C2W1_10GRAPH_SEARCH_AND_CONNECTIVITY.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Introduction \n",
    "We will start with Generic Graph search. Graph Search is an algorithm where, given a graph, G(V, E), with a starting vertex s \n",
    "∈\n",
    " V, we identify all vertices of G reachable from s\n",
    "The generic graph search algorithm does the following, with the input grapg G = (V, E) and start vertex s\n",
    "\n",
    "    mark s as explored and all other vertices as unexplored\n",
    "\n",
    "    while there is an edge (v, w) in E where v is explored and w is not explored:\n",
    "        mark w as explored \n",
    "\n",
    "***\n",
    "\n",
    "The above algorithm is very abstract and the strategy to pick the edge is not specified. In the above algorithm, the vertex v is in the frontier and we wish to find all edges with one vertex in frontier and other outside. Two strategies that we can look at are\n",
    "- For all vertices in the frontier, explore all edges that go from them to vertices outside the frontier\n",
    "- Another strategy would be\n",
    "    - Pick one vertex in the frontier\n",
    "    - Pick one edge from the this vertex to another vertex not in frontier\n",
    "    - Add this another vertex to the frontier and aggressively keep adding the vertices in the frontier before looking at another vertex, unlike the first strategy\n",
    "These two approches are BFS(Breadth First Search) and DFS (Depth First Search) algorothms for graph exploration. While they both work for finding all possible vertices reachable from a starting vertex s in time \n",
    "θ(m+n), for some use cases we choose one over another.\n",
    "For example, for computing the shortst distance to a vertex from a starting vertex s can be done only using BFS efficiently. Where as some use cases like finding strongly connected components in case of directed graphs, we need DFS.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118f0e890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "import numpy as np\n",
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "A = np.matrix([\n",
    "   [0, 1, 1, 0, 0, 0],\n",
    "   [1, 0, 0, 1, 0, 0],\n",
    "   [1, 0, 0, 1, 0, 0],\n",
    "   [0, 1, 1, 0, 1, 1],\n",
    "   [0, 0, 0, 1, 0, 1],\n",
    "   [0, 0, 0, 1, 1, 0],\n",
    "])\n",
    "\n",
    "G = nx.from_numpy_matrix(A)\n",
    "nx.draw(G, with_labels = True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d00f3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5)]\n",
    "G.add_edges_from(edges)\n",
    "nx.draw(G, with_labels = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the above graph, starting from vertex 0 which is at level 0, vertices 1 and 2 are at level 1, vertex 3 at level 2 and vertices 4 and 5 at level 3.\n",
    "<br>BFS will discover these edges in this very order.<br>\n",
    "Following Python code makes use of queues and implements BFS and also give is the level of each of the vertex from a provided starting vertex. We saw how addacency lists can be defined in Python in previous notebook, to avoid copying over the code, we will make use of the graph defined in above cell from python package networkx version 2.2 to get adjacent edges of a given vertex.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [1, 2], 1: [0, 3], 2: [0, 3, 5], 3: [1, 2, 4, 5], 4: [3, 5], 5: [2, 3, 4]}"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adjacency_dict = {v: [e for e in ae] for v, ae in G.adjacency()}\n",
    "adjacency_dict"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we look at the above dictionary, the key of the dictionary is the vertex and the value is the list of edges connected to it. With this dictionary, let us use bfs to find the level of vertices starting with vertex 0 as the starting vertex and see of we get the levels we expect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Breadth First Search (BFS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nodes: [0, 1, 2, 3, 4, 5]\n",
      "Order: [0, 0, 0, 1, 0, 0]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import networkx as nx\n",
    "from multiprocessing import Queue\n",
    "import matplotlib.pyplot as plt\n",
    "import urllib3\n",
    "\n",
    "def bfs(G, start):\n",
    "    print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    nodes = list(G.nodes)\n",
    "    order = 1\n",
    "    explored = [0]*len(list(G.nodes))\n",
    "    explored[nodes.index(start)] = order\n",
    "    q = Queue()\n",
    "    q.put(int(start))\n",
    "    while not q.empty():\n",
    "        v = q.get()\n",
    "        w = list(G.adj[v])\n",
    "        for adjnode in w:\n",
    "            if explored[nodes.index(adjnode)] == 0:\n",
    "                order += 1\n",
    "                explored[nodes.index(adjnode)] = order\n",
    "                q.put(int(adjnode))\n",
    "    print(\"Order: {}\".format(explored))\n",
    "\n",
    "    \n",
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5)]\n",
    "G.add_edges_from(edges)\n",
    "nx.draw(G, with_labels = True)\n",
    "bfs(G, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### BFS tracking Layers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nodes: [0, 1, 2, 3, 4, 5]\n",
      "Node 0 is at layer 0\n",
      "Node 1 is at layer 1\n",
      "Node 2 is at layer 1\n",
      "Node 3 is at layer 2\n",
      "Node 4 is at layer 3\n",
      "Node 5 is at layer 2\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d5a5690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def bfs_layer(G, src):\n",
    "    print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    nodes = list(G.nodes)\n",
    "    explored = [0]*len(list(G.nodes))\n",
    "    i = 0\n",
    "    layer = 0\n",
    "    explored[nodes.index(src)] = 1\n",
    "    q = Queue()\n",
    "    q.put(int(src))\n",
    "    i += 1\n",
    "    for node in nodes:\n",
    "        G.nodes[node]['layer'] = float('inf')\n",
    "    G.nodes[src]['layer'] = layer\n",
    "\n",
    "\n",
    "    while i != 0:\n",
    "        v = q.get()\n",
    "        i -= 1\n",
    "        w = list(G.adj[v])\n",
    "        for adjnode in w:\n",
    "            if explored[nodes.index(adjnode)] == 0:\n",
    "                explored[nodes.index(adjnode)] = 1\n",
    "                q.put(int(adjnode))\n",
    "                i += 1\n",
    "                G.nodes[adjnode]['layer'] = int(G.nodes[int(v)]['layer']) + 1\n",
    "    for n, nlay in G.nodes.data():\n",
    "        print(\"Node {} is at layer {}\".format(n, nlay['layer']))\n",
    "\n",
    "    \n",
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5)]\n",
    "G.add_edges_from(edges)\n",
    "nx.draw(G, with_labels = True)\n",
    "bfs_layer(G, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### BFS tracking Shortest Path from source node to a destination node"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nodes: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
      "Destination Node 10 is at layer 3 having path ['0', '2', '5', '10']\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d598690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def bfs_shortestpath(G, src, dest):\n",
    "    print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    nodes = list(G.nodes)\n",
    "    explored = [0]*len(list(G.nodes))\n",
    "    i = 0\n",
    "    layer = 0\n",
    "    foundat =  float('inf')\n",
    "    explored[nodes.index(src)] = 1\n",
    "    q = Queue()\n",
    "    q.put(int(src))\n",
    "    i += 1\n",
    "    for node in nodes:\n",
    "        G.nodes[node]['layer'] = float('inf')\n",
    "        G.nodes[node]['path'] = []\n",
    "    G.nodes[src]['layer'] = layer\n",
    "    G.nodes[src]['path'].append(str(src))\n",
    "\n",
    "    while i != 0:\n",
    "        v = q.get()\n",
    "        i -= 1\n",
    "        w = list(G.adj[v])\n",
    "        for adjnode in w:\n",
    "            if explored[nodes.index(adjnode)] == 0:\n",
    "                explored[nodes.index(adjnode)] = 1\n",
    "                if adjnode == dest:\n",
    "                    foundat = int(G.nodes[int(v)]['layer']) + 1\n",
    "                q.put(int(adjnode))\n",
    "                i += 1\n",
    "                G.nodes[adjnode]['layer'] = int(G.nodes[int(v)]['layer']) + 1\n",
    "                G.nodes[adjnode]['path'].append(str(G.nodes[int(v)]['path'][0]) +','+ str(adjnode))\n",
    "    #print(G.nodes.data())\n",
    "    if foundat !=  float('inf'):\n",
    "        for n, ndict in G.nodes.data():\n",
    "            if n == dest:\n",
    "                print(\"Destination Node {} is at layer {} having path {}\".format(n, ndict['layer'], ndict['path'][0].split(',')))  \n",
    "    else:\n",
    "        print(\"Destination Node {} not found\".format(dest))\n",
    "    \n",
    "\n",
    "    \n",
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5), (5, 6), (6, 7), (4, 8), (4, 9), (5, 9), (5, 10)]\n",
    "G.add_edges_from(edges)\n",
    "nx.draw(G, with_labels = True)\n",
    "bfs_shortestpath(G, 0, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nodes: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
      "Destination Node 7 is at layer 4 having path ['0', '2', '5', '6', '7']\n"
     ]
    }
   ],
   "source": [
    "bfs_shortestpath(G, 0, 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###### BFS tracking  Connected Components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Node \t Connected Component \t explored\n",
      "0 \t 0 \t\t\t 1\n",
      "1 \t 0 \t\t\t 1\n",
      "2 \t 0 \t\t\t 1\n",
      "3 \t 0 \t\t\t 1\n",
      "4 \t 0 \t\t\t 1\n",
      "5 \t 0 \t\t\t 1\n",
      "6 \t 0 \t\t\t 1\n",
      "7 \t 0 \t\t\t 1\n",
      "8 \t 8 \t\t\t 1\n",
      "9 \t 8 \t\t\t 1\n",
      "10 \t 8 \t\t\t 1\n",
      "11 \t 11 \t\t\t 1\n",
      "12 \t 11 \t\t\t 1\n",
      "13 \t 11 \t\t\t 1\n",
      "14 \t 8 \t\t\t 1\n",
      "15 \t 15 \t\t\t 1\n",
      "16 \t 15 \t\t\t 1\n",
      "17 \t 15 \t\t\t 1\n",
      "18 \t 15 \t\t\t 1\n",
      "19 \t 15 \t\t\t 1\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d2d9d50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def bfs_connectedcomponents(G, start):\n",
    "    #print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    i = 0\n",
    "    q = Queue()\n",
    "    q.put(int(start))\n",
    "    G.nodes[start]['explored'] = 1\n",
    "    G.nodes[start]['cc'] = start\n",
    "    i += 1\n",
    "    while i != 0:\n",
    "        v = q.get()\n",
    "        i -= 1\n",
    "        w = list(G.adj[v])\n",
    "        for adjnode in w:\n",
    "            if G.nodes[adjnode]['explored'] == 0:\n",
    "                G.nodes[adjnode]['explored'] = 1\n",
    "                G.nodes[adjnode]['cc'] = start\n",
    "                q.put(int(adjnode))\n",
    "                i += 1\n",
    "\n",
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5), (5, 6), (6, 7), (10, 8), (10, 9), (11, 12), (11, 13), (10, 14), (15, 16), (16, 17), (17, 18), (17, 19)]\n",
    "G.add_edges_from(edges)\n",
    "for node in list(G.nodes):\n",
    "    G.nodes[node]['explored'] = 0\n",
    "    G.nodes[node]['cc'] =  float('inf')\n",
    "nx.draw(G, with_labels = True)\n",
    "\n",
    "for newnode in list(G.nodes):\n",
    "    if G.nodes[newnode]['explored'] == 0:\n",
    "        bfs_connectedcomponents(G, newnode)\n",
    "\n",
    "print(\"Node \\t Connected Component \\t explored\")\n",
    "for n, ndict in G.nodes.data():\n",
    "    print(\"{} \\t {} \\t\\t\\t {}\".format(n, ndict['cc'], ndict['explored']))  \n",
    "    #print(\"Node {} is in connected component {} and explored is {}\".format(n, ndict['cc'], ndict['explored']))  \n",
    "\n",
    "            "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see above, we have four connected components made of vertices {0, 1, 2, 3, 4, 5, 6, 7}, {8, 9, 10, 14} , {11, 12, 13} and {15, 16, 17, 18, 19}\n",
    "\n",
    "These three mentioned above are the connected components of the undirected graph above. In general for a set X, if x, y $\\in$ X, then we write x ~ y if x is related to y.\n",
    "\n",
    "For our graph, the set of the ser of vertices V. We write v ~ w if there is an edge between the vertices v and w. The equivalence relation has to satisfy the following three properties\n",
    "\n",
    "- The relations are reflexive, which means x ~ x. In our graph each vertex if related to itself.\n",
    "- The relation is symmetric, which means if x ~ y then y ~ x. In case of undirected graph, this is true as, if there is an edge between w and v there is also an edge between v and w\n",
    "- The relation is transitive. Which means if x ~ y and y ~ z then x ~ z. In case of our graph, if we can reach from vertex x to y and we can reach from y to z, then we can reach from x to z.\n",
    "\n",
    "The equivalence relations partitions the set into some sub sets. Each element in a class is related to each other element in that class and only to elements in that class.\n",
    "\n",
    "In case of graph these classes are connected components.\n",
    "\n",
    "***\n",
    "\n",
    "##### Running time BFS\n",
    "\n",
    "Lets look at the running time of this algorithm.\n",
    "\n",
    "For BFS, the algorithm never processes a vertex more than once and thus one component of the complexity if linear in terms of the number of vertices n. If it processes a vertex, it will iterate over each edge of the graph which involves this vertex. Since each vertex if processed only once and each time we process all edges of that vertex, the complexity if also linear in the number of edges in that graph and thus the time complxity is $O(m + n)$\n",
    "\n",
    "For the connected components (UCC) algorithm, we do a linear scan on all vertices and invoke BFS from a vertex only if it does not yet belong to a UCC, Thus for a connected component i, the complexity if $\\theta(m_i + n_i)$\n",
    "\n",
    "The total complexity therefore is $\\theta(\\sum_i{m_i} + \\sum_i{n_i})$\n",
    "\n",
    "The ucc algoritm discovers all the vertices across all the connected components and by definition of ucc, one vertex belongs to only one connected component. Therefore $\\sum{n_i}$ will be same as the total vertices in the graph(s). Similarly, the edges in the connected components belong entirely to the vertices within that connected component and thus $\\sum{m_i}$ is same as the number of edges in the graph and therefore the complexity of ucc is $\\theta(m + n)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Depth First Search (DFS)\n",
    "We will now look at another graph traversal algorithm, DFS.\n",
    "\n",
    "Below, is an implementation using stack and a recursive implementation of DFS. \n",
    "\n",
    "##### DFS using Stack"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Node \t Connected Component \t starting time \t finishing time \t explored\n",
      "0 \t 0 \t\t\t 1 \t\t 1 \t\t\t 1\n",
      "1 \t 0 \t\t\t 2 \t\t 8 \t\t\t 1\n",
      "2 \t 0 \t\t\t 3 \t\t 2 \t\t\t 1\n",
      "3 \t 0 \t\t\t 4 \t\t 7 \t\t\t 1\n",
      "4 \t 0 \t\t\t 6 \t\t 6 \t\t\t 1\n",
      "5 \t 0 \t\t\t 5 \t\t 3 \t\t\t 1\n",
      "6 \t 0 \t\t\t 7 \t\t 4 \t\t\t 1\n",
      "7 \t 0 \t\t\t 8 \t\t 5 \t\t\t 1\n",
      "8 \t 8 \t\t\t 1 \t\t 1 \t\t\t 1\n",
      "9 \t 8 \t\t\t 3 \t\t 4 \t\t\t 1\n",
      "10 \t 8 \t\t\t 2 \t\t 2 \t\t\t 1\n",
      "11 \t 11 \t\t\t 1 \t\t 1 \t\t\t 1\n",
      "12 \t 11 \t\t\t 2 \t\t 3 \t\t\t 1\n",
      "13 \t 11 \t\t\t 3 \t\t 2 \t\t\t 1\n",
      "14 \t 8 \t\t\t 4 \t\t 3 \t\t\t 1\n",
      "15 \t 15 \t\t\t 1 \t\t 1 \t\t\t 1\n",
      "16 \t 15 \t\t\t 2 \t\t 2 \t\t\t 1\n",
      "17 \t 15 \t\t\t 3 \t\t 3 \t\t\t 1\n",
      "18 \t 15 \t\t\t 4 \t\t 5 \t\t\t 1\n",
      "19 \t 15 \t\t\t 5 \t\t 4 \t\t\t 1\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d598a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def dfs_stack(G, start):\n",
    "    #print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    i = 0\n",
    "    q = []\n",
    "    order = 1\n",
    "    ft = 0\n",
    "    q.append(int(start))\n",
    "    G.nodes[start]['explored'] = 1\n",
    "    G.nodes[start]['cc'] = start\n",
    "    G.nodes[start]['starting time'] = order\n",
    "    i += 1\n",
    "    while i != 0:\n",
    "        v = q.pop()\n",
    "        ft += 1\n",
    "        G.nodes[v]['finishing time'] = ft\n",
    "        i -= 1\n",
    "        w = list(G.adj[v])\n",
    "        #print(q)\n",
    "        for adjnode in w:\n",
    "            if G.nodes[adjnode]['explored'] == 0:\n",
    "                G.nodes[adjnode]['explored'] = 1\n",
    "                G.nodes[adjnode]['cc'] = start\n",
    "                order += 1\n",
    "                G.nodes[adjnode]['starting time'] = order\n",
    "                q.append(int(adjnode))\n",
    "                #print(q)\n",
    "                i += 1\n",
    "\n",
    "G = nx.Graph()\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5), (5, 6), (6, 7), (10, 8), (10, 9), (11, 12), (11, 13), (10, 14), (15, 16), (16, 17), (17, 18), (17, 19)]\n",
    "G.add_edges_from(edges)\n",
    "for node in list(G.nodes):\n",
    "    G.nodes[node]['explored'] = 0\n",
    "    G.nodes[node]['cc'] =  float('inf')\n",
    "    G.nodes[node]['starting time'] =  float('inf')\n",
    "    G.nodes[node]['finishing time'] =  float('inf')\n",
    "nx.draw(G, with_labels = True)\n",
    "\n",
    "for newnode in list(G.nodes):\n",
    "    if G.nodes[newnode]['explored'] == 0:\n",
    "        dfs_stack(G, newnode)\n",
    "\n",
    "print(\"Node \\t Connected Component \\t starting time \\t finishing time \\t explored\")\n",
    "for n, ndict in G.nodes.data():\n",
    "    print(\"{} \\t {} \\t\\t\\t {} \\t\\t {} \\t\\t\\t {}\".format(n, ndict['cc'], ndict['starting time'], ndict['finishing time'], ndict['explored']))  \n",
    "    #print(\"Node {} is in connected component {} and explored is {}\".format(n, ndict['cc'], ndict['explored']))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### DFS using Recursion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Node \t Connected Component \t starting time \t finishing time \t explored\n",
      "0 \t 0 \t\t\t 1 \t\t 8 \t\t\t 1\n",
      "1 \t 0 \t\t\t 2 \t\t 7 \t\t\t 1\n",
      "2 \t 0 \t\t\t 4 \t\t 5 \t\t\t 1\n",
      "3 \t 0 \t\t\t 3 \t\t 6 \t\t\t 1\n",
      "4 \t 0 \t\t\t 6 \t\t 1 \t\t\t 1\n",
      "5 \t 0 \t\t\t 5 \t\t 4 \t\t\t 1\n",
      "6 \t 0 \t\t\t 7 \t\t 3 \t\t\t 1\n",
      "7 \t 0 \t\t\t 8 \t\t 2 \t\t\t 1\n",
      "8 \t 8 \t\t\t 1 \t\t 4 \t\t\t 1\n",
      "9 \t 8 \t\t\t 3 \t\t 1 \t\t\t 1\n",
      "10 \t 8 \t\t\t 2 \t\t 3 \t\t\t 1\n",
      "11 \t 11 \t\t\t 1 \t\t 3 \t\t\t 1\n",
      "12 \t 11 \t\t\t 2 \t\t 1 \t\t\t 1\n",
      "13 \t 11 \t\t\t 3 \t\t 2 \t\t\t 1\n",
      "14 \t 8 \t\t\t 4 \t\t 2 \t\t\t 1\n",
      "15 \t 15 \t\t\t 1 \t\t 5 \t\t\t 1\n",
      "16 \t 15 \t\t\t 2 \t\t 4 \t\t\t 1\n",
      "17 \t 15 \t\t\t 3 \t\t 3 \t\t\t 1\n",
      "18 \t 15 \t\t\t 4 \t\t 1 \t\t\t 1\n",
      "19 \t 15 \t\t\t 5 \t\t 2 \t\t\t 1\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x151d9c0150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ft = 1\n",
    "order = 1\n",
    "\n",
    "def dfs_recursion(G, startvert, start):\n",
    "    global ft\n",
    "    global order\n",
    "    #print(\"Nodes: {}\".format(list(G.nodes)))\n",
    "    G.nodes[startvert]['explored'] = 1\n",
    "    G.nodes[startvert]['cc'] = start\n",
    "    G.nodes[startvert]['starting time'] = order\n",
    "    w = list(G.adj[startvert])\n",
    "    for adjnode in w:\n",
    "        if G.nodes[adjnode]['explored'] == 0:\n",
    "            order = order+1\n",
    "            dfs_recursion(G, adjnode, start)\n",
    "            G.nodes[adjnode]['finishing time'] = ft\n",
    "            ft += 1\n",
    "    \n",
    "def dfs(G, startvert, start):\n",
    "    global ft\n",
    "    global order\n",
    "    ft = 1\n",
    "    order = 1\n",
    "    dfs_recursion(G, newnode, newnode)\n",
    "    G.nodes[newnode]['finishing time'] = ft\n",
    "            \n",
    "        \n",
    "G = nx.Graph()\n",
    "#edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5), (5, 6), (6, 7)]\n",
    "edges = [(0, 1), (0, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5), (2, 5), (5, 6), (6, 7), (10, 8), (10, 9), (11, 12), (11, 13), (10, 14), (15, 16), (16, 17), (17, 18), (17, 19)]\n",
    "G.add_edges_from(edges)\n",
    "for node in list(G.nodes):\n",
    "    G.nodes[node]['explored'] = 0\n",
    "    G.nodes[node]['cc'] =  float('inf')\n",
    "    G.nodes[node]['starting time'] =  float('inf')\n",
    "    G.nodes[node]['finishing time'] =  float('inf')\n",
    "nx.draw(G, with_labels = True)\n",
    "\n",
    "for newnode in list(G.nodes):\n",
    "    if G.nodes[newnode]['explored'] == 0:\n",
    "        dfs(G, newnode, newnode)\n",
    "\n",
    "print(\"Node \\t Connected Component \\t starting time \\t finishing time \\t explored\")\n",
    "for n, ndict in G.nodes.data():\n",
    "    print(\"{} \\t {} \\t\\t\\t {} \\t\\t {} \\t\\t\\t {}\".format(n, ndict['cc'], ndict['starting time'], ndict['finishing time'], ndict['explored']))  \n",
    "    #print(\"Node {} is in connected component {} and explored is {}\".format(n, ndict['cc'], ndict['explored']))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Running time DFS\n",
    "For running time of DFS, it is similar to BFS, each vertex is visited only once and we travse all edges from the vertex currently processed. Therefore the running time is $\\theta(m + n)$\n",
    "\n",
    "## Topological Sort\n",
    "We will now look at an application of DFS, which is topological sort.<br><br>\n",
    "**Topological ordering is implemented in the above graph and is given by it's \"finishing time\"**\n",
    "\n",
    "\n",
    "\n",
    "Topological ordering of a graph G = (V, E) is assignment $f(v)$ to every vertex v $\\in$ V such that for every edge (v, w) $\\in$ E, $f(v) < f(w)$\n",
    "\n",
    "Topological ordering is useful when we want to sequence tasks when precence constraints are respected.\n",
    "\n",
    "<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kosaraju's Two-Pass Algorithm :  DFS - Strongly Connected Component\n",
    "\n",
    "##### Computing Strongly connected components\n",
    "Strongly connected component of a directed graph G with vertices V is a maximal set of S $\\subseteq$ V, such that we can travese from any given vertex to any other vertex in these set of vertices.\n",
    "\n",
    "The following directed graph has three strongly connected, components with each SCC as a vertex in the graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Kosaraju's Two-Pass Algorithm using reverse deep copied  graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 275,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Node \t Strongly Connected Component \t starting time \t finishing time \t explored\n",
      "1 \t\t 6 \t\t\t 5 \t\t 5 \t\t\t 1\n",
      "2 \t\t 4 \t\t\t 8 \t\t 8 \t\t\t 1\n",
      "3 \t\t 4 \t\t\t 9 \t\t 7 \t\t\t 1\n",
      "4 \t\t 4 \t\t\t 7 \t\t 9 \t\t\t 1\n",
      "5 \t\t 6 \t\t\t 6 \t\t 4 \t\t\t 1\n",
      "6 \t\t 6 \t\t\t 4 \t\t 6 \t\t\t 1\n",
      "7 \t\t 9 \t\t\t 2 \t\t 2 \t\t\t 1\n",
      "8 \t\t 9 \t\t\t 3 \t\t 1 \t\t\t 1\n",
      "9 \t\t 9 \t\t\t 1 \t\t 3 \t\t\t 1\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118ea70d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ft = 1\n",
    "order = 1\n",
    "\n",
    "def dfs_scc(G, startvert, start):\n",
    "    global ft\n",
    "    global order\n",
    "    G.nodes[startvert]['explored'] = 1\n",
    "    G.nodes[startvert]['scc'] = start\n",
    "    G.nodes[startvert]['starting time'] = order\n",
    "    w = list(G.adj[startvert])\n",
    "    for adjnode in w:\n",
    "        if G.nodes[adjnode]['explored'] == 0:\n",
    "            order = order+1\n",
    "            dfs_scc(G, adjnode, start)\n",
    "            G.nodes[adjnode]['finishing time'] = ft\n",
    "            ft += 1\n",
    "    \n",
    "def kosarajuSCC(DG):\n",
    "    # 1st Pass\n",
    "    reverseDG = DG.reverse() # default copy=True\n",
    "    for node in list(reverseDG.nodes):\n",
    "        reverseDG.nodes[node]['explored'] = 0\n",
    "        reverseDG.nodes[node]['scc'] =  float('inf')\n",
    "        reverseDG.nodes[node]['starting time'] =  float('inf')\n",
    "        reverseDG.nodes[node]['finishing time'] =  node\n",
    "    global ft\n",
    "    global order\n",
    "    ft = 0\n",
    "    order = 0\n",
    "    reversed_graph = reverseDG.nodes.data()\n",
    "    nodes_reverse = []\n",
    "    for n, ndict in reversed_graph:\n",
    "        nodes_reverse.append(ndict['finishing time'])\n",
    "    nodes_reverse.reverse()\n",
    "    for node_ft in nodes_reverse:\n",
    "        for n, ndict in reversed_graph:\n",
    "            if ndict['finishing time'] == node_ft:\n",
    "                if reverseDG.nodes[n]['explored'] == 0:\n",
    "                    order = order+1\n",
    "                    ft += 1\n",
    "                    dfs_scc(reverseDG, n, n)\n",
    "                    reverseDG.nodes[n]['finishing time'] = ft\n",
    "    # 2nd Pass\n",
    "    DG = reverseDG.reverse()\n",
    "    ft = 0\n",
    "    order = 0\n",
    "    for node in list(DG.nodes):\n",
    "        DG.nodes[node]['explored'] = 0\n",
    "    orig_graph = DG.nodes.data()\n",
    "    for node_ft in nodes_reverse:\n",
    "        for n, ndict in orig_graph:\n",
    "            if ndict['finishing time'] == node_ft:\n",
    "                if DG.nodes[n]['explored'] == 0:\n",
    "                    order = order+1\n",
    "                    ft += 1\n",
    "                    dfs_scc(DG, n, n)\n",
    "                    DG.nodes[n]['finishing time'] = ft\n",
    "                    \n",
    "    print(\"Node \\t Strongly Connected Component \\t starting time \\t finishing time \\t explored\")\n",
    "    for n, ndict in DG.nodes.data():\n",
    "        print(\"{} \\t\\t {} \\t\\t\\t {} \\t\\t {} \\t\\t\\t {}\".format(n, ndict['scc'], ndict['starting time'], ndict['finishing time'], ndict['explored'])) \n",
    "    \n",
    "            \n",
    "        \n",
    "DG = nx.DiGraph()\n",
    "reverseDG = nx.DiGraph()\n",
    "edges = [(9, 7), (7, 8), (8, 9), (6, 9), (5, 6), (6, 1), (1, 5), (4, 5), (3, 4), (4, 2), (2, 3)]\n",
    "DG.add_edges_from(edges)\n",
    "nx.draw(DG, with_labels = True)\n",
    "\n",
    "kosarajuSCC(DG)    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Todo:  Challenging Problem:\n",
    "\n",
    "Download the following text file:  SCC.txt\n",
    "<br>\n",
    "The file contains the edges of a directed graph. Vertices are labeled as positive integers from 1 to 875714. Every row indicates an edge, the vertex label in first column is the tail and the vertex label in second column is the head (recall the graph is directed, and the edges are directed from the first column vertex to the second column vertex). So for example, the $11^{th}$ row looks liks : \"2 47646\". This just means that the vertex with label 2 has an outgoing edge to the vertex with label 47646\n",
    "\n",
    "Your task is to code up the algorithm from the video lectures for computing strongly connected components (SCCs), and to run this algorithm on the given graph.\n",
    "\n",
    "Output Format: You should output the sizes of the 5 largest SCCs in the given graph, in decreasing order of sizes, separated by commas (avoid any spaces). So if your algorithm computes the sizes of the five largest SCCs to be 500, 400, 300, 200 and 100, then your answer should be \"500,400,300,200,100\" (without the quotes). If your algorithm finds less than 5 SCCs, then write 0 for the remaining terms. Thus, if your algorithm computes only 3 SCCs whose sizes are 400, 300, and 100, then your answer should be \"400,300,100,0,0\" (without the quotes).\n",
    "\n",
    "WARNING: This is the most challenging programming assignment of the course. Because of the size of the graph you may have to manage memory carefully. The best way to do this depends on your programming language and environment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Test case\n",
    "http = urllib3.PoolManager()\n",
    "r1 = http.request('GET', \"https://d3c33hcgiwev3.cloudfront.net/_410e934e6553ac56409b2cb7096a44aa_SCC.txt?Expires=1561507200&Signature=b-X~V6kXqfyz3PUZ6frDKWfBOdTiNqzgLRrzd3UwTZ9XGBPhztKN0ZALk7pK-01DcpKHnyLPUZK0YHduav3xof2xNchlXUz5vJ67WgvEjevxo0Kf8F1jwKVUjEFqVnsTxxedXejOHcXYtoU-MVFJtYxiWp2veOruppHBmzYidAc_&Key-Pair-Id=APKAJLTNE6QMUY6HBC5A\")\n",
    "IntegerMatrixStringJoin = r1.data.split('\\r\\n')\n",
    "IntegerMatrixString = IntegerMatrixStringJoin[0].split(' \\n')\n",
    "for i in IntegerMatrixString:\n",
    "    from_to = i.split(' ')\n",
    "    fromint = int(from_to[0])\n",
    "    toint = int(from_to[1])\n",
    "    print(\"from: {} \\t to: {}\".format(fromint, toint))\n",
    "    DG.add_edge(fromint, toint)\n",
    "    \n",
    "kosarajuSCC(DG)  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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