{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A common question in computer science is to measure the [computational complexity of algorithms](https://en.wikipedia.org/wiki/Computational_complexity_theory). I've recently taken a programming test where the two aspects of programming that were measured as part of the test were **correctness** and **speed**, as each programming task had an assigned target complexity ($\\mathcal{O}(n\\log{}n)$ or $\\mathcal{O}(n)$ for example).\n",
    "\n",
    "This is obviously a very important topic, for instance when working on scaling a particular algorithm for a size of data. Usually, the algorithm itself can be analyzed to determine its complexity. For instance, if your input is an array and you're looping over all elements, then this will have a $\\mathcal{O}(n)$ complexity, where $n$ is the size of the array. If you're doing two nested loops, then the complexity is $\\mathcal{O}(n^2)$. \n",
    "\n",
    "But it's not always possible to actually be sure of the complexity before you write the algorithm. For instance when you use built-in functions for which the complexity is not straightforward. A simpler alternative is then to measure the timings of your function as a function of the input array size. I like this practical benchmarking idea, which we will demonstrate on three implementations that compute the [Fibonacci numbers](https://en.wikipedia.org/wiki/Fibonacci_number). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# First implementation: recursion "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first implementation I'd like to use if the recursive one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def fib_rec(n):\n",
    "    \"Computes Fibonacci number recursively.\"\n",
    "    if n == 0: \n",
    "        return 0\n",
    "    elif n == 1: \n",
    "        return 1\n",
    "    else:\n",
    "        return fib_rec(n-1) + fib_rec(n-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's test this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_rec(i) for i in range(10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Second implementation: loop style "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second one is a loop style implementation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fib_loop(n):\n",
    "    \"Computes Fibonacci number by iterating over all elements.\"\n",
    "    if n == 0: \n",
    "        return 0\n",
    "    elif n == 1: \n",
    "        return 1\n",
    "    else: \n",
    "        a, b = 0, 1\n",
    "        for i in range(n - 1):\n",
    "            a, b = b, a + b\n",
    "        return b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_loop(i) for i in range(10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Third implementation: log time "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The last implementation makes use of the fact that the Fibonacci numbers can be computed a [matrix multiplication in log time](http://kukuruku.co/hub/algorithms/the-nth-fibonacci-number-in-olog-n). The idea here is to decompose the number $n$ in powers of two to yield a speedup.\n",
    "\n",
    "It is based on the following things:\n",
    "\n",
    "- decompose $n$ in a sum of powers of 2\n",
    "- compute the $Q$ matrix exponents up to the highest power of two needed\n",
    "- assemble the result\n",
    "\n",
    "Let's look at this in a step by step process. First, let's decompose any number in powers of 2. We use a simple iterative algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import log2\n",
    "\n",
    "def powers_of_two(n):\n",
    "    \"Returns a list of the powers of 2 that sum to equal n.\"\n",
    "    powers = []\n",
    "    remainder = n\n",
    "    while remainder != 0:\n",
    "        powers.append(int(log2(remainder)))\n",
    "        remainder -= 2 ** powers[-1]\n",
    "    return powers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1, [0]),\n",
       " (2, [1]),\n",
       " (3, [1, 0]),\n",
       " (4, [2]),\n",
       " (5, [2, 0]),\n",
       " (6, [2, 1]),\n",
       " (7, [2, 1, 0]),\n",
       " (8, [3]),\n",
       " (9, [3, 0])]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[(i, powers_of_two(i)) for i in range(1, 10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second step is to compute the matrices needed for rapid multiplication. We will store each 2x2 matrix as a tuple. We also need to define a matrix multiplication operation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def precompute_matrix(max_power):\n",
    "    \"Precomputes matrix until highest power of 2 needed.\"\n",
    "    matrix_powers = {}\n",
    "    matrix_powers[0] = (1, 1, 1, 0)\n",
    "    for i in range(1, max_power + 1):\n",
    "        matrix_powers[i] = matrix_mult(matrix_powers[i-1], matrix_powers[i-1])\n",
    "    return matrix_powers\n",
    "\n",
    "def matrix_mult(mat1, mat2):\n",
    "    \"Multiplication of 2x2 matrices.\"\n",
    "    a, b, c, d = mat1\n",
    "    e, f, g, h = mat2\n",
    "    return (a*e + b*g, \n",
    "            a*f + b*h,\n",
    "            c*e + d*g,\n",
    "            c*f + d*h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check that the results are correct for the first few powers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: (1, 1, 1, 0),\n",
       " 1: (2, 1, 1, 1),\n",
       " 2: (5, 3, 3, 2),\n",
       " 3: (34, 21, 21, 13),\n",
       " 4: (1597, 987, 987, 610)}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precompute_matrix(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy.linalg import matrix_power"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 1],\n",
       "       [1, 1]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_power(np.array([[1, 1], [1, 0]]), 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5, 3],\n",
       "       [3, 2]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_power(np.array([[1, 1], [1, 0]]), 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[34, 21],\n",
       "       [21, 13]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_power(np.array([[1, 1], [1, 0]]), 8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1597,  987],\n",
       "       [ 987,  610]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix_power(np.array([[1, 1], [1, 0]]), 16)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This seems good, so we can now assemble the final parts of our log fibonacci function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "\n",
    "def fib_log(n):\n",
    "    \"Computes Fibonacci number in logarithmic time.\"\n",
    "    if n == 0: return 0\n",
    "    terms = powers_of_two(n)\n",
    "    matrix_terms = precompute_matrix(max(terms))\n",
    "    result = (1, 0, 0, 1)\n",
    "    for term in terms:\n",
    "        result = matrix_mult(result, matrix_terms[term])\n",
    "    return result[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_log(i) for i in range(10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Measuring timings"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this point, one can wonder: what is the complexity of these three implementations? Except for the loop one, which is linear, it's not easy to tell right away. That's why, I suggest, we just measure this."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Measuring is quite easy, in fact. Let's just create a timing function, that accepts a function as an argument and a number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import time\n",
    "\n",
    "def time_func(func, *arg):\n",
    "    \"Times the execution of a function. Result is in seconds.\"\n",
    "    tic = time.clock()\n",
    "    func(*arg)\n",
    "    toc = time.clock()\n",
    "    return toc - tic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see if it works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2619419745651634"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "time_func(fib_rec, 32)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4757292146374297"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "time_func(fib_loop, 200000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.47338340004959445"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "time_func(fib_log, 1000000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comparison of the three implementations using a plot "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this tool, we can devise a test case for which we can compare all three implementations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "test_case_rec = [2, 4, 8, 16, 24, 26, 28, 30, 32]\n",
    "test_case_loop = [10, 100, 200, 300, 500, 1000, 2000, 5000, 10000, 20000, 50000, 75000, 100000]\n",
    "test_case_power = [10, 100, 200, 300, 500, 1000]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use different test cases for each algorithm as will become clear in a second."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('bmh')\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2Uatw7uDO9M1pHceojEgxDZv/qkAtc5Y6Ld++P6YnHdtkeBpTPPBlkmwYLaGq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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7738ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 4))\n",
    "for ind, func, tc in zip(range(1, 4), \n",
    "                        [fib_rec, fib_loop, fib_log],\n",
    "                        [test_case_rec, test_case_loop, test_case_loop]):\n",
    "    timings = [time_func(func, test) for test in tc]\n",
    "    plt.subplot(1, 3, ind)\n",
    "    plt.plot(tc, timings, '-o')\n",
    "    plt.xlabel('size of input (n)')\n",
    "    plt.ylabel('time (s)')\n",
    "    plt.title(\"{}\".format(func.__name__))\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can put that same plot on the same axis to see things more concretely:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1000, 100000.0)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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AIJgZoOlA20UXzi6yt4387SJ7hM3MAA0AAAAEwcwATQfaLrpwdpG9beRvF9kj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\nTYAWERGRLoUYnEWyRT3QUvDUC+cv1d5vqn96iR7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RJR/pGSLT/ep9ACE2n05I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\nP/xDm7yR6wm5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x874af28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10, 10))\n",
    "for ind, func, tc in zip(range(1, 4), \n",
    "                        [fib_rec, fib_loop, fib_log],\n",
    "                        [test_case_rec, test_case_loop, test_case_loop]):\n",
    "    timings = [time_func(func, test) for test in tc]\n",
    "    plt.plot(tc, timings, '-o', label=\"{}\".format(func.__name__))\n",
    "    plt.xlabel('size of input (n)')\n",
    "    plt.ylabel('time (s)')\n",
    "plt.tight_layout()\n",
    "plt.legend()\n",
    "plt.ylim(-0.01, 0.4)\n",
    "plt.xlim(-1000, 1e5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we see above is quite interesting:\n",
    "\n",
    "- the recursive implementation explodes so much it doesn't even show on the plot (effectively making any usage above n=32 unpractical)\n",
    "- the loop style scales better\n",
    "- the log time implementation fares even better than the loop style"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusions "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Time for some takeaways. As the last plot has shown, the log implementation is definitely the fastest. The recursive implementation, while true to the spirit of the \"sequence style definition\" of the Fibonacci numbers is **very** impractical. It turns out that a straightforward way of making it faster is to memoize it, as [this blog post shows in the context of dynamic programming](https://jeremykun.com/2012/01/12/a-spoonful-of-python/). Another thought is that the algorithm's complexity effectively makes it tractable for large cases or just don't. If all you can come up with is something like the recursive implementation, there's many problems that will not be tractable (I'm thinking in terms of [project Euler problems](https://projecteuler.net/) here). "
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    "This post was entirely written using the IPython notebook. Its content is BSD-licensed. You can see a static view or download this notebook with the help of nbviewer at [20160531_MeasuringComplexityAlgorithm.ipynb](http://nbviewer.ipython.org/urls/raw.github.com/flothesof/posts/master/20160531_MeasuringComplexityAlgorithm.ipynb)."
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