20160913_RandomNumbersProgrammingPearls.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# First method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try that highlevel routine found in programming pearls. The example was: we want to select 2 numbers from the 1..5 range. I will select 1 with a 2 out of 5 chance. \n",
    "Possible outcomes are:\n",
    "\n",
    "- I selected 1 and now for 2 chances are 1 out of 4\n",
    "- I didn't select 1 and now for 2 chances are 2 out of 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def genrandom(m, n):\n",
    "    \"Returns m random numbers from 0..n range.\"\n",
    "    needed, remaining_in_list = m, n\n",
    "    numbers = []\n",
    "    for i in range(n):\n",
    "        if bigrand() % remaining_in_list < needed:\n",
    "            numbers.append(i)\n",
    "            needed -=1\n",
    "        remaining_in_list -= 1\n",
    "    return numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 4]"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "genrandom(2, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also need to define something like bigrand():"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from random import randint\n",
    "from functools import partial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "bigrand = partial(randint, a=0, b=2**32) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1087683378"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bigrand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now to some real questions: does this function work correctly?\n",
    "\n",
    "Let's try to run it a couple of times always with the same input and count the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "\n",
    "c = Counter()\n",
    "for _ in range(20000):\n",
    "    numbers = genrandom(2, 5)\n",
    "    c.update(numbers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(3, 8084), (1, 8040), (0, 7989), (4, 7950), (2, 7937)]"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.most_common()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the expected result versus the real result from our experiment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('bmh')\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 11000)"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Xe+y3/+rljJISLgl579XLdf0+SkoygEuaYASLKCnZQWFhn1rHE/7e43oE32vL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\nQFLdqQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x94e7780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(range(5), [c[key] for key in range(5)],  width=0.4, label='experimental result')\n",
    "plt.bar([_ + 0.5 for _ in range(5)], [20000*2/5 for _ in range(5)], width=0.4, label='perfect world', color='r')\n",
    "plt.legend()\n",
    "plt.ylim(0, 11000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks correct. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 2, 4, 6, 7, 9, 11, 13, 14, 19]"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "genrandom(10, 21)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Second method "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second method is also easy. We build a set of numbers and randomly try to insert a new one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def genset(m, n):\n",
    "    \"Selects m numbers from 0..n-1.\"\n",
    "    numbers = set()\n",
    "    while len(numbers) < m:\n",
    "        candidate = bigrand() % n\n",
    "        numbers.update([candidate])\n",
    "    return list(numbers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 3]"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "genset(2, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, let's try to see if the outcome are what we expect:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "c = Counter()\n",
    "for _ in range(20000):\n",
    "    numbers = genset(2, 5)\n",
    "    c.update(numbers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(3, 8203), (2, 8024), (4, 7996), (0, 7932), (1, 7845)]"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.most_common()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 11000)"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LKCnZQWFhn1rHE/7e43oE32vLY9689wmm7UIfUfxvsCpl/x+LgB0lJfSp5f8D\nYN683wC9Qx6BX071/0fi40UsFmPWrFkA5Ofnk52dTf/+/Tko51yjv4j/DRQFlycBo4LLo4BfBpdP\nAlYARwAdgXWABde9DfQkfqTxMnBhbfuZPHmyA9ckX0VFlW7btm21flUWFTXJICqLiuocw/jxLymL\nNMqiqXJQFsqiIVksWLDA1eexvDFHCHX5JTDHzG4CNhJ/ZRHOuVVmNof4K5K+Boa5oBoAw4HHgZbA\ny865V2q7Y80heOqPesrCUxaeskjeISkIzrnXgdeDy9uA8+rYbgIwoZb1y4Buh2IsIiLSMJF6p7Le\nh+DpNdaesvCUhacskhepgiAiIuGJVEHQHIKn/qinLDxl4SmL5EWqIIiISHgiVRA0h+CpP+opC09Z\neMoieZEqCCIiEp5IFQTNIXjqj3rKwlMWnrJIXqQKgoiIhCdSBUFzCJ76o56y8JSFpyySF6mCICIi\n4YlUQdAcgqf+qKcsPGXhKYvkRaogiIhIeCJVEDSH4Kk/6ikLT1l4yiJ5kSoIIiISnkgVBM0heOqP\nesrCUxaeskhepAqCiIiEJ1IFQXMInvqjnrLwlIWnLJIXqYIgIiLhiVRB0ByCp/6opyw8ZeEpi+RF\nqiCIiEh4IlUQNIfgqT/qKQtPWXjKInmRKggiIhKeSBUEzSF46o96ysJTFp6ySF6kCoKIiIQnUgVB\ncwie+qOesvCUhacskhepgiAiIuGJVEHQHIKn/qinLDxl4SmL5EWqIIiISHgiVRA0h+CpP+opC09Z\neMoieZEqCCIiEp5IFQTNIXjqj3rKwlMWnrJIXqQKgoiIhKfBBcHM8sxsoZl9YGYlZjYiWP8tM5tv\nZh+a2atmlplwm7vNbK2ZrTazCxLW9zCzlWa2xsym1LVPzSF46o96ysJTFp6ySF5jjhB2A3c6574H\n9AaGm9m/AaOBBc65E4GFwN0AZnYSMBjoCgwEppmZBfc1HRjqnOsCdDGzAY0Yl4iINECDC4JzrsI5\nVxxc/iewGsgDLgFmBpvNBC4NLg8CZjvndjvnNgBrgZ5mlgu0cc4tDbZ7IuE2e9Ecgqf+qKcsPGXh\nKYvkHZI5BDMrAE4F3gZynHNbIV40gOxgs/ZAWcLNNgXr2gPlCevLg3UiItKEDmvsHZjZ0cBc4KfO\nuX+amdtnk32XG+zBBx8EngQKgjXHEq9DPwiWFwXfD81yLBYDoLCwcK/lfqHsbf/lN0tKqKpl/4WF\nhUF/tHUYfaf3AAAGgklEQVTII6he7rHf/quXM0pKuDjkvVcv1/X7KCnJAC5ughEsoqRkB4WFfWod\nT/h7j+sRfK8tj3nz3ifo0oY+ovjfYFXK/j8WATtKSuhTy/8HwLx5vyHezQ5zBH451f8fiY8XsViM\nWbNmAZCfn092djb9+/fnoJxzDf4iXlBeIV4MqtetJn6UAJALrA4ujwZGJWz3CtArcZtg/VXA9Nr2\nN3nyZAeuSb6Kiirdtm3bav2qLCpqkkFUFhXVOYbx419SFmmURVPloCyURUOyWLBggavPY3pjW0aP\nAquccw8mrCsCbgwu3wDMS1h/lZkdYWYdgc7AO0FbqdLMegaTzNcn3GYvmkPw1B/1lIWnLDxlkbwG\nt4zMrC9wLVBiZisAB4wBJgJzzOwmYCPxVxbhnFtlZnOAVcDXwDAXHBIAw4HHgZbAy865Vxo6LhER\naZgGFwTn3FtAizquPq+O20wAJtSyfhnQ7WD7jL8PoR59sGYg3r+9+KDbNQfKwlMWnrJInt6pLCIi\nQMQKguYQPPVHPWXhKQtPWSQvUgVBRETCE6mCoM8y8vQ5LZ6y8JSFpyySF6mCICIi4YlUQdAcgqf+\nqKcsPGXhKYvkRaogiIhIeCJVEDSH4Kk/6ikLT1l4yiJ5kSoIIiISnkgVBM0heOqPesrCUxaeskhe\npAqCiIiEJ1IFQXMInvqjnrLwlIWnLJIXqYIgIiLhiVRB0ByCp/6opyw8ZeEpi+RFqiCIiEh4IlUQ\nNIfgqT/qKQtPWXjKInmRKggiIhKeSBUEzSF46o96ysJTFp6ySF6kCoKIiIQnUgVBcwie+qOesvCU\nhacskhepgiAiIuGJVEHQHIKn/qinLDxl4SmL5EWqIIiISHgiVRA0h+CpP+opC09ZeMoieZEqCCIi\nEp5IFQTNIXjqj3rKwlMWnrJIXqQKgoiIhCdSBUFzCJ76o56y8JSFpyySF6mCICIi4YlUQdAcgqf+\nqKcsPGXhKYvkRaogiIhIeNKmIJjZhWb2NzNbY2ajattGcwie+qOesvCUhacskpcWBcHMMoCHgAHA\n94Crzezf9t1u3bp1TT20tLV+/cpUDyFtKAtPWXjKwqvvk+m0KAhAT2Ctc26jc+5rYDZwyb4b7dix\no8kHlq527Nie6iGkDWXhKQtPWXjvvfdevbZLl4LQHihLWC4P1omISBNJl4JQLxUVFakeQtr45JON\nqR5C2lAWnrLwlEXyzDmX6jFgZmcB9zrnLgyWRwPOOTcxcbtbb73VJbaNTjnllGb7UtTi4uJm+7Pv\nS1l4ysJrzlkUFxfv1SZq3bo106dPt4PdLl0KQgvgQ6A/sAV4B7jaObc6pQMTEWlGDkv1AACcc3vM\n7DZgPvE21gwVAxGRppUWRwgiIpJ6kZlUrs8b15oDM5thZlvNrNm/yNrM8sxsoZl9YGYlZjYi1WNK\nFTM70sz+amYrgiz+O9VjSiUzyzCz5WZWlOqxpJqZbTCz94K/jXcOuG0UjhCCN66tIT7HsBlYClzl\nnPtbSgeWAmZWCPwTeMI51z3V40klM8sFcp1zxWZ2NLAMuKQ5/l0AmFkr59zOYE7uLWCEc+6ADwDf\nVGZ2B3A6cIxzblCqx5NKZrYeON059/nBto3KEUK93rjWHDjnYsBBf7HNgXOuwjlXHFz+J7CaZvz+\nFefczuDikcTnB9P/2V4IzCwPuAj4XarHkiaMej7WR6Ug6I1rckBmVgCcCvw1tSNJnaBNsgKoAP7s\nnFua6jGlyAPAf9FMC2ItHPBnM1tqZjcfaMOoFASROgXtornAT4MjhWbJOVflnDsNyAN6mdlJqR5T\nUzOzi4GtwZGjBV/NXV/nXA/iR03Dg7ZzraJSEDYB+QnLecE6aebM7DDixeBJ59y8VI8nHTjntgOv\nARemeiwp0BcYFPTNnwHOMbMnUjymlHLObQm+fwq8QLwFX6uoFISlQGczO97MjgCuAprzqwf0zMd7\nFFjlnHsw1QNJJTP7jpllBpePAs4Hmt3kunNujHMu3znXifjjxELn3PWpHleqmFmr4AgaM2sNXAC8\nX9f2kSgIzrk9QPUb1z4AZjfXN66Z2SxgMdDFzErN7MepHlOqmFlf4Frg3OAldcvNrDk+KwZoB7xm\nZsXE51Fedc69nOIxSerlALFgbult4A/Oufl1bRyJl52KiEj4InGEICIi4VNBEBERQAVBREQCKggi\nIgKoIIiISEAFQUREABUEEREJqCCIiAgA/w8Jv4BXD/SdjgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9674128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(range(5), [c[key] for key in range(5)],  width=0.4, label='experimental result')\n",
    "plt.bar([_ + 0.5 for _ in range(5)], [20000*2/5 for _ in range(5)], width=0.4, label='perfect world', color='r')\n",
    "plt.legend()\n",
    "plt.ylim(0, 11000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, this works nicely. Incidentaly, I would have expected this generation routine not to distribute things evenly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Third method "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The third method is also quite clever. We generate a shuffled sequence and take the first m elements. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from random import shuffle\n",
    "\n",
    "def genshuffle(m, n):\n",
    "    \"Generates a sequence of m integers in range 1..n.\"\n",
    "    return sorted(shuffled(range(n))[:m])\n",
    "\n",
    "def shuffled(seq):\n",
    "    \"Returns a shuffled version of a sequence.\"\n",
    "    copy = list(seq[:])\n",
    "    shuffle(copy)\n",
    "    return copy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1]"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "genshuffle(2, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, let's see if this is distributed in the expected way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "c = Counter()\n",
    "for _ in range(20000):\n",
    "    numbers = genshuffle(2, 5)\n",
    "    c.update(numbers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({0: 8041, 1: 7921, 2: 8026, 3: 8031, 4: 7981})"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 11000)"
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LQ9571XJtv4+Skgzg4iYYwSJKSnZQWNi3xvGEv/e4nsH3mvKYN+8Dgmm70EcU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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x96aae10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(range(5), [c[key] for key in range(5)],  width=0.4, label='experimental result')\n",
    "plt.bar([_ + 0.5 for _ in range(5)], [20000*2/5 for _ in range(5)], width=0.4, label='perfect world', color='r')\n",
    "plt.legend()\n",
    "plt.ylim(0, 11000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nice, a third algorithm that works. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# One that doesn't work "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To finish off this post, we'll use an algorithm that doesn't work. It's taken from Peter Norvig's CS212 class on Udacity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'll '"
   ]
  }
 ],
 "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}