{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sebastian Raschka \n",
      "last updated: 2016-06-01 \n",
      "\n",
      "CPython 3.5.1\n",
      "IPython 4.2.0\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a 'Sebastian Raschka' -u -d -v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Fibonacci Numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A Fibonacci number F(n) is computed as the sum of the two numbers preceeding it in a Fibonacci sequence\n",
    "\n",
    "(0), 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...,\n",
    "\n",
    "for example, F(10) = 55.\n",
    "\n",
    "More formally, we can define a Fibonacci number F(n) as \n",
    "\n",
    "$F(n) = F(n-1) + F(n-2)$, for integers $n > 1$:\n",
    "\n",
    "$$F(n)=\n",
    "\\begin{cases} \n",
    "      0 & n=0, \\\\\n",
    "      1, & n=1, \\\\\n",
    "      F(n-1) + F(n-2), & n > 1.\n",
    "   \\end{cases}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Fibonacci sequence was named after Leanardo Fibonacci, who used the Fibonacci sequence to study rabit populations in the 12th century. I highly recommend reading the excellent articles on [Wikipedia](https://en.wikipedia.org/wiki/Fibonacci_number) and [Wolfram](http://mathworld.wolfram.com/FibonacciNumber.html), which discuss the interesting facts about the Fibonacci number in great detail."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The recursive Fibonacci number computation is a typical text book example of a recursive algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "1\n",
      "55\n"
     ]
    }
   ],
   "source": [
    "def fibo_recurse(n):\n",
    "    if n <= 1:\n",
    "        return n\n",
    "    else:\n",
    "        return fibo_recurse(n-1) + fibo_recurse(n-2)\n",
    "    \n",
    "print(fibo_recurse(0))\n",
    "print(fibo_recurse(1))\n",
    "print(fibo_recurse(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, it is unfortunately a terribly inefficient algorithm with an exponential running time of $O(2^n)$. The main problem why it is so slow is that we recompute Fibonacci number $F(n) = F(n-1) + F(n-2)$ repeatedly as shown in the recursive tree below:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Fibonacci Number Recursive Tree](../../images/efficiency/fibonacci_tree.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, assuming $n \\geq 2$ we have \n",
    "\n",
    "$O(2^{n-1}) + O(2^{n-2}) + O(1) = O(2^n)$\n",
    "\n",
    "for $F(n) = F(n-1) + F(n-2)$, where $O(1)$ is for adding to Fibonacci numbers together.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A more efficient approach to compute a Fibonacci number is a dynamic approach with linear runtime, $O(n)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "1\n",
      "55\n"
     ]
    }
   ],
   "source": [
    "def fibo_dynamic(n):\n",
    "    f, f_minus_1 = 0, 1\n",
    "    for i in range(n):\n",
    "        f_minus_1, f = f, f + f_minus_1\n",
    "    return f\n",
    "\n",
    "print(fibo_dynamic(0))\n",
    "print(fibo_dynamic(1))\n",
    "print(fibo_dynamic(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(If you are interested in other approaches, I recommend you take a look at the pages on [Wikipedia](https://en.wikipedia.org/wiki/Fibonacci_number) and [Wolfram](http://mathworld.wolfram.com/FibonacciNumber.html).)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get a rough idea of the running times of each of our implementations, let's use the `%timeit` magic for F(30)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 499 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit -r 3 -n 10 fibo_recurse(n=30) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 4.05 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit -r 3 -n 10 fibo_dynamic(n=30) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, let's benchmark our implementations for varying sizes of n:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "\n",
    "funcs = ['fibo_recurse', 'fibo_dynamic']\n",
    "orders_n = list(range(0, 50, 10))\n",
    "times_n = {f:[] for f in funcs}\n",
    "\n",
    "for n in orders_n:\n",
    "    for f in funcs:\n",
    "        times_n[f].append(min(timeit.Timer('%s(n)' % f, \n",
    "                'from __main__ import %s, n' % f)\n",
    "                    .repeat(repeat=3, number=5)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def plot_timing():\n",
    "\n",
    "    labels = [('fibo_recurse', 'fibo_recurse'), \n",
    "              ('fibo_dynamic', 'fibo_dynamic')]\n",
    "\n",
    "    plt.rcParams.update({'font.size': 12})\n",
    "\n",
    "    fig = plt.figure(figsize=(10, 8))\n",
    "    for lb in labels:\n",
    "        plt.plot(orders_n, times_n[lb[0]], \n",
    "             alpha=0.5, label=lb[1], marker='o', lw=3)\n",
    "    plt.xlabel('sample size n')\n",
    "    plt.ylabel('time per computation in milliseconds [ms]')\n",
    "    plt.legend(loc=2)\n",
    "    plt.ylim([-1, 300])\n",
    "    plt.grid()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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MHp29Y6sHTkRERKQOCu3yKexhAmdmD0V8TclmwVI3+d57kCvKJZUyCadcwimX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\n1q5Nti+5BM49N9t4bHTG+nvgzMzMrErs2wft7f3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AUwo1MzkivljZ0DLTBGwALoyIKcCf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ca86908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_timing()"
   ]
  }
 ],
 "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}