{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Searching, Sorting, and Timing\n",
    "==============================\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Agenda\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "1.  Timing\n",
    "2.  Prelude: Timing list indexing\n",
    "3.  Linear search\n",
    "4.  Binary search\n",
    "5.  Insertion sort\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 1.  Timing\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "IndentationError",
     "evalue": "unexpected indent (<ipython-input-1-88c2d8f0fad2>, line 2)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-1-88c2d8f0fad2>\"\u001b[0;36m, line \u001b[0;32m2\u001b[0m\n\u001b[0;31m    time.time()\u001b[0m\n\u001b[0m    ^\u001b[0m\n\u001b[0;31mIndentationError\u001b[0m\u001b[0;31m:\u001b[0m unexpected indent\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "  time.time()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t1 = time.time()\n",
    "  time.sleep(1)\n",
    "  t2 = time.time()\n",
    "  t2 - t1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 1.  Prelude: Timing list indexing\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = [0] * 10**5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  timeit.timeit(stmt='lst[0]', globals=globals())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "timeit.timeit(stmt='lst[10**5-1]', globals=globals())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "times = [timeit.timeit(stmt='lst[{}]'.format(i),\n",
    "                         globals=globals(),\n",
    "                         number=100)\n",
    "           for i in range(10**5)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "times[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "  import matplotlib.pyplot as plt\n",
    "  plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Observation: accessing an element in a list by index takes a constant\n",
    "amount of time, regardless of position.\n",
    "\n",
    "How? **A Python list uses an array as its underlying data storage\n",
    "mechanism.** To access an element in an array, the interpreter:\n",
    "\n",
    "1.  Computes an *offset* into the array by multiplying the element's\n",
    "    index by the size of each array entry (which are uniformly sized,\n",
    "    since they are merely *references* to the actual elements)\n",
    "2.  Adds the offset to the *base address* of the array\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 1.  Linear Search\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Task: to locate an element with a given value in a list (array).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def index(lst, x):\n",
    "      pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = list(range(100))\n",
    "  index(lst, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, 99)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  lst = list(range(1000))\n",
    "  times = [timeit.timeit(stmt='index(lst, {})'.format(x),\n",
    "                         globals=globals(),\n",
    "                         number=100)\n",
    "           for x in range(1000)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "  plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 1.  Binary search\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Task: to locate an element with a given value in a list (array) whose\n",
    "contents are *sorted in ascending order*.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def index(lst, x):\n",
    "      # assume that lst is sorted!!!\n",
    "      pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = list(range(1000))\n",
    "  index(lst, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, 999)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  lst = list(range(1000))\n",
    "  times = [timeit.timeit(stmt='index(lst, {})'.format(x),\n",
    "                         globals=globals(),\n",
    "                         number=1000)\n",
    "           for x in range(1000)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "  plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  times = []\n",
    "  for size in range(1000, 100000, 100):\n",
    "      lst = list(range(size))\n",
    "      times.append(timeit.timeit(stmt='index(lst, -1)',\n",
    "                                 globals=globals(),\n",
    "                                 number=1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "  plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  times = []\n",
    "  for e in range(5, 20):\n",
    "      lst = list(range(2**e))\n",
    "      times.append(timeit.timeit(stmt='index(lst, -1)',\n",
    "                                 globals=globals(),\n",
    "                                 number=100000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "  plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 1.  Insertion sort\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Task: to sort the values in a given list (array) in ascending order.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import random\n",
    "  lst = list(range(1000))\n",
    "  random.shuffle(lst)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(lst, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def insertion_sort(lst):\n",
    "      pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "insertion_sort(lst)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(lst, 'ro')\n",
    "  plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "  import random\n",
    "  times = [timeit.timeit(stmt='insertion_sort(lst)',\n",
    "                         setup='lst=list(range({})); random.shuffle(lst)'.format(size),\n",
    "                         globals=globals(),\n",
    "                         number=1)\n",
    "           for size in range(100, 5000, 250)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(times, 'ro')\n",
    "  plt.show()"
   ]
  }
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