3 Time Complexity.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Collecting sympy\n",
      "  Downloading sympy-1.8-py3-none-any.whl (6.1 MB)\n",
      "\u001b[K     |████████████████████████████████| 6.1 MB 2.9 MB/s eta 0:00:01     |██████████                      | 1.9 MB 2.9 MB/s eta 0:00:02     |███████████████████▋            | 3.7 MB 2.9 MB/s eta 0:00:01\n",
      "\u001b[?25hCollecting mpmath>=0.19\n",
      "  Downloading mpmath-1.2.1-py3-none-any.whl (532 kB)\n",
      "\u001b[K     |████████████████████████████████| 532 kB 11.9 MB/s eta 0:00:01\n",
      "\u001b[?25hInstalling collected packages: mpmath, sympy\n",
      "Successfully installed mpmath-1.2.1 sympy-1.8\n"
     ]
    }
   ],
   "source": [
    "!pip3 install sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import symbolic capability to Python\n",
    "from sympy import *\n",
    "from sympy.abc import *\n",
    "from sympy.plotting import plot\n",
    "from IPython.display import display\n",
    "# print things all pretty\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Size of the problem: n "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = symbols(\"n\", positive=True, real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Different Functions f with different Time complexity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_constant = 20\n",
    "f_linear = 4 * n + 10\n",
    "f_linear2 = 10 * n +1\n",
    "f_quadratic = 1/4 * n**2 + 1/2*n +1 \n",
    "f_quadratic2 = 1/8 * n**2 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Definition of Order of Growth (O) (\"Big O\")\n",
    "\n",
    "Order of growth (O) is a set of functions whose asymptotic growth behavior is considered equivalent:\n",
    "\n",
    "Two functions f(n) and g(n) have the equivalent order of growth if $\\displaystyle \\lim_{n \\rightarrow \\infty} \\frac{f(n)}{g(n)} = c$ with $c > 0$ and $c < \\infty $.\n",
    "\n",
    "\n",
    "f(n) has a higher order of growth than g(n) if\n",
    "$\\displaystyle \\lim_{n \\rightarrow \\infty} \\frac{f(n)}{g(n)} = \\infty $.\n",
    "\n",
    "f(n) has a smaller order of growth than g(n) if $\\displaystyle \\lim_{n \\rightarrow \\infty} \\frac{f(n)}{g(n)} = 0 $.\n",
    "\n",
    "$f(n) \\in O(g(n))$ if $\\displaystyle \\lim_{n \\rightarrow \\infty} \\frac{f(n)}{g(n)} = c$ with $c < \\infty $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Order of Growth with the Sympy limit function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2.0$"
      ],
      "text/plain": [
       "2.00000000000000"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(f_quadratic / f_quadratic2, n, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAKCAYAAABWiWWfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIklEQVQoFX2S4U3DQAyFEyaIYIOwAYwAI6R0A7oBVX8l/6qyQdgAygbtCKQjpBMgsUH4vstFojoFS6+2n1/t+O7yYRgyrWmaO9wjuAUd+Rs+MfgC8gncgw58wP3gs9xmJM/EC3AAN8BcW1A7jmEYWBFvwBp8gRIswTu6U1bXdQlam04gL0AHBlDJ68F+0vz18Dvz3ICuTrowONc5A71fvYFztcTgPaLrK36+kyoEAs/hIdb2+GRgrKk9EZc2m7Uoeo2C1axwLBT/Not/9oadXNF8uphYGh28F9HbrCdx58TgXW8LXLcH7YzWrz5OT6MlWSMM7wXvodvoQBzWxDvdp6P3/D6BZhzeZWgmg3iHU2gjm27hXO/C4F4glsBtrKsLjX8B1i+VmyjeP9EAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(f_quadratic / f_linear, n, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit( f_linear2 / f_quadratic2, n, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plotting Time Complexity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr_range = (n,0,30)\n",
    "p = plot(\n",
    "    f_constant,\n",
    "    f_linear,\n",
    "    f_quadratic,\n",
    "    expr_range,\n",
    "    show = False,\n",
    "    legend = True\n",
    ");\n",
    "\n",
    "p[0].line_color = 'g'\n",
    "p[1].line_color = 'b'\n",
    "p[2].line_color = 'r'\n",
    "\n",
    "\n",
    "\n",
    "p[0].label = 'constant'\n",
    "p[1].label = 'linear'\n",
    "p[2].label = 'quadratic'\n",
    "\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Crossover point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.25 n^{2} + 0.5 n + 1 = 4 n + 10$"
      ],
      "text/plain": [
       "      2                       \n",
       "0.25⋅n  + 0.5⋅n + 1 = 4⋅n + 10"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = Eq(f_quadratic, f_linear)\n",
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 16.2195444572929\\right]$"
      ],
      "text/plain": [
       "[16.2195444572929]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(e1, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPCAYAAAD6fR2jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFnUlEQVRoBeWa0XUVNxCGNzku4MapANOBgQ6cDkKowNABebTffKADoIIAHYRUQKADSAVJ3IHzfbqrtXZXuzt7vefwkDlHV9Lo12g0Go2ktZubm5vm/5AuLi5Oh/OEtyOdDPnfqh7VUZ1JT0m7b6XrFuMeNQVdXl6eUn1HekD5umjqFWl70WM0zW/wPg94oyqYE5i/tg0Pyf+xPtUX/qI+YF618syOSefwarr/AX9He9bTsvRgn9V/Wx2ekT+rI/bcKE70DDaqo3Zx3q+QRTaia/g/yCUP2xxsyJZbyTxCkIvwhqQj6BAqW6V2UJ1Th/kgiNy66b71KWr7aqyfMoayTvxJHinLC+kDXtwnkjJfKpPcRfmL3M3zVV5Bzk8SY9t70hW4a/I5cm7ZYbfAKWNKZlTHR8jQXsM5KvuMlDY4c3MtozYP2XJLmTqfxn9Maig/J3NxpkijGeWSo7QgnaBmhKEMHa0XPZCjEz+Fr9y8U6P6uGGO6Z8cj3JD+TPpT4ru4M7JbYNsS/PcV5d/wWuPRYriFLSADeuInOH8lJ0CB7kbSwrZHNwaW24m8/ukYuCHCf0MTMd8XcLhG7VGhigxbdkd+QWszlqSjryDPxlxS3BRVp+a0xulzirjFF2Xi/R3rm4E0yRFcQpYg50ccN/wcaL9BWOUGzxq8zW23Exm2PmYrJPyLjG7GBNGkZ2OiZn+Q6ecFIWMjM3HVIn9u614hbgLPWGc3kabEBbF2X0NdmK45MQ5snUYdDUiXXWMfWHR5gfYcjOZRwNl56ou5leUNSI8IbnI3vPewVOhWQIzdeSlY572yL0qjQHWTWD5ODH6Pz+21VEkpY9HvI4rxnbvfKNx4XnclpdvqmOK4uwZxYIL6VhqQx/nckqeH3OpmXrI5uDEh2y5pcw1kS9Hm4dOkvSSZDTU+Qzbq4l+Op6G6xktKMjdb98hJWeGmfXN7dbftno7nsnHjsdIR9SVqXPXjvTVODtEZQIN6dgpcVsw6pkWCV1qNl9ry944h8oMOR/C80K6u4ZH0Vs0eVNgeootVHxovKdv92hYwJfN51bo2zkPZQ173YJ6zkObd9Pc1lC23Yg9jHB+VhnOsRXZy6I4O4WwjBvVsVOEPm6WM/LF06ftVLP5Klt2g98WDpIZcr7bMaoXfJ/oOueqOxbGctE9xqeOhmLYcZF+OtI90mPKz0nufBciX8Z7zge/RmJO6Gu/htwjb+iMNvUoirPTGmxvkNtKT8dbdlfy9InMVV2qNod/sC3vIvOom8JMQeVIIrrIUYGnBazwRyxkuch+Jom8kkf9M0O9KGv8juDl46dbEHi/A3C8qQ/KO9rU37zr1wktClGcXVZiF3Us1CiLXnlmdRaMLrM2pz1ky3Lgu8oMOV87oGF9zsEWDaAcFNZY98m7iEc5R56QDOXMkEfvB2SWG8WoXHsZHysHrN/XPL4fkXuElJTuSC1f/XSSRRx475TOK4pd1BFZPWKMHQzHGD2aSiC4Q21es2USvYXMNc5nyB4ujIoYTYyMi3cOME7GxRg+MDRO5J4FbE/t5P04eo9ycjRyF0MnGka415UxFSQ26U27+WgO8P+VT95tlihupcxFHRl3SDqsVNtYqQEdFm0ORvtHbdlsJXPofH6CkI5JZeRwQB8GLoIfMpPzkLvYv5DShZW8aXkumNGkcwLK7lCdVxnDe9UZvNqjY1If5ChvaHTle7kfRgL/xGTqjmjK+a8XpVPRfUTO0bREUZxyathDdMx6XdeUY45Rm4dtuaXM7/zvBATmiGYkcEIuXjpiaOtFJOreqfKkddIreL3Fpv4Fvk5WLrY8J1mjoaOG9Gl1UV7WZ/KbI1jHzhFXvXVcPxlNLZwbxD7aRPJzxEfwvU1CPYRTwBKW9rU6ivfBd07f2ofnNTbPd+VZWzLOZjL/A5zZVYqywNiaAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 16.2195444572929$"
      ],
      "text/plain": [
       "16.2195444572929"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(e1, n)[0].evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Todo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# TODO: Define \n",
    "# - a logarithmic function\n",
    "# - a log-linear (n log n) function\n",
    "# - a cubic function\n",
    "# - an exponential function\n",
    "# - a factorial function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# TODO create multiple plots with different combinations of functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# TODO create a log-log plot with your factorial, exponential, quadratic, log-linear, and linear function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# TODO Calculate one crossover point for two functions"
   ]
  }
 ],
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