class06_bfs_R.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CSX46\n",
    "\n",
    "## Class session 6:  BFS\n",
    "Objective: write and test a function that can compute single-vertex shortest paths in an unweighted simple graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's load the packages that we will need"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "suppressPackageStartupMessages(library(igraph))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the random number seed to 1337 using `set.seed`. Make a simple 10-vertex random (Barabasi-Albert model) graph.  Set the random number seed so that the graph is always the same, for purposes of reproducibility (we want to know that the \"hub\" vertex will be vertex 2, and we will test your BFS function starting at that \"hub\" vertex)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ROp3OuUMAZ0SwA1xLq9VO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",
      "text/plain": [
       "Plot with title “”"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      },
      "text/plain": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "set.seed(1337)\n",
    "g <- barabasi.game(10, directed=FALSE)\n",
    "plot(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'igraph.vs'"
      ],
      "text/latex": [
       "'igraph.vs'"
      ],
      "text/markdown": [
       "'igraph.vs'"
      ],
      "text/plain": [
       "[1] \"igraph.vs\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "is(V(g))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a function `bfs_single_vertex` that accepts two arguments, `p_graph` and `p_vertex`, and returns the vector of distances from vertex `p_vertex` to all the other vertices in the graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create an igraph.vs object by calling the V function and passing g to it; assign the vs object to name \"vertices\"\n",
    "# compute N, the number of vertices by calling length on the vertices object\n",
    "# initialize \"queue\" array (length N, containing NA)\n",
    "# initialize distances array (length N, containing NA)\n",
    "# set \"p_vertex\" entry of distances array to be 0\n",
    "# while write_ptr is gerater than read_ptr:\n",
    "# obtain the vertex ID of the entry at index \"read_ptr\" in the queue array, as cur_vertex_num\n",
    "# increment read_ptr\n",
    "# get the distance to cur_vertex_num, from the \"distances\" array\n",
    "# get the neighbors of vertex cur_vertex_num in the graph, using the igraph \"neighbors\" func\n",
    "# for each vertex_neighbor in the array vertex_neighbors\n",
    "#    if the distances[vertex_neighbor] is NA:\n",
    "#         (1) set the distance to vertex_neighbor (in \"distances\" vector) to the distance to\n",
    "#           cur_vertex_num, plus one\n",
    "#         (2) add neighbor to the queue\n",
    "#         put vertex_neighbor at position write_ptr in the queue array\n",
    "#         increment write_ptr\n",
    "# end-while\n",
    "# return \"distances\"\n",
    "\n",
    "bfs_single_vertex <- function(p_graph, p_vertex) {\n",
    "    vertices <- V(p_graph)\n",
    "    stopifnot(p_vertex %in% vertices)\n",
    "    N <- length(vertices)\n",
    "    distances <- rep(NA, N)\n",
    "    distances[p_vertex] <- 0\n",
    "    queue <- c(p_vertex, rep(NA, N-1))\n",
    "    read_ptr <- 1\n",
    "    write_ptr <- 2\n",
    "\n",
    "    while(write_ptr > read_ptr) {\n",
    "        cur_vertex_num <- queue[read_ptr]\n",
    "        read_ptr <- read_ptr + 1\n",
    "        cur_vertex_distance <- distances[cur_vertex_num]\n",
    "        stopifnot(! is.na(cur_vertex_distance))\n",
    "        vertex_neighbors <- neighbors(p_graph, cur_vertex_num)\n",
    "        for (vertex_neighbor in vertex_neighbors) {\n",
    "            if (is.na(distances[vertex_neighbor])) {\n",
    "                distances[vertex_neighbor] <- cur_vertex_distance + 1\n",
    "                queue[write_ptr] <- vertex_neighbor\n",
    "                write_ptr <- write_ptr + 1\n",
    "            }\n",
    "        }\n",
    "    }\n",
    "\n",
    "    distances\n",
    " }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test out your function, starting at vertex 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 0\n",
       "3. 1\n",
       "4. 1\n",
       "5. 2\n",
       "6. 1\n",
       "7. 1\n",
       "8. 1\n",
       "9. 2\n",
       "10. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 1 0 1 1 2 1 1 1 2 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bfs_single_vertex(g, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 1 × 10 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>0</td><td>1</td><td>1</td><td>2</td><td>1</td><td>1</td><td>1</td><td>2</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 1 × 10 of type dbl\n",
       "\\begin{tabular}{llllllllll}\n",
       "\t 1 & 0 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 1 × 10 of type dbl\n",
       "\n",
       "| 1 | 0 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]\n",
       "[1,] 1    0    1    1    2    1    1    1    2    1    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shortest.paths(g, v=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's define a function sp_matrix that computes the all-pairs shortest-paths distances and puts them in a matrix. You will have to use V, bfs_single_vertex, length, lapply, cbind, and do.call."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "sp_matrix <- function(p_graph) {\n",
    "    do.call(cbind, lapply(1:length(V(p_graph)), function(vi) { bfs_single_vertex(g, vi) }))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's test out our function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 10 × 10 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>0</td><td>1</td><td>2</td><td>2</td><td>1</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>1</td><td>0</td><td>1</td><td>1</td><td>2</td><td>1</td><td>1</td><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>0</td><td>2</td><td>3</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>0</td><td>3</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>1</td><td>2</td><td>3</td><td>3</td><td>0</td><td>3</td><td>3</td><td>3</td><td>4</td><td>3</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>3</td><td>0</td><td>2</td><td>2</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>3</td><td>2</td><td>0</td><td>2</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td><td>0</td><td>1</td><td>2</td></tr>\n",
       "\t<tr><td>3</td><td>2</td><td>3</td><td>3</td><td>4</td><td>3</td><td>3</td><td>1</td><td>0</td><td>3</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td><td>2</td><td>3</td><td>0</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 10 × 10 of type dbl\n",
       "\\begin{tabular}{llllllllll}\n",
       "\t 0 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 3 & 2\\\\\n",
       "\t 1 & 0 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1\\\\\n",
       "\t 2 & 1 & 0 & 2 & 3 & 2 & 2 & 2 & 3 & 2\\\\\n",
       "\t 2 & 1 & 2 & 0 & 3 & 2 & 2 & 2 & 3 & 2\\\\\n",
       "\t 1 & 2 & 3 & 3 & 0 & 3 & 3 & 3 & 4 & 3\\\\\n",
       "\t 2 & 1 & 2 & 2 & 3 & 0 & 2 & 2 & 3 & 2\\\\\n",
       "\t 2 & 1 & 2 & 2 & 3 & 2 & 0 & 2 & 3 & 2\\\\\n",
       "\t 2 & 1 & 2 & 2 & 3 & 2 & 2 & 0 & 1 & 2\\\\\n",
       "\t 3 & 2 & 3 & 3 & 4 & 3 & 3 & 1 & 0 & 3\\\\\n",
       "\t 2 & 1 & 2 & 2 & 3 & 2 & 2 & 2 & 3 & 0\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 10 × 10 of type dbl\n",
       "\n",
       "| 0 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 2 |\n",
       "| 1 | 0 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 |\n",
       "| 2 | 1 | 0 | 2 | 3 | 2 | 2 | 2 | 3 | 2 |\n",
       "| 2 | 1 | 2 | 0 | 3 | 2 | 2 | 2 | 3 | 2 |\n",
       "| 1 | 2 | 3 | 3 | 0 | 3 | 3 | 3 | 4 | 3 |\n",
       "| 2 | 1 | 2 | 2 | 3 | 0 | 2 | 2 | 3 | 2 |\n",
       "| 2 | 1 | 2 | 2 | 3 | 2 | 0 | 2 | 3 | 2 |\n",
       "| 2 | 1 | 2 | 2 | 3 | 2 | 2 | 0 | 1 | 2 |\n",
       "| 3 | 2 | 3 | 3 | 4 | 3 | 3 | 1 | 0 | 3 |\n",
       "| 2 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 0 |\n",
       "\n"
      ],
      "text/plain": [
       "      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]\n",
       " [1,] 0    1    2    2    1    2    2    2    3    2    \n",
       " [2,] 1    0    1    1    2    1    1    1    2    1    \n",
       " [3,] 2    1    0    2    3    2    2    2    3    2    \n",
       " [4,] 2    1    2    0    3    2    2    2    3    2    \n",
       " [5,] 1    2    3    3    0    3    3    3    4    3    \n",
       " [6,] 2    1    2    2    3    0    2    2    3    2    \n",
       " [7,] 2    1    2    2    3    2    0    2    3    2    \n",
       " [8,] 2    1    2    2    3    2    2    0    1    2    \n",
       " [9,] 3    2    3    3    4    3    3    1    0    3    \n",
       "[10,] 2    1    2    2    3    2    2    2    3    0    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sp_matrix(g)"
   ]
  }
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