAbjNPhOpiMMV81xhQbYyZhdwVsMMb8JUM8bgARyRKREX3b2P3bexkGnxdjzDGgTkSmOkXXAvsZBrH3cyfHu45gMGJP9UWU4bg4/0mNQAw7Q38Ku8/3FeCQs/Y7dQX4IXbf9x5gQYpjvwr7tHI3sMtZVg2H+IE5wNtO7HuBrznlpcAOoBL7NNvnlKc7+5XO66VD4LOzAlg/XOJ2Yvyzs+wDHnDKh/znxYlnHrDT+cw8D+QNo9gzgVYgp1/ZgMeuw1wopZRK0u4jpZRSSZoUlFJKJWlSUEoplaRJQSmlVJImBaWUUkmaFJQaZCKySUSG5WTx6uKnSUEppVSSJgWlSD65+zux52rYKyK3i8jXROQNZ/8J52nwvm/6j4nIq84Y/VeIyG+cMe6/6dSZ5Izh/1NnfPvnRCTzND/3L0Rkq4i8JSK/csakQkS+LSL7nWMfHdzfhrqUaVJQynYdcNQYM9cYMwt4EfiBMeYKZz8DuKFf/agxZjnwn9hDDawBZgF3i0jfiKZTgSeMMXOATuCz/X+giBQADwIrjT3g3E7gfhHxAx8FZjrHfnNg/slKnUqTglK2PcBKEXlERJYZYzqAq8We+WwP9iB2M/vVX9vvuH3GmEZjTC9QzfGByeqMMZud7aexhxjpbzEwA9jsDAd+FzARO4FEgCdF5GPYE6woNSg8Z6+i1MXPGPOuiFyOPQ7UwyLyMva3/wXGmDoR+Qb2mER9ep211W+7b7/v7+rkMWRO3hfgD8aYO0+OR0QWYg/gdgdwH3ZSUmrA6ZmCUoCIjANCxpingUexh1gGaHH6+W8548FnNkFEljjbdwKvn/T6NmCpiJQ5MWSKyBTn5+UYY14APo89qJtSg0LPFJSyzQb+XUQs7NFvPwPcjN09VAu8cR7veQC4S0Qexx7V8oR5c40xzSJyN/CMiPic4geBLqBCRNKxzya+cB4/W6nzoqOkKjUAnKlO1zsXqZUaNrT7SCmlVJKeKSillErSMwWllFJJmhSUUkolaVJQSimVpElBKaVUkiYFpZRSSf8PtAlCHXto5JUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "samples = [50,100,200,300,400,500,600,700]\n",
    "cplex = [0.01,0.20,2.39,9.36,27.98,65.56,140.08,  None]\n",
    "gurob = [0.02,0.42,6.91,33.31,113.38,251.72,None, None]\n",
    "trans = [0.04,0.54,6.81,22.85,80.53,200.60,379.14,648.51]\n",
    "unary = [0.07,0.81,6.32,14.12,69.23,170.45,265.98,417.42]\n",
    "binar = [0.12,0.79,6.06,12.79,56.67,121.76,229.31,375.54]\n",
    "cplex_line, = plt.plot(samples, cplex, label='ILP-CPLEX')\n",
    "plt.plot(600, 140.08, marker='x', c=cplex_line.get_color())\n",
    "gurob_line, = plt.plot(samples, gurob, label='ILP-Gurobi')\n",
    "plt.plot(500, 251.72, marker='x', c=gurob_line.get_color())\n",
    "plt.plot(samples, trans, label='MaxSAT-Transitive')\n",
    "plt.plot(samples, unary, label='MaxSAT-Unary')\n",
    "plt.plot(samples, binar, label='MaxSAT-Binary')\n",
    "plt.xlabel('samples')\n",
    "plt.ylabel('time (s)')\n",
    "plt.legend()\n",
    "plt.title('Execution Time')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "samples = [50,100,200,300,400,500,600,700]\n",
    "cplex = [None,None,302,1038,2444,5593,10699,None]\n",
    "gurob = [None,None,425,1480,3643,7241,None,None]\n",
    "trans = [13.66,21.80,91.75,336.73,543.36,1059.09,2227.91,3830.15]\n",
    "unary = [17.12,32.61,92.16,187.04,342.77,513.50,735.01,924.57]\n",
    "binar = [16.82,33.40,95.45,210.08,318.43,491.46,729.07,891.93]\n",
    "cplex_line, = plt.plot(samples, cplex, label='ILP-CPLEX')\n",
    "plt.plot(600, 10699, marker='x', c=cplex_line.get_color())\n",
    "gurob_line, = plt.plot(samples, gurob, label='ILP-Gurobi')\n",
    "plt.plot(500, 7241, marker='x', c=gurob_line.get_color())\n",
    "plt.plot(samples, trans, label='MaxSAT-Transitive')\n",
    "plt.plot(samples, unary, label='MaxSAT-Unary')\n",
    "plt.plot(samples, binar, label='MaxSAT-Binary')\n",
    "plt.xlabel('samples')\n",
    "plt.ylabel('memory (MB)')\n",
    "plt.legend()\n",
    "plt.title('Memory Usage')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As can be seen here, the MaxSat algorithms utilize far less memory than ILP. As expected, limiting the maximum number of clusters using the unary or binary MaxSat encodings significantly reduces memory usage. While ILP-CPLEX was faster for smaller data sets, memory consumption becomes prohibitive for larger data sets. The reduced memory consumption of MaxSAT algorithms allows us to solve problems on larger data sets without exhausting resources."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References**\n",
    "\n",
    "Bansal, Nikhil, Avrim Blum, and Shuchi Chawla. 2004. “Correlation Clustering.” *Mach. Learn.* 56 (1-3): 89–113. doi:10.1023/B:MACH.0000033116.57574.95.\n",
    "\n",
    "Berg, Jeremias, and Matti Järvisalo. 2017. “Cost-Optimal Constrained Correlation Clustering via Weighted Partial Maximum Satisfiability.” *Artificial Intelligence* 244: 110–42. doi:10.1016/j.artint.2015.07.001."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "cite2c": {
   "citations": {
    "7765392/6HBV4DQ9": {
     "DOI": "10.1016/j.artint.2015.07.001",
     "URL": "https://linkinghub.elsevier.com/retrieve/pii/S0004370215001022",
     "abstract": "Integration of the ﬁelds of constraint solving and data mining and machine learning has recently been identiﬁed within the AI community as an important research direction with high potential. This work contributes to this direction by providing a ﬁrst study on the applicability of state-of-the-art Boolean optimization procedures to cost-optimal correlation clustering under constraints in a general similarity-based setting. We develop exact formulations of the correlation clustering task as Maximum Satisﬁability (MaxSAT), the optimization version of the Boolean satisﬁability (SAT) problem. For obtaining cost-optimal clusterings, we apply a state-of-the-art MaxSAT solver for solving the resulting MaxSAT instances optimally, resulting in cost-optimal clusterings. We experimentally evaluate the MaxSAT-based approaches to cost-optimal correlation clustering, both on the scalability of our method and the quality of the clusterings obtained. Furthermore, we show how the approach extends to constrained correlation clustering, where additional user knowledge is imposed as constraints on the optimal clusterings of interest. We show experimentally that added user knowledge allows clustering larger datasets, and at the same time tends to decrease the running time of our approach. We also investigate the effects of MaxSAT-level preprocessing, symmetry breaking, and the choice of the MaxSAT solver on the efﬁciency of the approach.",
     "accessed": {
      "day": 16,
      "month": 4,
      "year": 2020
     },
     "author": [
      {
       "family": "Berg",
       "given": "Jeremias"
      },
      {
       "family": "Järvisalo",
       "given": "Matti"
      }
     ],
     "container-title": "Artificial Intelligence",
     "container-title-short": "Artificial Intelligence",
     "id": "7765392/6HBV4DQ9",
     "issued": {
      "year": 2017
     },
     "journalAbbreviation": "Artificial Intelligence",
     "language": "en",
     "page": "110-142",
     "page-first": "110",
     "title": "Cost-optimal constrained correlation clustering via weighted partial Maximum Satisfiability",
     "type": "article-journal",
     "volume": "244"
    },
    "7765392/HTRNQ2G7": {
     "ISBN": "978-3-540-40770-6 978-3-540-45198-3",
     "URL": "http://link.springer.com/10.1007/978-3-540-45198-3_1",
     "abstract": "We consider the following general correlation-clustering problem [1]: given a graph with real edge weights (both positive and negative), partition the vertices into clusters to minimize the total absolute weight of cut positive edges and uncut negative edges. Thus, large positive weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster; large negative weights encourage the endpoints to belong to diﬀerent clusters; and weights with small absolute value represent little information. In contrast to most clustering problems, correlation clustering speciﬁes neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are eﬀectively chosen to be the best possible by the problem deﬁnition.",
     "accessed": {
      "day": 16,
      "month": 4,
      "year": 2020
     },
     "author": [
      {
       "family": "Demaine",
       "given": "Erik D."
      },
      {
       "family": "Immorlica",
       "given": "Nicole"
      }
     ],
     "collection-editor": [
      {
       "family": "Goos",
       "given": "Gerhard"
      },
      {
       "family": "Hartmanis",
       "given": "Juris"
      },
      {
       "family": "van Leeuwen",
       "given": "Jan"
      }
     ],
     "container-title": "Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques",
     "editor": [
      {
       "family": "Arora",
       "given": "Sanjeev"
      },
      {
       "family": "Jansen",
       "given": "Klaus"
      },
      {
       "family": "Rolim",
       "given": "José D. P."
      },
      {
       "family": "Sahai",
       "given": "Amit"
      }
     ],
     "event-place": "Berlin, Heidelberg",
     "id": "7765392/HTRNQ2G7",
     "issued": {
      "year": 2003
     },
     "language": "en",
     "note": "Series Title: Lecture Notes in Computer Science\nDOI: 10.1007/978-3-540-45198-3_1",
     "page": "1-13",
     "page-first": "1",
     "publisher": "Springer Berlin Heidelberg",
     "publisher-place": "Berlin, Heidelberg",
     "title": "Correlation Clustering with Partial Information",
     "type": "chapter",
     "volume": "2764"
    },
    "7765392/S23E2LPM": {
     "DOI": "10.1023/B:MACH.0000033116.57574.95",
     "abstract": "We consider the following clustering problem: we have a complete graph on n vertices (items), where each edge (u, v) is labeled either + or − depending on whether u and v have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as much as possible with the edge labels. That is, we want a clustering that maximizes the number of + edges within clusters, plus the number of − edges between clusters (equivalently, minimizes the number of disagreements: the number of − edges inside clusters plus the number of + edges between clusters). This formulation is motivated from a document clustering problem in which one has a pairwise similarity function f learned from past data, and the goal is to partition the current set of documents in a way that correlates with f as much as possible; it can also be viewed as a kind of “agnostic learning” problem.",
     "author": [
      {
       "family": "Bansal",
       "given": "Nikhil"
      },
      {
       "family": "Blum",
       "given": "Avrim"
      },
      {
       "family": "Chawla",
       "given": "Shuchi"
      }
     ],
     "container-title": "Mach. Learn.",
     "id": "7765392/S23E2LPM",
     "issue": "1-3",
     "issued": {
      "month": 6,
      "year": 2004
     },
     "language": "en",
     "page": "89-113",
     "page-first": "89",
     "title": "Correlation Clustering",
     "type": "article-journal",
     "volume": "56"
    }
   }
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