experiments/notebooks/figure-3-fldr-gap.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from math import ceil\n",
    "from math import log2\n",
    "\n",
    "import matplotlib.ticker as ticker\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Compute the terms in the gap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_rejection_precision(m):\n",
    "    return ceil(log2(m))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_gap(m):\n",
    "    k = ceil(log2(m))\n",
    "    term1 = log2(2**k/m)\n",
    "    term2 = (2**k-m)/m * log2(2**k/(2**k-m)) if m != 2**k else 0\n",
    "    term3 = 2 * 2**k/m\n",
    "    return (term1, term2, term3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ms = np.arange(2, 2**13, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "gaps = list(map(get_gap, Ms))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot the three elements of the gap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams['text.usetex'] = True\n",
    "plt.rcParams['font.family'] = 'sans-serif'\n",
    "\n",
    "plt.rcParams['font.sans-serif'] = 'Computer Modern Sans Serif'\n",
    "plt.rcParams['font.serif'] = 'Computer Modern Roman'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "colors = plt.rcParams['axes.prop_cycle'].by_key()['color']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "colors = ['black'] * 10\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "for i in range(1, 13):\n",
    "    labels = [None] * 4\n",
    "    if i == 1:\n",
    "        labels = [\n",
    "            'Eq~(3), Term 1',\n",
    "            'Eq~(3), Term 2',\n",
    "            'Eq~(3), Term 3',\n",
    "            'Eq~(3); Total',\n",
    "        ]\n",
    "    xs = np.arange(2**i+1, 2**(i+1)+1)\n",
    "    gaps = list(map(get_gap, xs))\n",
    "    ax.plot(xs, [sum(g) for g in gaps], label=labels[3], color=colors[3])\n",
    "    ax.plot(xs, [g[2] for g in gaps], label=labels[2], color=colors[2], linestyle='--', linewidth=1)\n",
    "    ax.plot(xs, [g[1] for g in gaps], label=labels[1], color=colors[1], linestyle='-.', linewidth=1)\n",
    "    ax.plot(xs, [g[0] for g in gaps], label=labels[0], color=colors[0], linestyle=':', linewidth=1)\n",
    "\n",
    "# ax.scatter(Zs, [g[0] for g in gaps], marker='.', label='Rejection Dist. Entropy', s=2)\n",
    "# ax.scatter(Zs, [g[1] for g in gaps], marker='.', label='Rejection Probability', s=2)\n",
    "# ax.scatter(Zs, [g[2] for g in gaps], marker='.', label='Sampling Toll', s=2)\n",
    "# ax.scatter(Zs, [g[0] + g[1] + g[2] for g in gaps], marker='.', label='Total Gap', s=2)\n",
    "\n",
    "ax.set_xscale('log', basex=2)\n",
    "ax.set_xlabel('Sum of Integers in Input Instance ($m$)')\n",
    "ax.set_ylabel('FLDR Entropy Gap (Upper Bound)')\n",
    "\n",
    "ax.legend(loc='upper left', framealpha=0, handlelength=1, handletextpad=0.5)\n",
    "\n",
    "ax.yaxis.set_major_locator(\n",
    "    ticker.MultipleLocator(1))\n",
    "ax.yaxis.set_minor_locator(\n",
    "    ticker.MultipleLocator(.5))\n",
    "\n",
    "ax.xaxis.set_major_locator(\n",
    "    ticker.LogLocator(base=2., subs=(1.,), numticks=24))\n",
    "\n",
    "ax.xaxis.set_minor_locator(\n",
    "    ticker.LogLocator(base=2., subs=np.arange(2, 10)*.1, numticks=200))\n",
    "ax.xaxis.set_minor_formatter(ticker.NullFormatter())\n",
    "\n",
    "ax.set_xticks([2**i for i in range(1, 14)])\n",
    "ax.set_xlim([.5, 2**10])\n",
    "ax.set_ylim([None, 6.5])\n",
    "ax.axhline(y=6, xmin=0.3, color='red', linewidth=1)\n",
    "\n",
    "fig.set_size_inches(5, 3)\n",
    "fig.set_tight_layout(True)\n",
    "fig.set_facecolor('white')\n",
    "for ext in ['pdf', 'png']:\n",
    "    fig.savefig('figure-3-fldr-gap.%s' % (ext,))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
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