wk9/MY470_wk9_lecture.ipynb

{
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   "source": [
    "### MY470 Computer Programming\n",
    "# ⚡️ Algorithms and Order of Growth\n",
    "### Week 9 Lecture"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Overview\n",
    "\n",
    "* Algorithms and computational complexity\n",
    "* Time complexity\n",
    "    * Big-O notation\n",
    "    * Important computational complexity classes\n",
    "    * Performance of common Python data structures and methods\n",
    "* Space complexity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## From Week 1: What Is an Algorithm?\n",
    "\n",
    "A well-defined computational procedure that takes value(s) as input and produces value(s) as output\n",
    "\n",
    "  * \"Recipe\" or \"instructions\" for solving a well-defined computational problem\n",
    "  * Consists of a sequence of simple steps, control flow, and a stopping rule\n",
    "  * Can be specified in human language or programming language"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Algorithm Analysis\n",
    "\n",
    "Helps compare algorithms based on the amount of **computing resources** they use\n",
    "\n",
    "  * Time (execution or **running time**)\n",
    "  * Space (**memory**)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity: Example 1\n",
    "\n",
    "How long does this program take to run?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def sum_of_n(n):\n",
    "    \"\"\"Assume n is a positive integer. \n",
    "    Calculate the sum of all integers from 1 to n, inclusive.\n",
    "    \"\"\"\n",
    "    res = 0\n",
    "    for i in range(1, n + 1):\n",
    "        res += i\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "## Benchmarking"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "500000500000 0.053186893463134766\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "start = time.time()\n",
    "res = sum_of_n(1000000)\n",
    "end = time.time()\n",
    "print(res, end - start)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Limitations to Benchmarking \n",
    "\n",
    "Running time depends on:\n",
    "\n",
    "  * The speed of the computer\n",
    "  * The programming language\n",
    "  * The compiler/programming language implementation\n",
    "  * The value of the input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5000050000 0.006281852722167969\n",
      "500000500000 0.05945992469787598\n",
      "50000005000000 0.5503628253936768\n"
     ]
    }
   ],
   "source": [
    "for i in [100000, 1000000, 10000000]:\n",
    "    start = time.time()\n",
    "    res = sum_of_n(i)\n",
    "    end = time.time()\n",
    "    print(res, end - start)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Measure Algorithm Efficiency Inpendently of Computer and Program\n",
    "\n",
    "* Count **number of steps**, or basic operations (not seconds)\n",
    "* Express measure as **function of the size of inputs**\n",
    "* Consider the **upper bound** on the running time (the worst-case scenario) \n",
    "* Use an **approximation** as the size of the input gets extremely large"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity: Example 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def sum_of_n(n):\n",
    "    \"\"\"Assume n is a positive integer. \n",
    "    Calculate the sum of all integers from 1 to n, inclusive.\n",
    "    \"\"\"\n",
    "    res = 0\n",
    "    for i in range(1, n + 1):\n",
    "        res += i\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "* The basic operation is assignment\n",
    "* The number of steps is $T(n) = 1 + n$\n",
    "* This is on the order of n: $O(n)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Big-O Notation\n",
    "\n",
    "* The mathematical way to express running time\n",
    "* Gives an **upper bound** on the asymptotic growth (also called **order of growth**) of an algorithm\n",
    "\n",
    "   >The complexity of `func(n)` is $O(n^2)$\n",
    "\n",
    "   * This means that in the worst case `func` will take on the order of $n^2$ steps to run."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Important Computational Complexity Classes\n",
    "\n",
    "| Big-O notation      | Running time   \n",
    "| :------------ |:------\n",
    "| $O(1)$     | constant           \n",
    "| $O(\\log n)$ | logarithmic  \n",
    "| $O(n)$     | linear\n",
    "| $O(n \\log n)$ | log-linear \n",
    "| $O(n^c)$ | polynomial\n",
    "| $O(c^n)$ | exponential"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Comparison of Complexity Classes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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fZMuWLUyaNImzZ8/y2WefER0dzbhx41i/fj1xcXHs2LGDP/3pT2zbtu2euk+fPs3rr7/O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\n+CDaaMSrkw+ra61J3J3NwX+cw7FfT2b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ll+NPtzM5Eqai0c97PnE1VdvVu7CK5REbG7PG\nrFtbCbe0WsLd2pqw+AdC4Kqpxt3QSP4tt1i964ZG3PUBtLy8tW2AQrFOUUK+wYmEpnn+u9/i7K9+\nQfVVO7n7U58hv3hzWnszZvLOsYu8+tN2JoZClAcKue0jV1J15SY1rqhYlNmAs5YWIi2thNtarX1r\nK7HBueU1ha6j19mLf9x3H+6GevT6evS6OrUmtkKxRJYt5EKIauBvgXJAAt+WUn5NCPF54A+AAdv0\nc1LKpy+1ooql09/eylNf+xIjvd0c+J0Pc8P7PoimpZ6YxTQl51+xBHxsYJqy2gJufmg7NVcXKwFX\nzEMaBpHOLiKtljs80tpCuLWNSGtrQsCZVliIu76e/Jtvwl1vibW7oQFXZaVarUuhWCEupUceBf69\nlPKkEKIAOCGEeM7Oe1RK+deXXj3FcpBScvroM7zw+Hfw5Bfw/j//AjU7dqW2NSXNJ/p55ak2RvuC\nbK7K5+5/s5O6XWp2KgXEJieJtFkCPSPU4dZWIh0diePX5eW4G+rxP/ig1bsO1Fvj12qWM4Vi1Vm2\nkEspe4FeOz0hhHgLqFypiimWR2hqkuf+19c5d+wl6nbv4a4//Hf4/PMD06QpaX19gFeeamO4Z4ri\nrXkcfngH9btLEZp68OYS1rKavZZgt9lC3dZOpLWV6MDAnKHDgV5djd7QQMFtt1mvczU0oAfqceSr\n8WuFYq1YkTFyIUQdcC1wDHgX8CkhxEeB41i99pGVuI5iYXqb3+Gpr36JyeFBbvrdj7P3yIMITUuw\nCU0ZvP0vvZz5ZTdjA9Ns2uLjjt+/msbrypSAb3DMqSnC7e2zIh1uswW7vT3hda4Zd3jewYPo9QHL\nJR4IoFdVIdRa2ArFukPImWjR5Z5AiHzgl8AXpJQ/EkKUA4NY4+b/GaiQUn4iRbmHgYcBampq9ly4\ncOGS6pHLSNPkxE9/wq//8XHyizdzz7/9U7ZuuyLBZqBjgjde7OLcq33EDJOKBj87b6miYU8ZmhLw\nDcNMsJkl1G1EWu1edls70d7eOUNNw1VVhR6owx2otwQ7EECvr8dRrOIiFIrLgRDihJRy4VeIMjnP\npQi5EMIFPAX8TEr5lRT5dcBTUsoFpw7bu3evPH78+LLrkcsEx8f42be+SuvJV2ncd4A7H/kTPPnW\nPOdRI0bLiX7e+GU3fW3jOHWNbddvYefNlZRUqalUsxlr7NrqTUfa2oi0txNubyPSfgEZDM7aaXl5\nVm96tmddjx6oQ6+tRXO7164BCoVixYT8UqLWBfBd4K14ERdCVNjj5wAPAmcurYqKdHSdPcNPv/5X\nTI+PcdvHP8nuO48ghGB8cJo3f93N2Zd6CU0aFJX7OPiBJq64YQtun2utq63IEGkYRLq6Ugp2bGDu\nVS6EwFVZiR4I4NuzF3d9wBLvQL2ajlShyAEuZYz8XcBHgDeEEK/bxz4HPCSE2I3lWm8HPnlJNVTM\nwzRjvPLjJ/jtE9/DX17OQ//ly5TV1tNxdpgzL3bRfmYIAQSuKWXHzZVUXaHeAV+vSCmJDgzYQp0o\n2JGuLohGZ20dmzah19WRf+gm9Lo6yy1eV4erpkb1rhWKHOZSotZ/A6RSB/XO+CoyOTLMM//jr+k4\nc5orD97CwYcepuW1UX7+2MuMD0zjLXCx53AtVx+qpKBYTayxXohNTBBpv0DkwgUiF+JEu70dc3Jy\n1k643ei1tbi3baPgzjvR6+pwB+rQ6+pwFKlpcRWK9YSUklAsRNAIEowGmY5Oz6WNaYLRIEHDPm6n\ng9HgbHqlUDO7ZRHtp07yzDe+QmR6mgPvf5jgVCPf+8uTVvBao5/r7wvQcG0ZDqe2+MkUK44ZDBLp\n6LAEu73dFm1riw0NzRkKgauiAr2uDv/999u96wB6XR2urRXz3jRQKBSXjmEac6JqBNOLbApRjhfm\n5PKSzOPMvE4vPqcPn8uH1+ldsbYpIc8CzFiMl37w97zykyfI37yV0oYP89pRD069n+03qOC1y4kZ\nDmN0dMyJdJxoR/v7E2ydpaXotbXk33oLem2tJdi1teg1NWoaUoUiDaY0CUVDKcU1pcgmiXOyAM+k\no2Z08YvbuDTXrNjmOfPwuXz4nD625G1JEOP4vdfpnXfM5/ThdVn2HqcHTST+ky5SOrWXjhLydc74\nYD9PfvmL9LW+g553DUbsJqT0c/ADlVxxoAK3V93ClWY2yOzCnEgbtmgbvb1zC3wAjuJi9Npa8m68\nEb2udk6wa2rUIh+KDY2UkogZSXAhx4tnck93QVdznAhPR6czroNApBRUv8dPhbMipeDOCHRCGVdi\nT9mlZVdQsFKBdYoRMnj5Jz/n+D//DWY0hivvbhr3H2LHLZVUbVfBa5eKGQphdHZarvALHUQ6OzAu\ndBDp6LDEOhabtdUKC9Fra/Hu2YO/dkasrb2jsHANW6FQZEbUjKZ1FS9FZJNFOSZji1/cxuPwzApl\nfO+12FOcUlzjj8ULbXyex+FRz0KUkK8rjEiMMy+c4vQvjjLUcQJpTuHQy9lz38PsuesaFby2RGKT\nk5YbPIVYR/v6Emw1vx+9pgbvNddQeO8RS6zt3rWjqEg9LBSXBSnlrJDOCOdCApxqrDe5hzwdnSYc\nC2dcB6dwphTOEm8JvoL5ruTFXM0z+Y40CzYpLh0l5GtMJBTl3LF2Tj13lP62VzCj/YBGYfmVXHXo\nVvbdexu6R02LmQopJbHRUcvt3dlpiXXHBYwOq6cdGx5OsHeUlKDX1JB34ACummr0mlr02hr06moV\nEa5YMkbMyGj8drFgqfge8HR0etnBUzP7Ar2AMl/ZnIjGjdemGs/1Or3kuebGgV2O7HIrK5SQrwmh\nKYOWk72c/sVv6Gs+RizSBpj4/FVsO/Cv2P/AnRRs2rTW1VwXSMPAuHjRcoN3dmF02fuODiKdnZgT\nE3PGQuDcsgW9poaCd787QaxdVdVqYY8cxZRmxpHKM3npAqbixXopwVO6ps/1cuME1Z/vzyh4alZs\nFwmeUuQmSsgvE8HxCC2v9fPWr0/Se+4Y0cjbIMO4PIU0HbyLfffdRVlt3VpX87IjpcQcG0sU6c5O\nIl2dGJ1d88arcbnQt27FVV2Nf/duS6Sra2yxrlITo2QxUkrCsXD6Xm0qkV0gUnnGdinBU5rQEsR2\nZjx3k2cTlc7KtAFTM0Kbakw3G4OnFNmFEvJVZHIkRMtrA5x7uZmec8eIhc8izWE0h4u6a/Zx7eE7\nqNt1LZpjY48dyUgEo7c3TqwtkZ4R64ReNVYkuKu6yhqvPnIPenU1rqpq9OoqnOXliA3+98oGDNNI\nGfy0kABn0vM1pZlxHeKDp+LdxyXekvSBU4uM6bodbhUPocg6lJCvMGMD07S81k/z8W4unj9JLHIW\nM9oBQFn9FVzzng+z/cAh3L6N4+aVsRjRgQGM7u7ZLdLdbfWoOzsxLl4Ec+4BLVwuXFVVuKqr8O2+\nFle1JdKu6mpclVXKBb6CJARPpYpSXsC9nCpwaiZtmEbGdUgXPFXmK5vnPl7wndy4vcfhUcFTCoXN\nJS9juhIU114pb//cY2tdjWXjDZmUjsYoGYninejAjLxF1DiHkAYRdxEDFTsZ3LKTsC87x72FaVIw\nNUrR+CBF40P2fpCiMeuzf2IIh5n4Gsqkr5ARf2nKbSK/CKnG9hKQSCRRJBFMQpgijEnY3ocwicSl\nw0n5M+nQbFrOHo+AyPw3rkk3Gm406UHDjUCfTVvH3Wh40KRuH/PMlZnNj7PDjVD9BYUiJT945Ma1\nXf0sl9EjksKpGP4pk6IJE+/UMLHIWaLGWxixcWIOnaEtVzG0ZRcTRdWwzl11yxHqCZ+fUX8J3Vvq\neHPbXkYLS+xtM2OFm4k6N26kvcSME894oQ0h54lrCqFNmR8CkblbWUgXGnqikEo3DvJx4EGY+qyQ\nJoprolDHHxe4EKh/sBSKbGNd9MjX83rkpikZ7pniYssoPc1j9DaPMj54ERkbAAZxiG5Ck50IoVG7\nazdX3XQbjftuwOVeP+98m9PTGL0XiV7sxejtxejpxbjYi9HTg9HdYwWUGYmuUkdJCa7KreiVlbiS\nt61bs2KK0fjgqXTBUYu9KpQ8BeR0dJpQLJRxHdIFTy3kNo4Plkr1+pDX6cWpqf/BFYpsZ83XI9+o\nGOEYfe3jXGwZpfudAXrOtxCZ7kNG+xFiCDM6gBmLACA0jaKaOvbd9zGuPHQLBcUll72+s+PTvb1E\ne22h7r1o73uI9l4kNjIyr5yztBRnRQXeHVdTeOcdiUJdUYHmXbkJ/TNhpYKnktNLCZ6aEdl48cxz\n5lHqLU0bqbzYmK6u6Sp4SqFQrCo5L+RTY2F6zo/ScaaD7rfPMXKxEzPaj4wNIM05AXR5vJQF6imr\nvY7SugBltfVsrqrBqa+eC1lKiTk+br1H3dNji/WMSNvC3deX+HoWoOXn46qowLm1Au/OXbgqKnBV\nbMFZUYFr61ZcZWWIZdY7eUGDjFYOihPndMv5LSV4yqW5UvZUy33liZHJC8ytnCzAKnhKoVBkKzkl\n5NKUDHSO0XzibbrOnmOws53wZC9mbADknLs0b1Mp5YFtlNc3UFZXT2ltgMLSshXtWclIxOpJ9/UT\n7e8n2t9HtL/f+txnp/v7kcGkNWtdLlzl5bgqKvDt24tzS4Ul1FsrcG7ZgquiAkdBAVLK2WX7gtEg\no7PC2UXw4rm0Cxos1gNe6oIGqXqqhe7CBVcRWmzeZTXzlEKhUMyxIYXcNGNMjY4x2NHHQEcfPeda\nGbjQxuRIN6YxCFjuVqE5KdhcSVndAaquaqK8voHSmgBun2/Z15ZSEhsZscS5rw+jr49o/0CCOEf7\n+uZNHwqAy4Uo3YxZUkQsUI5xXQOhzfkEN/uY2ORhfJPOmE8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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b0fcc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import math\n",
    "\n",
    "x = [i for i in range(1, 21)]\n",
    "\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(x, [10 for i in x], label=r'$O(1)$ constant')\n",
    "plt.plot(x, [math.log(i) for i in x], label=r'$O(\\log n)$ logarithmic')\n",
    "plt.plot(x, [i for i in x], label=r'$O(n)$ linear')\n",
    "plt.plot(x, [i*math.log(i) for i in x], label=r'$O(n \\log n)$ log-linear')\n",
    "plt.plot(x, [i**2 for i in x], label=r'$O(n^c)$ polynomial')\n",
    "plt.plot(x, [2**i for i in x], label=r'$O(c^n)$ exponential')\n",
    "plt.legend(loc='upper right')\n",
    "plt.ylim(0, 200)\n",
    "plt.xlim(1, 20)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Approximation\n",
    "\n",
    "* Keep the term with the largest growth rate\n",
    "* Drop any constants from the remaining term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WQ6f4pFcTKpUOcDqOiIjkglNJqXy1NJL3yv2Cn02Hf73idKQ8keUi2RgzALgd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\nEREREZEzqCSLiIiIiJzh/wDk9lgtA2GnTAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fba2278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import math\n",
    "\n",
    "maxx = [11, 501]\n",
    "fig = plt.figure(figsize=(12, 6))\n",
    "\n",
    "for k in range(len(maxx)):\n",
    "    plt.subplot(1, 2, k+1)\n",
    "    x = [i for i in range(1, maxx[k])]\n",
    "    plt.plot(x, [i*i + 5*i + 8 for i in x], label=r'$T(n) = n^2 + 5n + 8$')\n",
    "    plt.plot(x, [i*i for i in x], label=r'$O(n^2)$')\n",
    "plt.legend(loc='upper left')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Constant Complexity: $O(1)$\n",
    "\n",
    "* Running time is independent of the input size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5000050000 3.0994415283203125e-06\n",
      "500000500000 2.6226043701171875e-06\n",
      "50000005000000 1.6689300537109375e-06\n"
     ]
    }
   ],
   "source": [
    "def sum_of_n(n):\n",
    "    \"\"\"Assume n is a positive integer. \n",
    "    Calculate the sum of all integers from 1 to n, inclusive.\n",
    "    \"\"\"\n",
    "    return int((n*(n+1))/2)\n",
    "\n",
    "for i in [100000, 1000000, 10000000]:\n",
    "    start = time.time()\n",
    "    res = sum_of_n(i)\n",
    "    end = time.time()\n",
    "    print(res, end - start)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Logarithmic Complexity: $O(\\log n)$\n",
    "\n",
    "* Running time grows as the log of at least one of the inputs\n",
    "* Typically, we use base 2 but the base of the logarithm does not matter as $\\log_2 x = log_2 10 * log_{10} x$ and we ignore constants\n",
    "* If we use base 2, then we can say: **Every time the size of the input doubles, the algorithm performs one additional step.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "source": [
    "*To install R in Anaconda, run the following in the Terminal:* `conda install r-essentials`. *Then restart the Jupyter server.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "5.00030517578125"
      ],
      "text/latex": [
       "5.00030517578125"
      ],
      "text/markdown": [
       "5.00030517578125"
      ],
      "text/plain": [
       "[1] 5.000305"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "### R code ###\n",
    "\n",
    "# Program using bisection search to find an approximation \n",
    "# within epsilon to the square root of a number x.\n",
    "bisec_search <- function(x, epsilon) {\n",
    "  # Define interval for search\n",
    "  low <- 0\n",
    "  high <- max(1, x)\n",
    "\n",
    "  # Start in the middle\n",
    "  guess <- (low + high) / 2\n",
    "\n",
    "  # Narrow down search interval until guess close enough\n",
    "  while (abs(guess ** 2 - x) >= epsilon) {\n",
    "    if (guess ** 2 < x) {\n",
    "      low <- guess\n",
    "    } else {\n",
    "      high <- guess\n",
    "    }\n",
    "    guess <- (low + high) / 2\n",
    "  }\n",
    "  return(guess)\n",
    "}\n",
    "\n",
    "bisec_search(25, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Linear Complexity: $O(n)$\n",
    "\n",
    "* Usually, algorithms dealing with lists and sequences as they touch each element of the sequence a constant number of times\n",
    "    * E.g., comparing two strings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1 2 3 4 5 6 7 8 9 "
     ]
    }
   ],
   "source": [
    "for x in range(10):\n",
    "    print(x, end=' ')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "120"
      ],
      "text/latex": [
       "120"
      ],
      "text/markdown": [
       "120"
      ],
      "text/plain": [
       "[1] 120"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "### R code ###\n",
    "        \n",
    "# Program to find n!        \n",
    "factorial <- function(n) {    \n",
    "  if (n == 1) {\n",
    "    return(n)\n",
    "  } else {\n",
    "    return(n * factorial(n-1))\n",
    "  }\n",
    "}\n",
    "\n",
    "factorial(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Log-Linear Complexity: $O(n \\log n)$\n",
    "\n",
    "* Many practical algorithms are log-linear\n",
    "* For example, **sorting a list**\n",
    "* Next week, we'll study different sorting algorithms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Polynomial Complexity: $O(n^c)$\n",
    "\n",
    "* E.g. quadratic $O(n^2)$ or cubic $O(n^3)$\n",
    "* Usually, algorithms with nested loops"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3025\n"
     ]
    }
   ],
   "source": [
    "n = 10\n",
    "test = 0\n",
    "for i in range(1, n + 1):\n",
    "    for j in range(1, n + 1):\n",
    "        test += i * j\n",
    "print(test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exponential Complexity: $O(c^n)$\n",
    "\n",
    "* Many important problems are inherently exponential\n",
    "* Exponential algorithms are considered too slow to be practical\n",
    "* In such cases, we can find algorithms that provide approximate solutions or at least address specific instances of the problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 55\n"
     ]
    }
   ],
   "source": [
    "### R code ###\n",
    "\n",
    "# Program to find the n-th Fibonacci number using recursion\n",
    "recurse_fibonacci <- function(n) {\n",
    "    if(n <= 1) {\n",
    "        return(n)\n",
    "    } else {\n",
    "        return(recurse_fibonacci(n-1) + recurse_fibonacci(n-2))\n",
    "    }\n",
    "}\n",
    "\n",
    "print(recurse_fibonacci(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Important Computational Complexity Classes\n",
    "\n",
    "\n",
    "![Big O](figs/big-o.jpg \"Big O\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Analyzing Time Complexity\n",
    "\n",
    "Where to look:\n",
    "\n",
    "* Function and method calls \n",
    "* Recursive calls\n",
    "* Loops\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity of Python List Methods\n",
    "\n",
    "* Growing a list\n",
    "    * `L.append(k)` is $O(1)$\n",
    "    * `L + [k]` is $O(l)$ where $l$ is the size of list `L`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "source": [
    "*The method `append()` adds an item in place and since it adds to the end of the list, it does not have to touch the existing items*. In contrast, the list concatenation operation touches every item of the first list since it creates a copy of it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "l_concatenate \t 0.11763811111450195\n",
      "l_append \t 0.0005648136138916016\n"
     ]
    }
   ],
   "source": [
    "def l_concatenate(k):\n",
    "    lst = []\n",
    "    for i in range(k):\n",
    "        lst = lst + [i]\n",
    "    return lst\n",
    "\n",
    "def l_append(k):\n",
    "    lst = []\n",
    "    for i in range(k):\n",
    "        lst.append(i)        \n",
    "    return lst\n",
    "\n",
    "def timit(fnc, func_arg): # Note that functions can be passed as arguments\n",
    "    start = time.time()\n",
    "    fnc(*func_arg)  # Pass items from iterable as function arguments\n",
    "    end = time.time()\n",
    "    print(fnc.__name__, '\\t', end - start)\n",
    "    \n",
    "timit(l_concatenate, (10000,))\n",
    "timit(l_append, (10000,))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity of Python List Methods\n",
    "\n",
    "* Creating a list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "l_concatenate \t 0.12314295768737793\n",
      "l_append \t 0.0005640983581542969\n",
      "l_comprehen \t 0.00026702880859375\n",
      "l_range \t 0.00014495849609375\n"
     ]
    }
   ],
   "source": [
    "'''\n",
    "def l_concatenate(k):\n",
    "    lst = []\n",
    "    for i in range(k):\n",
    "        lst = lst + [i]\n",
    "    return lst\n",
    "\n",
    "def l_append(k):\n",
    "    lst = []\n",
    "    for i in range(k):\n",
    "        lst.append(i)        \n",
    "    return lst\n",
    "'''\n",
    "\n",
    "def l_comprehen(k):\n",
    "    return [i for i in range(k)]\n",
    "\n",
    "def l_range(k):\n",
    "    return list(range(k))\n",
    "\n",
    "timit(l_concatenate, (10000,))\n",
    "timit(l_append, (10000,))\n",
    "timit(l_comprehen, (10000,))\n",
    "timit(l_range, (10000,))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity of Python List Methods\n",
    "\n",
    "$O(1)$\n",
    "\n",
    "* Indexing `[]`, `append()`, `pop()`\n",
    "\n",
    "\n",
    "$O(n)$\n",
    "\n",
    "* Iteration, `pop(i)`, `insert()`, `del`, `remove()`, `in`, `reverse()`, `min()`, `max()`\n",
    "\n",
    "$O(n \\log n)$\n",
    "\n",
    "* `sort()`\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Time Complexity of Python Dictionary Methods\n",
    "\n",
    "$O(1)$\n",
    "\n",
    "* Indexing `[]`, `in`, `del`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "List:\n",
      "in_seq \t 0.00011086463928222656\n",
      "in_seq \t 0.001123189926147461\n",
      "in_seq \t 0.012640953063964844\n",
      "Dictionary:\n",
      "in_seq \t 9.5367431640625e-07\n",
      "in_seq \t 1.9073486328125e-06\n",
      "in_seq \t 2.86102294921875e-06\n"
     ]
    }
   ],
   "source": [
    "def in_seq(i, seq):\n",
    "    return i in seq\n",
    "\n",
    "print('List:')\n",
    "for n in [10000, 100000, 1000000]:\n",
    "    last = n - 1\n",
    "    lst = [i for i in range(n)]\n",
    "    timit(in_seq, (last, lst))\n",
    "\n",
    "print('Dictionary:')\n",
    "for n in [10000, 100000, 1000000]:\n",
    "    last = n - 1\n",
    "    dic = {i: None for i in range(n)}\n",
    "    timit(in_seq, (last, dic))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$O(n)$\n",
    "\n",
    "* Iteration\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Space Complexity \n",
    "\n",
    "Use Big-O notation for the amount of memory required by an algorithm.\n",
    "\n",
    "Compare the total size of any new variables the algorthm is allocating, relative to the size of the input.\n",
    "\n",
    "Where to look:\n",
    "\n",
    "* New lists, dictionaries, etc.\n",
    "* New stacks in recursive calls"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Space Complexity: Examples 1-2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "* $O(1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def print_n_times(s, n):\n",
    "    for i in range(n):\n",
    "        print(s)        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* $O(n)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def combine_n_times(s, n):\n",
    "    ls = [s for i in range(n)]\n",
    "    return ls"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "source": [
    "*To be precise, the answer here is $O(sn)$ since the required memory depends on the size of both inputs.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Space Complexity: Example 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4950"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def list_sum(lst):\n",
    "    if len(lst) == 1:  \n",
    "        return lst[0]\n",
    "    else:\n",
    "        return lst[0] + list_sum(lst[1:])     \n",
    "\n",
    "list_sum(list(range(100)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Each recursive call is added to the call stack and takes up actual memory. Thus, the algorithm takes $O(n)$ space, where $n$ is the length of `lst`.\n",
    "\n",
    "![Recursion](figs/recursion.png \"Recursion\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Space Complexity: Exercise 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4950"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# What is the space complexity of the function pair_list_sum()?\n",
    "\n",
    "def pair_list_sum(lst):\n",
    "    \"\"\"Sum all elements of a list using pair_sum.\n",
    "    Assume elements of the list are numeric type.\n",
    "    \"\"\"\n",
    "    res = 0\n",
    "    for i in range(len(lst) - 1):\n",
    "        if i % 2 == 0:\n",
    "            res += pair_sum(lst[i], lst[i+1])\n",
    "        # if the one but last index is odd, just add the last number to the sum\n",
    "        elif i == len(lst) - 2:\n",
    "            res += lst[i+1]\n",
    "    return res \n",
    "    \n",
    "def pair_sum(a, b):\n",
    "    \"\"\"Take numbers a and b and return their sum a + b.\"\"\"\n",
    "    return a + b\n",
    "\n",
    "pair_list_sum(list(range(100)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Space Complexity: Exercise 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "55\n"
     ]
    }
   ],
   "source": [
    "# What is the space complexity of the fib() function?\n",
    "\n",
    "def fib(n):\n",
    "    \"\"\"Assume n is a non-negative integer.\n",
    "    Find the n-th Fibonacci number using recursion.\n",
    "    \"\"\"\n",
    "    if n <= 1:\n",
    "        return n\n",
    "    else:\n",
    "        return fib(n - 1) + fib(n - 2)\n",
    "    return answer\n",
    "\n",
    "print(fib(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Complexity Analysis in Practice\n",
    "\n",
    "* There is no fixed list of possible runtimes, e.g. $O(\\sqrt n)$, $O(n!)$, ...\n",
    "* Runtimes can be a function of more than one input, e.g. $O(nk)$, $O(n + k)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Big-O describes the rate of increase in runtime, not the actual runtime\n",
    "    * Benchmarking is still a useful tool"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Big-O ignores contsants but they often matter in practice\n",
    "    * If you reduce your code from three loops to one loop, Big-O will not change but the difference may be on the order of hours or days! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* There is often a tradeoff between time and space efficiency\n",
    "* **You need a balance between runtime, space, implementation time, maintainability, and readability**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Analyzing and Benchmarking Your Code: Exercise 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[12000.297727272728, 5796.265909090909, 7951.277272727273, 3071.931818181818, 2881.4931818181817, 1524.8704545454545]\n",
      "[12000.297727272728, 5796.265909090909, 7951.277272727273, 3071.931818181818, 2881.4931818181817, 1524.8704545454545]\n"
     ]
    }
   ],
   "source": [
    "# What is the time complexity and actual runtime \n",
    "# of the two get_centroid functions?\n",
    "\n",
    "import csv\n",
    "\n",
    "def get_data():\n",
    "    \"\"\"Read the file Wholesale customers data.csv \n",
    "    and return part of the data as a list of lists.\n",
    "    \"\"\"\n",
    "    with open('Wholesale customers data.csv') as f:\n",
    "        reader = csv.reader(f)\n",
    "        data = [[int(i) for i in row[2:]] for row in reader if row[0] != 'Channel']\n",
    "    return data\n",
    "\n",
    "def get_centroid(pointLists):\n",
    "    \"\"\"Estimate the centroid for a collection of n-dimensional points.\n",
    "    Assume pointLists is a collection of lists of numerical values.\n",
    "    Return a list of numerical values (the coordinates of the centroid).\n",
    "    \"\"\"\n",
    "    num = len(pointLists)\n",
    "    centroids = []\n",
    "    demension = len(pointLists[0])\n",
    "    \n",
    "    for a in range(demension):\n",
    "        centroids.append(0.0)\n",
    "\n",
    "    for i in range(num):\n",
    "        point =  pointLists[i]\n",
    "        for d in range(demension):\n",
    "            centroids[d] = centroids[d] + point[d]\n",
    "            \n",
    "    for a in range(demension):\n",
    "        centroids[a] = centroids[a]/num\n",
    "\n",
    "    return centroids\n",
    "\n",
    "def get_centroid_2(points):\n",
    "    \"\"\"Estimate the centroid for a collection of n-dimensional points.\n",
    "    Assume pointLists is a collection of lists of numerical values.\n",
    "    Return a list of numerical values (the coordinates of the centroid).\n",
    "    \"\"\"\n",
    "    centroid = []\n",
    "    num_points = len(points)\n",
    "    num_dims = len(points[0])\n",
    "    for dim in range(num_dims):\n",
    "        coord = [i[dim] for i in points]\n",
    "        centroid.append(sum(coord)/num_points)\n",
    "    return centroid\n",
    "\n",
    "data = get_data()\n",
    "print(get_centroid(data))\n",
    "print(get_centroid_2(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Analyzing and Benchmarking Your Code: Exercise 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "# The algorithms take two inputs: list coauthors of length c \n",
    "# and dictionary author_dic of length d.\n",
    "# What is the time complexity of the two algorithms?\n",
    "\n",
    "# Algorithm 1\n",
    "for k, v in author_dic.items():\n",
    "    vlst = [sub_list[1] for sub_list in coauthors if sub_list[0] == k and sub_list[1] != k]\n",
    "    vlst = sorted(vlst)\n",
    "    author_dic[k] = vlst\n",
    "    \n",
    "# Algorithm 2\n",
    "for i, j in coauthors:\n",
    "    if j != i:\n",
    "        author_dic[i].append(j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Algorithms and Order of Growth\n",
    "\n",
    "* **Algorithmic complexity**, also called running time and order of growth, refers to the number of steps a program takes as a function of the size of its inputs  \n",
    "\n",
    "* We use **Big-O** notation to express how, in the worst-case scenario, an algorithm's **running time or space requirements** grow as the input size grows\n",
    "\n",
    "* **Complexity analysis and benchmarking** are essential for designing programs that can efficiently manage large amounts of data and large numbers of users "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "-------\n",
    "\n",
    "* **Lab**: Practicing order of growth analysis\n",
    "* **Next week**: Basic searching and sorting algorithms"
   ]
  }
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