frechet_distance_recursion_vs_dp.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fréchet distance\n",
    "\n",
    "This will compare the performance of calculating the discrete Fréchet distance with the recursive and dynamic programming algorithms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import similaritymeasures as sm\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import toleranceinterval as ti\n",
    "from scipy.spatial import minkowski_distance\n",
    "from time import time\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dp_frechet_dist(P, Q, pnorm=2):\n",
    "    \"\"\" dynamic programming for discrete frechet distance\n",
    "    \n",
    "    by SenZHANG https://github.com/SenZHANG-GitHub\n",
    "    https://github.com/cjekel/similarity_measures/issues/6#issuecomment-544350039\n",
    "    \"\"\"\n",
    "    n = len(P)\n",
    "    m = len(Q)\n",
    "    ca = np.ones((n, m))\n",
    "    ca = np.multiply(ca, -1)\n",
    "    ca[0, 0] = minkowski_distance(P[0], Q[0], p=pnorm)\n",
    "    for i in range(1, n):\n",
    "        ca[i, 0] = max(ca[i-1, 0], minkowski_distance(P[i], Q[0], p=pnorm))\n",
    "    for j in range(1, m):\n",
    "        ca[0, j] = max(ca[0, j-1], minkowski_distance(P[0], Q[j], p=pnorm))\n",
    "    for i in range(1, n):\n",
    "        for j in range(1, m):\n",
    "            ca[i, j] = max(min(ca[i-1, j], ca[i, j-1], ca[i-1, j-1]), minkowski_distance(P[i], Q[j], p=pnorm))\n",
    "    return ca[n-1, m-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def benchmark_frechet(sizes, method, seed):\n",
    "    np.random.seed(seed)\n",
    "    n_dim = 5\n",
    "    n_rep = 100\n",
    "    times = np.zeros((len(sizes), n_rep))\n",
    "    for i, size in enumerate(sizes):\n",
    "        for j in range(n_rep):\n",
    "            P = np.random.random((size, n_dim))\n",
    "            Q = np.random.random((size, n_dim))\n",
    "            t0 = time()\n",
    "            d = method(P, Q)\n",
    "            t1 = time()\n",
    "            times[i, j] = t1 - t0\n",
    "    return times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "sizes = [20, 40, 50, 100, 200, 400, 500, 1000]\n",
    "# recursion times\n",
    "times_recursion = benchmark_frechet(sizes, sm.frechet_dist, 123)\n",
    "# dynamic programming times\n",
    "times_dp = benchmark_frechet(sizes, dp_frechet_dist, 123)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x22182296668>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# compute means of runtimes\n",
    "times_recursion_mean = times_recursion.mean(axis=1)\n",
    "times_dp_mean = times_dp.mean(axis=1)\n",
    "\n",
    "# compute the 99th percntile to 99% confidence\n",
    "bounds_recursion = ti.twoside.normal(times_recursion, 0.99, 0.99)\n",
    "bounds_dp = ti.twoside.normal(times_dp, 0.99, 0.99)\n",
    "\n",
    "# get the - and plus values for error bars\n",
    "k_sigma_recursion = bounds_recursion[:, 1] - times_recursion_mean\n",
    "k_sigma_dp = bounds_dp[:, 1] - times_dp_mean\n",
    "\n",
    "fig = plt.figure()\n",
    "\n",
    "plt.errorbar(sizes, times_recursion_mean,\n",
    "             yerr=k_sigma_recursion,\n",
    "             capsize=2.0,\n",
    "             label='Recursion')\n",
    "plt.errorbar(sizes, times_dp_mean,\n",
    "             yerr=k_sigma_dp,\n",
    "             capsize=2.0,\n",
    "             label='Dynamic Programming')\n",
    "\n",
    "plt.semilogx(True)\n",
    "plt.semilogy(True)\n",
    "plt.xlabel(r'size of both $n$ and $m$')\n",
    "plt.ylabel('Run time, seconds')\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.44202367e-03, 4.50518283e-03],\n",
       "       [1.43606370e-02, 1.71280056e-02],\n",
       "       [2.35760593e-02, 2.60690833e-02],\n",
       "       [9.53312417e-02, 1.02528656e-01],\n",
       "       [3.84261025e-01, 4.05437513e-01],\n",
       "       [1.56694557e+00, 1.62862086e+00],\n",
       "       [2.46232444e+00, 2.52240927e+00],\n",
       "       [9.83034253e+00, 1.01670652e+01]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bounds_dp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.00053158, 0.00138368, 0.00124651, 0.00359871, 0.01058824,\n",
       "       0.03083765, 0.03004241, 0.16836134])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_sigma_dp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.69427109e-03, 1.95678020e-02, 3.02675223e-02, 1.21580572e-01,\n",
       "       4.73871052e-01, 1.94466767e+00, 2.88474394e+00, 1.18369269e+01])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "times_recursion_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.97360325e-03, 1.57443213e-02, 2.48225713e-02, 9.89299488e-02,\n",
       "       3.94849269e-01, 1.59778322e+00, 2.49236686e+00, 9.99870387e+00])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "times_dp_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.18136381, 1.24284823, 1.21935484, 1.22895618, 1.20013152,\n",
       "       1.21710358, 1.15743151, 1.18384613])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "times_recursion_mean/times_dp_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x221833ea908>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "sns.set()\n",
    "sns.distplot(times_recursion[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x2218346fb38>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(times_dp[-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusion\n",
    "\n",
    "**The iterative code was about 1.2 times faster (on average, with 5 dimensions) than the recursive code.** The iterative code also has the benefit of not dealing with recursion limits. I think it's time to replace the recursive code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.3.4'"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}