maximal independent set.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Maximal Independent Set\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "An independent set (IS) is a subset of a graph such that no two vertices inside are connected. A maximal independent set (MIS) is an IS that cannot include any extra vertices without violating the definition. A maximum independent set is one of the MIS with maximum cardinality. The definition of maximal and maximum is somehow similar to clique. MIS is somewhat connected to graph coloring as well.\n",
    "\n",
    "Reference to clique\n",
    "\n",
    "https://github.com/je-suis-tm/graph-theory/blob/master/maximal%20clique.ipynb\n",
    "\n",
    "Reference to graph coloring\n",
    "\n",
    "https://github.com/je-suis-tm/graph-theory/blob/master/graph%20coloring.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os\n",
    "os.chdir('K:/ecole/github/televerser')\n",
    "import matplotlib.pyplot as plt\n",
    "import random as rd\n",
    "import copy\n",
    "\n",
    "#graph adt\n",
    "#check the below link for more details\n",
    "# https://github.com/je-suis-tm/graph-theory/blob/master/graph.py\n",
    "import graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graph ADT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#create graph adt\n",
    "#exactly the same adt in kcore and graph coloring\n",
    "ADT=graph.graph()\n",
    "ADT.append(1,2,0)\n",
    "ADT.append(1,3,0)\n",
    "ADT.append(1,4,0)\n",
    "ADT.append(1,6,0)\n",
    "ADT.append(2,1,0)\n",
    "ADT.append(3,1,0)\n",
    "ADT.append(4,1,0)\n",
    "ADT.append(6,1,0)\n",
    "\n",
    "ADT.append(2,3,0)\n",
    "ADT.append(2,4,0)\n",
    "ADT.append(2,6,0)\n",
    "ADT.append(3,2,0)\n",
    "ADT.append(4,2,0)\n",
    "ADT.append(6,2,0)\n",
    "\n",
    "ADT.append(3,4,0)\n",
    "ADT.append(3,7,0)\n",
    "ADT.append(4,3,0)\n",
    "ADT.append(7,3,0)\n",
    "\n",
    "ADT.append(4,5,0)\n",
    "ADT.append(5,4,0)\n",
    "\n",
    "ADT.append(5,6,0)\n",
    "ADT.append(6,5,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: {2: 0, 3: 0, 4: 0, 6: 0},\n",
       " 2: {1: 0, 3: 0, 4: 0, 6: 0},\n",
       " 3: {1: 0, 2: 0, 4: 0, 7: 0},\n",
       " 4: {1: 0, 2: 0, 3: 0, 5: 0},\n",
       " 6: {1: 0, 2: 0, 5: 0},\n",
       " 7: {3: 0},\n",
       " 5: {4: 0, 6: 0}}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ADT.reveal()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![alt text](./preview/kcore1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fast Randomized Distributed MIS Algorithm\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "There are tons of algorithms to find MIS. MIS is an NP-complete problem for computer scientists. Currently the most popular one is randomized distributed algorithm. \n",
    "\n",
    "Reference to the original paper in 2009\n",
    "\n",
    "https://www.researchgate.net/publication/220567138_An_Optimal_Bit_Complexity_Randomized_Distributed_MIS_Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#fast mis algorithm after 2009\n",
    "def get_maximal_independent_set(ADT):\n",
    "    \"\"\"fast randomized algorithm to fetch one of the maximal independent sets\"\"\"\n",
    "\n",
    "    #assign random value from uniform distribution to every vertex\n",
    "    random_val=dict(zip(ADT.vertex(),\n",
    "        [rd.random() for _ in range(ADT.order())]))\n",
    "\n",
    "    #initialize\n",
    "    maximal_independent_set=[]\n",
    "    queue=[i for i in random_val]\n",
    "    \n",
    "    while len(queue)>0:\n",
    "        for node in queue:\n",
    "            \n",
    "            #select the vertex which has larger value than all of its neighbors\n",
    "            neighbor_vals=[random_val[neighbor] for neighbor in ADT.edge(node) if neighbor in random_val]\n",
    "            if len(neighbor_vals)==0 or random_val[node]<min(neighbor_vals):\n",
    "                \n",
    "                #add to mis\n",
    "                maximal_independent_set.append(node)\n",
    "                \n",
    "                #remove the vertex and its neighbors\n",
    "                queue.remove(node)\n",
    "                for neighbor in ADT.edge(node):\n",
    "                    if neighbor in queue:\n",
    "                        queue.remove(neighbor)\n",
    "        \n",
    "        #reassign random values to existing vertices\n",
    "        random_val=dict(zip(queue,\n",
    "                            [rd.random() for _ in range(len(random_val))]))\n",
    "        \n",
    "    return maximal_independent_set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#run\n",
    "mis=get_maximal_independent_set(ADT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#assign coordinates for viz\n",
    "coordinates={1: (0.5, 3), 2: (0.5, 1),\n",
    " 3: (1.5, 2), 4: (2, 1),\n",
    " 5: (1, 0), 6: (0, 0), 7: (3, 3)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot edge\n",
    "for i in ADT.vertex():\n",
    "    for j in ADT.edge(i):\n",
    "        plt.plot([coordinates[i][0],coordinates[j][0]],\n",
    "                 [coordinates[i][1],coordinates[j][1]],c='grey',alpha=0.5)\n",
    "\n",
    "#plot vertex\n",
    "plt.scatter([coordinates[i][0] for i in coordinates if i in mis],\n",
    "           [coordinates[i][1] for i in coordinates if i in mis],\n",
    "           s=500,label=\"Maximal Independent Set\",c='gold')\n",
    "plt.scatter([coordinates[i][0] for i in coordinates if i not in mis],\n",
    "           [coordinates[i][1] for i in coordinates if i not in mis],\n",
    "           s=500)\n",
    "\n",
    "#plot vertex label\n",
    "for i in coordinates:\n",
    "    plt.text(coordinates[i][0],coordinates[i][1],i,\n",
    "             va='center',ha='center',color='white',fontsize=20)\n",
    "\n",
    "#tidy up the chart\n",
    "plt.axis('off')\n",
    "plt.xticks([])\n",
    "plt.yticks([])\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ramsey Algorithm\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "Finding maximal independent set is NP-complete but finding maximum independent set (MaxIS) is NP-hard. Intuitively, the easiest way to find a MaxIS is a brute force algorithm to select MaxIS from all the MIS. `NetworkX` prefers Ramsey algorithm to approximate MaxIS. Thus, it is worth the effort to make a little introduction here as well.\n",
    "\n",
    "Reference to `NetworkX`\n",
    "\n",
    "https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.approximation.independent_set.maximum_independent_set.html#networkx.algorithms.approximation.independent_set.maximum_independent_set\n",
    "\n",
    "Reference to the original paper\n",
    " https://www.researchgate.net/publication/227294736_Approximating_Maximum_Independent_Sets_by_Excluding_Subgraphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#choice of networkx\n",
    "def ramsey(graph_adt):\n",
    "    \"\"\"Ramsey algorithm to approximate the maximum independent set\"\"\"\n",
    "\n",
    "    #base case of the recursion\n",
    "    if graph_adt.order()==0:\n",
    "        return []\n",
    "    \n",
    "    #select a random vertex\n",
    "    #this can be optimized by selecting the vertex with the smallest degree\n",
    "    selected=graph_adt.vertex()[0]\n",
    "\n",
    "    #find its neighbors and non neighbors\n",
    "    neighbors=graph_adt.edge(selected)\n",
    "    non_neighbors=[i for i in graph_adt.vertex() if i not in neighbors and i!=selected]\n",
    "\n",
    "    #create a subset which only contains its neighbors\n",
    "    subset_neighbors=copy.deepcopy(graph_adt)\n",
    "    for vertex in non_neighbors:\n",
    "        subset_neighbors.remove(vertex)\n",
    "    subset_neighbors.remove(selected)\n",
    "\n",
    "    #create a subset which only contains its non neighbors\n",
    "    subset_non_neighbors=copy.deepcopy(graph_adt)\n",
    "    for vertex in neighbors:\n",
    "        subset_non_neighbors.remove(vertex)\n",
    "    subset_non_neighbors.remove(selected)\n",
    "        \n",
    "    #recursively check other vertices\n",
    "    neighbors_mis=ramsey(subset_neighbors)\n",
    "    non_neighbors_mis=ramsey(subset_non_neighbors)\n",
    "    \n",
    "    #add to approximated maximum independent set\n",
    "    non_neighbors_mis.append(selected)\n",
    "\n",
    "    #only export the larger set between the two\n",
    "    return max(neighbors_mis,non_neighbors_mis,key=len)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#run\n",
    "approximation=ramsey(ADT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot edge\n",
    "for i in ADT.vertex():\n",
    "    for j in ADT.edge(i):\n",
    "        plt.plot([coordinates[i][0],coordinates[j][0]],\n",
    "                 [coordinates[i][1],coordinates[j][1]],c='grey',alpha=0.5)\n",
    "\n",
    "#plot vertex\n",
    "plt.scatter([coordinates[i][0] for i in coordinates if i in approximation],\n",
    "           [coordinates[i][1] for i in coordinates if i in approximation],\n",
    "           s=500,label=\"≈Maximum Independent Set\",c='red')\n",
    "plt.scatter([coordinates[i][0] for i in coordinates if i not in approximation],\n",
    "           [coordinates[i][1] for i in coordinates if i not in approximation],\n",
    "           s=500)\n",
    "\n",
    "#plot vertex label\n",
    "for i in coordinates:\n",
    "    plt.text(coordinates[i][0],coordinates[i][1],i,\n",
    "             va='center',ha='center',color='white',fontsize=20)\n",
    "\n",
    "#tidy up the chart\n",
    "plt.axis('off')\n",
    "plt.xticks([])\n",
    "plt.yticks([])\n",
    "plt.legend(loc=4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {
    "height": "calc(100% - 180px)",
    "left": "10px",
    "top": "150px",
    "width": "154px"
   },
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}