Complexity.ipynb

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   "source": [
    "## Algorithm Analysis"
   ]
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   "source": [
    "* An **algorithm** is a generic, step-by-step list of instructions for solving a problem. It is a method for solving any instance of the problem such that given a particular input, the algorithm produces the desired result.\n",
    "\n",
    "* A **program**, on the other hand, is an algorithm that has been encoded into some programming language. There may be many programs for the same algorithm, depending on the programmer and the programming language being used."
   ]
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   "source": [
    "* **Algorithm analysis** is concerned with comparing algorithms based upon the amount of **computing resources** that each algorithm uses. We want to be able to consider two algorithms and say that one is better than the other because it is more efficient in its use of those resources or perhaps because it simply uses fewer.\n",
    "\n",
    "* There are two different ways to look at resources. One way is to consider **the amount of space or memory** an algorithm requires to solve the problem. The amount of space required by a problem solution is typically dictated by the problem instance itself. Every so often, however, there are algorithms that have very specific space requirements.\n",
    "\n",
    "* As an alternative to space requirements, we can analyze and compare algorithms based on **the amount of time** they require to execute. One way we can measure the execution time for a function is to do a **benchmark analysis**. This means that we will track the actual time required for the program to compute its result."
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   "source": [
    "def sumOfN(n):\n",
    "    theSum = 0\n",
    "    for i in range(n):\n",
    "        theSum += i\n",
    "    return theSum"
   ]
  },
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   "source": [
    "In the function <code>sumOfN</code>, the number of assignment statements is 1 (<code>theSum = 0</code>) plus the value of $n$ (the number of times we perform <code>theSum =+ i</code>). We can denote this by a function, call it $T$, where $T(n)=1+n$. The parameter $n$ is often referred to as **the size of the problem**, and we can read this as \"$T(n)$ is the time it takes to solve a problem of size $n$, namely $1+n$ steps.\""
   ]
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   "source": [
    "**The order of magnitude** function describes the part of $T(n)$ that increases the fastest as the value of $n$ increases. Order of magnitude is often called **Big-O notation** and written as $O(f(n))$. It provides a useful approximation to the actual number of steps in the computation. The function $f(n)$ provides a simple representation of the dominant part of the original $T(n)$."
   ]
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   "source": [
    "## Time Complexity\n",
    "\n",
    "https://en.wikipedia.org/wiki/Time_complexity"
   ]
  },
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   "source": [
    "### linear scale"
   ]
  },
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   "source": [
    "import numpy as np\n",
    "import scipy.misc\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# draw figure in notebook, note: you can not add any comment after the following line.\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "x = np.linspace(0.000001, 4.0, 100)\n",
    "\n",
    "logarithmic = np.log(x)\n",
    "fractional_power = np.power(x, 0.5)\n",
    "linear = x\n",
    "quasilinear = x * np.log(x)\n",
    "quadratic = np.power(x, 2)\n",
    "exponential = np.exp(x)\n",
    "factorial = scipy.special.factorial(x)"
   ]
  },
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   "execution_count": 3,
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zzDOZPXt26Ir26quv5je/+Q2FhYWUl5eHjmOxWHjyySe58soryc/PR6fT9XuN6eFuu+02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\nC5sptMWYUk1mAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x179028c8e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, logarithmic, label='Logarithmic')\n",
    "plt.plot(x, fractional_power, label='Fractional Power, n^(1/2)')\n",
    "plt.plot(x, linear, label='Linear')\n",
    "plt.plot(x, quasilinear, label='Quasilinear, n log n')\n",
    "plt.plot(x, quadratic, label='Quadratic, n^2')\n",
    "#plt.plot(x, exponential, label='Exponential, e^n')\n",
    "#plt.plot(x, factorial, label='Factorial, e^n')\n",
    "\n",
    "plt.legend()\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### log scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "n = np.linspace(1.0, 20.0, 100)\n",
    "\n",
    "logarithmic = np.log(n)\n",
    "fractional_power = np.power(n, 0.5)\n",
    "linear = n\n",
    "quasilinear = x * np.log(n)\n",
    "quadratic = np.power(n, 2)\n",
    "exponential = np.exp(n)\n",
    "factorial = scipy.special.factorial(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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a2loGDBiARqPhjTfewOv1AjBr1iyWLFmC3W4HoKamBmhe7jwtLY3y8nJ27twJ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\nA4H3OM2WeaL3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x179049b1208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plt.plot(x, logarithmic, label='Logarithmic')\n",
    "#plt.plot(n, fractional_power, label='Fractional Power, n^(1/2)')\n",
    "plt.plot(n, linear, label='Linear')\n",
    "plt.plot(n, quasilinear, label='Quasilinear, n log n')\n",
    "plt.plot(n, quadratic, label='Quadratic, n^2')\n",
    "plt.plot(n, exponential, label='Exponential, e^n')\n",
    "plt.plot(n, factorial, label='Factorial, e^n')\n",
    "\n",
    "plt.yscale('log') # try close it\n",
    "plt.legend()\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Profiling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Linear"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem Size          Seconds    Iteration\n",
      "     1000000            0.237\n",
      "     2000000            0.508\n",
      "     4000000            0.847\n",
      "     8000000            1.678\n",
      "    16000000            4.082\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "problemSize = 1000000\n",
    "print(\"%12s %16s %12s\" % (\"Problem Size\", \"Seconds\", \"Iteration\"))\n",
    "\n",
    "x_li = list()\n",
    "y_li = list()\n",
    "\n",
    "for epoch in range(5):\n",
    "    \n",
    "    start = time.time()\n",
    "    \n",
    "    #########################\n",
    "    work = 0\n",
    "    for x in range(problemSize):\n",
    "        work += 1\n",
    "        work -= 1\n",
    "    #########################\n",
    "    \n",
    "    end = time.time()\n",
    "    elasped = end - start\n",
    "    \n",
    "    print(\"%12d %16.3f\" % (problemSize, elasped))\n",
    "    x_li.append(problemSize)\n",
    "    y_li.append(elasped)\n",
    "    \n",
    "    problemSize *= 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8NW8mpFAxqiy/+1krftOvGVWjo8IdTSQihfuYQh8zSwK2AHe5+9Iw55ESYteh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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17904b3ad30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_li, y_li, label='Computation Time')\n",
    "\n",
    "plt.legend()\n",
    "plt.xlabel('Problem Size')\n",
    "plt.ylabel('Time (s)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Double loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem Size          Seconds   Iterations\n",
      "         250            0.020        62500\n",
      "         500            0.082       250000\n",
      "        1000            0.362      1000000\n",
      "        2000            1.338      4000000\n",
      "        4000            5.847     16000000\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "problemSize = 250\n",
    "print(\"%12s %16s %12s\" % (\"Problem Size\", \"Seconds\", \"Iterations\"))\n",
    "\n",
    "x_li = list()\n",
    "y_li = list()\n",
    "\n",
    "for epoch in range(5):\n",
    "\n",
    "    start = time.time()\n",
    "    \n",
    "    #########################\n",
    "    iterations = 0\n",
    "    \n",
    "    work = 0\n",
    "    for x in range(problemSize):\n",
    "        for y in range(problemSize):\n",
    "            work += 1\n",
    "            work -= 1\n",
    "            \n",
    "            iterations += 1\n",
    "    #########################\n",
    "    \n",
    "    end = time.time()\n",
    "    elasped = end - start\n",
    "    \n",
    "    print(\"%12d %16.3f %12d\" % (problemSize, elasped, iterations))\n",
    "    x_li.append(problemSize)\n",
    "    y_li.append(elasped)\n",
    "    \n",
    "    problemSize *= 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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MujhJ515gGvAG1SxnIyIN0xdfwIABYS7OLbfA1VenHZFsw30mZu2r2aM/MJowsuxlzHbB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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17904b3acc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax1 = plt.subplots()\n",
    "ax1.plot(x_li, y_li, 'b-', label='Computation Time')\n",
    "ax1.set_xlabel('Problem Size')\n",
    "ax1.set_ylabel('Computation Time (s)', color='b')\n",
    "ax1.tick_params('s', colors='b')\n",
    "\n",
    "ax2 = ax1.twinx()\n",
    "ax2.plot(x_li, np.array(x_li)*np.array(x_li), 'r.', label='Quadratic, n^2')\n",
    "ax2.set_ylabel('Quadratic', color='r')\n",
    "ax2.tick_params('y', colors='r')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Tripple loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem Size          Seconds   Iterations\n",
      "          20            0.002         8000\n",
      "          40            0.017        64000\n",
      "          80            0.147       512000\n",
      "         160            1.363      4096000\n",
      "         320           11.610     32768000\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "problemSize = 20\n",
    "print(\"%12s %16s %12s\" % (\"Problem Size\", \"Seconds\", \"Iterations\"))\n",
    "\n",
    "x_li = list()\n",
    "y_li = list()\n",
    "\n",
    "for epoch in range(5):\n",
    "\n",
    "    start = time.time()\n",
    "    \n",
    "    #########################\n",
    "    iterations = 0\n",
    "    \n",
    "    work = 0\n",
    "    for x in range(problemSize):\n",
    "        for y in range(problemSize):\n",
    "            for z in range(problemSize):\n",
    "                work += 1\n",
    "                work -= 1\n",
    "            \n",
    "                iterations += 1\n",
    "    #########################\n",
    "    \n",
    "    end = time.time()\n",
    "    elasped = end - start\n",
    "    \n",
    "    print(\"%12d %16.3f %12d\" % (problemSize, elasped, iterations))\n",
    "    x_li.append(problemSize)\n",
    "    y_li.append(elasped)\n",
    "    \n",
    "    problemSize *= 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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efx1q1Yr4BBs3wiWXhFERXbpEfHCJSjYlo2Rq0/2iwJ0ZrYB7YotIRCrkvffC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\nAQO+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17904952518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax1 = plt.subplots()\n",
    "ax1.plot(x_li, y_li, 'b-', label='Computation Time')\n",
    "ax1.set_xlabel('Problem Size')\n",
    "ax1.set_ylabel('Computation Time (s)', color='b')\n",
    "ax1.tick_params('s', colors='b')\n",
    "\n",
    "ax2 = ax1.twinx()\n",
    "x = np.linspace(20.0, problemSize/2.0, 100) # hack\n",
    "ax2.plot(x, np.array(x)*np.array(x)*np.array(x), 'r--', label='Cubic, n^3')\n",
    "ax2.set_ylabel('Cubic', color='r')\n",
    "ax2.tick_params('y', colors='r')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Big O"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem Size          Seconds        Works\n",
      "         400            0.021       160400\n",
      "         800            0.105       640800\n",
      "        1600            0.474      2561600\n",
      "        3200            1.625     10243200\n",
      "        6400            7.206     40966400\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "initProblemSize = 400\n",
    "problemSize = initProblemSize\n",
    "print(\"%12s %16s %12s\" % (\"Problem Size\", \"Seconds\", \"Works\"))\n",
    "\n",
    "x_li = list()\n",
    "y_li = list()\n",
    "\n",
    "for epoch in range(5):\n",
    "\n",
    "    start = time.time()\n",
    "    \n",
    "    #########################\n",
    "    work = 0\n",
    "    for x in range(problemSize):\n",
    "        work += 1\n",
    "        for y in range(problemSize):\n",
    "            work += 1\n",
    "    #########################\n",
    "    \n",
    "    end = time.time()\n",
    "    elasped = end - start\n",
    "    \n",
    "    print(\"%12d %16.3f %12d\" % (problemSize, elasped, work))\n",
    "    x_li.append(problemSize)\n",
    "    y_li.append(elasped)\n",
    "    \n",
    "    problemSize *= 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CEbkNOIqqIlIV537/CU+Ek1jS2Qag6qwYaoxJv06ehCZNYONGmDoVOndOpPGW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\ncDUtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17904b61320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax1 = plt.subplots()\n",
    "ax1.plot(x_li, y_li, 'b-', label='Computation Time')\n",
    "ax1.set_xlabel('Problem Size')\n",
    "ax1.set_ylabel('Computation Time (s)', color='b')\n",
    "ax1.tick_params('s', colors='b')\n",
    "\n",
    "ax2 = ax1.twinx()\n",
    "x = np.linspace(initProblemSize, problemSize/2.0, 100) # hack\n",
    "ax2.plot(x, np.array(x)*np.array(x), 'r-', label='Quadratic, n^2')\n",
    "ax2.plot(x, np.array(x)*np.array(x)+np.array(x), 'g--', label='n^2+n')\n",
    "ax2.set_ylabel('Cubic', color='r')\n",
    "ax2.tick_params('y', colors='r')\n",
    "ax2.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Try n^2 + 2*n\n",
    "# How about 2^n + 4*n^2 + 5n ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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A24A/8L61dpS1dhPwiDHGD3iviJmlKDLTYc1U+Pktd+E3iOJk5zE8v74OX83Z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\nAAquAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x179049975f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0.0, 40, 100)\n",
    "\n",
    "plt.plot(x, np.power(2, x), label = '2^n')\n",
    "plt.plot(x, np.power(2, x) + 4*np.power(x, 2) + 5*x, label = '2^n + 4*n^2 + 5n')\n",
    "plt.yscale('log')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Anagram Detection"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def anagramSolution0(str1, str2):\n",
    "    mark_li = list(str2)\n",
    "    \n",
    "    for i in range(len(str1)):\n",
    "        found = False\n",
    "        for j in range(len(mark_li)):\n",
    "            if str1[i] == mark_li[j]:\n",
    "                mark_li[j] = None\n",
    "                found = True\n",
    "                break\n",
    "        if not found:\n",
    "            return False\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "print(anagramSolution0('abcd','dcba'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To analyze this <code>anagramSolution0</code>, we need to note that each of the $n$ characters in <code>str1</code> will cause an iteration through up to $n$ characters in the list from $str2$. It is a double loop, whcih is $O(n^2)$, with conditional break. Therefore, this solution is $O(n^2)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def anagramSolution1(str1, str2):\n",
    "    mark_li = list(str2)\n",
    "\n",
    "    pos1 = 0\n",
    "    stillOK = True\n",
    "\n",
    "    while pos1 < len(str1) and stillOK:\n",
    "        pos2 = 0\n",
    "        found = False\n",
    "        while pos2 < len(mark_li) and not found:\n",
    "            if str1[pos1] == mark_li[pos2]:\n",
    "                found = True\n",
    "            else:\n",
    "                pos2 += 1\n",
    "\n",
    "        if found:\n",
    "            mark_li[pos2] = None\n",
    "        else:\n",
    "            stillOK = False\n",
    "\n",
    "        pos1 = pos1 + 1\n",
    "\n",
    "    return stillOK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "print(anagramSolution1('abcd','dcba'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To analyze this <code>anagramSolution1</code>, we need to note that each of the $n$ characters in <code>str1</code> will cause an iteration through up to $n$ characters in the list from $str2$. Each of the $n$ positions in the list will be visited once to match a character from <code>str1</code>. The number of visits then becomes the sum of the integers from $1$ to $n$. This can be written as\n",
    "\n",
    "$\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2} = \\frac{1}{2} n^2 + \\frac{1}{2} n$\n",
    "\n",
    "As $n$ gets large, the $n^2$ term will dominate the $n$ term and the $1/2$ can be ignored. Therefore, this solution is $O(n^2)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def anagramSolution2(str1, str2):\n",
    "    li1 = list(str1)\n",
    "    li2 = list(str2)\n",
    "\n",
    "    li1.sort()\n",
    "    li2.sort()\n",
    "\n",
    "    pos = 0\n",
    "    matches = True\n",
    "\n",
    "    while pos < len(li1) and matches:\n",
    "        if li1[pos] == li2[pos]:\n",
    "            pos += 1\n",
    "        else:\n",
    "            matches = False\n",
    "\n",
    "    return matches"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "print(anagramSolution2('abcde','edcab'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "At first glance you may be tempted to think that this algorithm is $O(n)$, since there is one simple iteration to compare the $n$ characters after the sorting process. However, the two calls to the Python <code>sort</code> method are not without their own cost. As we will see later, sorting is typically either $O(n^2)$ or $O(n\\log{n})$, so the sorting operations dominate the iteration. In the end, this algorithm $2n^2+n$ or $2n\\log{n}+n$ will have the same order of magnitude as that of the sorting process."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def anagramSolution3(str1, str2):\n",
    "    ##  brute force \n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "A **brute force** technique for solving a problem typically tries to exhaust all possibilities. For the anagram detection problem, we can simply generate a list of all possible strings using the characters from <code>str1</code> and then see if <code>str2</code> occurs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def anagramSolution4(str1, str2):\n",
    "    c1 = [0]*26\n",
    "    c2 = [0]*26\n",
    "    \n",
    "    for i in range(len(str1)):\n",
    "        index = ord(str1[i]) - ord('a')\n",
    "        c1[index] += 1\n",
    "        \n",
    "    for j in range(len(str2)):\n",
    "        index = ord(str2[j]) - ord('a')\n",
    "        c2[index] += 1\n",
    "    \n",
    "    print(c1)\n",
    "    print(c2)\n",
    "    \n",
    "    for k in range(len(c1)):\n",
    "        if c1[k] != c2[k]:\n",
    "            return False\n",
    "    \n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]\n",
      "[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "print(anagramSolution4('abcde','edcab'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Again, the solution has a number of iterations. However, unlike the first solution, **none of them are nested.** The first two iterations used to count the characters are both based on $n$. The third iteration, comparing the two lists of counts, always takes 26 steps since there are 26 possible characters in the strings. Adding it all up gives us $T(n)=2n+26$ steps. That is $O(n)$. We have found **a linear order of magnitude** algorithm for solving this problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Performance of Python List"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "| Operation|Big-O Efficiency|\n",
    "|----------|---------------|\n",
    "|index|O(1)|\n",
    "|index assignment|O(1)|\n",
    "|append|O(1)|\n",
    "|pop()|O(1)|\n",
    "|pop(i)|O(n)|\n",
    "|insert(i,item)|O(n)|\n",
    "|del operator|O(n)|\n",
    "|iteration|O(n)|\n",
    "|contains (in)|O(n)|\n",
    "|get slice|O(k)|\n",
    "|del slice|O(n)|\n",
    "|set slice|O(n+k)|\n",
    "|reverse|O(n)|\n",
    "|concatenate|O(k)|\n",
    "|sort|O(n log n)|\n",
    "|multiply|O(nk)|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "122 ns ± 13.7 ns per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "1.79 ms ± 109 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "# Lets take a look at pop() and pop(i)\n",
    "x = 2000000\n",
    "li = list(range(x))\n",
    "%timeit -n 1000 li.pop()\n",
    "\n",
    "x = 2000000\n",
    "li = list(range(x))\n",
    "%timeit -n 1000 li.pop(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "\n",
    "pop = timeit.timeit('li.pop', setup = 'x = 2000000; li = list(range(x))', number = 1000)\n",
    "pop0 = timeit.timeit('li.pop(0)', setup = 'x = 2000000; li = list(range(x))', number = 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x_li = list()\n",
    "pop_li = list()\n",
    "pop0_li = list()\n",
    "\n",
    "for x in range(1000, 100001, 1000):\n",
    "    pop = timeit.timeit('li.pop()', setup = 'li = list(range({}))'.format(x), number = 1000)\n",
    "    pop0 = timeit.timeit('li.pop(0)', setup = 'li = list(range({}))'.format(x), number = 1000)\n",
    "    \n",
    "    x_li.append(x)\n",
    "    pop_li.append(pop)\n",
    "    pop0_li.append(pop0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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ubJnu7u7PAs3ufnF3d/fvgS+sWLHiYTM7C1gBnA08Aqzp7u6+e8WKFZH/29/b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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17905cacc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.plot(x_li, pop_li, \"r^\", label='pop')\n",
    "plt.plot(x_li, pop0_li, \"go\", label='pop0')\n",
    "\n",
    "plt.legend()\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Time (s)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Performance of Python Dictionary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "| Operation|Big-O Efficiency|\n",
    "|----------|---------------|\n",
    "|copy|O(n)|\n",
    "|get item|O(1)|\n",
    "|set item|O(1)|\n",
    "|delete item|O(1)|\n",
    "|contains (in)|O(1)|\n",
    "|iteration|O(n)|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "For our last performance experiment we will compare the performance of the **contains** operation between **lists** and **dictionaries**. In the process we will confirm that the contains operator for lists is $O(n)$ and the contains operator for dictionaries is $O(1)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "import random\n",
    "\n",
    "x_li = list()\n",
    "list_li = list()\n",
    "dict_li = list()\n",
    "\n",
    "for i in range(1000, 100001, 1000):\n",
    "    t = timeit.Timer(\"random.randrange({}) in x\".format(i),\n",
    "                     \"from __main__ import random, x\")\n",
    "    x = list(range(i))\n",
    "    lst_time = t.timeit(number=100)\n",
    "    x = {j:None for j in range(i)}\n",
    "    d_time = t.timeit(number=100)\n",
    "    \n",
    "    x_li.append(i)\n",
    "    list_li.append(lst_time)\n",
    "    dict_li.append(d_time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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XZgPBoDbADcAP3N3N7FTgO8BSd38mwzrWnrhpsseOBSk+TpwY/AlaRyEiZZVZ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\nIiKpKGCIiEgqChgiIpKKAoaIiKTy/wEnjeqBTZ4o7wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x17905caceb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.plot(x_li, list_li, \"r^\", label='in list')\n",
    "plt.plot(x_li, dict_li, \"go\", label='in dict')\n",
    "\n",
    "plt.legend()\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Time (s)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Summations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considering the case of executing a loop in which the *ith* setp require $i$ operations. In other words, we are looking for the summation\n",
    "\n",
    "$S_1(n) = \\sum_{i=1}^{n} i $\n",
    "\n",
    "It is clear that the summation is $1 + 2 + \\dots + n = \\frac{(n+1) \\times n}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Considering the case of executing a loop in which the *ith* setp require $i^2$ operations. The summation is\n",
    "\n",
    "$S_2(n) = \\sum_{i=1}^{n} i^2 $\n",
    "\n",
    "It is clear that $n^2 \\leq S_2(n) \\leq n^3 $, but what is the exact number?"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}