{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Searching, Sorting, and Timing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Agenda\n",
    "\n",
    "1. Timing\n",
    "2. Prelude: Timing list indexing\n",
    "3. Linear search\n",
    "4. Binary search\n",
    "5. Insertion sort"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Timing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import time\n",
    "time.time()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Prelude: Timing list indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "timeit.timeit(stmt='lst[0]',\n",
    "              setup='import random; lst=[0] * 10**6')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "timeit.timeit(stmt='lst[10**6-1]',\n",
    "              setup='import random; lst=[0] * 10**6')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import random\n",
    "size = 10**3\n",
    "times = [0] * size\n",
    "lst   = [0] * size\n",
    "for _ in range(100):\n",
    "    for i in range(size):\n",
    "        times[i] += timeit.timeit(stmt='lst[{}]'.format(i),\n",
    "                                  globals=globals(),\n",
    "                                  number=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "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": {},
   "source": [
    "Accessing an element in a list by index always takes the same amount of time, regardless of position. I.e., indexing incurs a *constant time* delay.\n",
    "\n",
    "How? **A Python list uses an array as its underlying data storage mechanism.** To access an element in an array, the interpreter:\n",
    "\n",
    "1. Computes an *offset* into the array by multiplying the element's index by the size of each array entry (which are uniformly sized, since they are merely *references* to the actual elements)\n",
    "2. Adds the offset to the *base address* of the array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Linear Search"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Task: to locate an element with a given value in a list (array)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def index(lst, x):\n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = list(range(100))\n",
    "index(lst, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, 99)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "index(lst, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def index(lst, x):\n",
    "    raise ValueError(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "try:\n",
    "    print('Value found at', index(lst, -1))\n",
    "except ValueError as e:\n",
    "    print('Value not found:', e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "times = []\n",
    "lst = list(range(1000))\n",
    "for x in lst:\n",
    "    times.append(timeit.timeit(stmt='index(lst, {})'.format(x),\n",
    "                               globals=globals(),\n",
    "                               number=100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot(times, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Binary search"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Task: to locate an element with a given value in a list (array) whose contents are *sorted in ascending order*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def index(lst, x):\n",
    "    # assume that lst is sorted!!!\n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = list(range(1000))\n",
    "index(lst, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, 999)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "index(lst, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(len(lst)):\n",
    "    assert(i == index(lst, i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "times = []\n",
    "lst = list(range(1000))\n",
    "for x in lst:\n",
    "    times.append(timeit.timeit(stmt='index(lst, {})'.format(x),\n",
    "                               globals=globals(),\n",
    "                               number=1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot(times, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "import random\n",
    "times = []\n",
    "for size in range(100, 10000, 100):\n",
    "    lst = list(range(size))\n",
    "    times.append(timeit.timeit(stmt='index(lst, -1)'.format(random.randrange(size)),\n",
    "                               globals=globals(),\n",
    "                               number=10000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot(times, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "import random\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": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot(times, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Insertion sort"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Task: to sort the values in a given list (array) in ascending order."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import random\n",
    "lst = list(range(1000))\n",
    "random.shuffle(lst)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(lst, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def insertion_sort(lst):\n",
    "    pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "insertion_sort(lst)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(lst, 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "import random\n",
    "times = []\n",
    "for size in range(100, 5000, 100):\n",
    "    lst = list(range(size))\n",
    "    times.append(timeit.timeit(stmt='insertion_sort(lst)',\n",
    "                               setup='random.shuffle(lst)',\n",
    "                               globals=globals(),\n",
    "                               number=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.plot(times, 'ro')\n",
    "plt.show()"
   ]
  }
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