Golden Ratio and time complexity.ipynb

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    "Golden ratio and time complexity\n",
    "======\n",
    "The time it takes a program to terminate as a function of the input is sometimes exponential. This is often expressed as $O(c^n)$, where $c$ is some constant.\n",
    "\n",
    "An example is the following code:"
   ],
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    "def powerset(n):\n",
    "    '''Returns the a list of all subsets of {1,2,...,n} for a postive integer n'''\n",
    "    if n==0:\n",
    "        return [[]]\n",
    "    subsets_without_n = powerset(n-1)\n",
    "    subsets_with_n = [subset + [n] for subset in  subsets_without_n]\n",
    "    return subsets_without_n + subsets_with_n\n",
    "\n",
    "def test_powerset():\n",
    "    assert powerset(0) == [[]]\n",
    "    assert powerset(3) == [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]\n",
    "test_powerset()"
   ]
  },
  {
   "source": [
    "Here the constant $c$ is $2$, as there are $2^n$ subsets of $\\{1,2,...,n \\}$. Recently, I encountered an interesting situation, where the constant happens to be the golden ratio.\n",
    "\n",
    "Consider for a moment the following program (and disregard the fact that it could have solved the problem in a much better way)"
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   "source": [
    "def fibonnachi(n):\n",
    "    '''\n",
    "    Returns the n'th term in the fibonnachi sequence\n",
    "    '''\n",
    "    if n <= 2:\n",
    "        return 1\n",
    "    return fibonnachi(n-1) + fibonnachi(n-2)"
   ]
  },
  {
   "source": [
    "Recall that the fibonacchi sequence $F_n$ is defined by:\n",
    "$$ F_1 = 1, F_2 = 1 $$\n",
    "and\n",
    "$$ F_n = F_{n-1} + F_{n-2}  \\ , n ≥ 2 $$\n",
    "\n",
    "(for more info see https://oeis.org/A000045 )\n",
    "\n",
    "The question now is \"what is the time complexity of this function?\"\n",
    "\n",
    "We see that if $n$ is 1 or 2, then the result is returned immediately, and if $n$ is greater, then we have to wait for fibonacchi(n-1) and fibonacchi(n-2) to finish.\n",
    "\n",
    "The time $T_n$ for the program to terminate thus satisfy the same identies as $F_n$ above. This roughly means that the time to compute the answer and hte answer itself is the same. Cool.\n",
    "\n",
    "But where does the golden ration come from? It turns out that there exist a closed form for the fibanacchi sequence, first discovered by Daniel Bernoulli in 1728:\n",
    "$$\n",
    "F_n = \\frac{1}{\\sqrt{5}}(\\phi^n-\\Phi^n)\n",
    "$$\n",
    "where $\\phi = \\frac{1+\\sqrt{5}}{2} = 1.6180339887...$ and $\\Phi = \\frac{1-\\sqrt{5}}{2} = -0.6180339887...$.\n",
    "\n",
    "Since the $|\\Phi|$ < 1, we see that  $\\lim_{n\\to\\infty}\\Phi^n = 0$, so \n",
    "$$\n",
    "F_n  \\approx \\frac{1}{\\sqrt{5}}(\\phi^n)\n",
    "$$\n",
    "But we can turn this approximation into an equality! Consider the following number $\\frac{\\Phi}{\\sqrt{5}}$:"
   ],
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       "-0.27639320225002106"
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   "source": [
    "import math\n",
    "Phi = -0.6180339887498949\n",
    "Phi/math.sqrt(5)"
   ]
  },
  {
   "source": [
    "Notice that $|\\frac{\\Phi}{\\sqrt{5}}| < 0.5$. Since furthermore $|\\frac{\\Phi^n}{\\sqrt{5}}| < 0.5$ for all $n$, we have the following identity:\n",
    "$$ F_n = \\left\\lfloor \\frac{\\phi^n}{\\sqrt{5}}=+\\frac{1}{2} \\right\\rfloor = \\text{round}\\left(\\frac{\\phi^n}{\\sqrt{5}}\\right) $$\n",
    "\n",
    "These identities show that $T_n$ is $O(\\phi^n)$.\n",
    "\n",
    "That is all well and good in theory, but does it also work in practice.\n",
    "\n",
    "Let us a define a method which plots data on the actual time complexity and an exponential regression of the data. We will test it on the \"powerset\" method from earlier."
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     },
     "metadata": {}
    },
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The regression curve is\n2.0726432487205365e-07 * exp(0.7789252629055129 x) + 0.0012709958961720125\nThe regression extimates the constant c as 2.179129015748139\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "def time_plot(method, start, end):\n",
    "    method_time = []\n",
    "    input_range = range(start,end)\n",
    "\n",
    "    for n in input_range:\n",
    "        start = time.perf_counter()\n",
    "        method(n)\n",
    "        end = time.perf_counter()\n",
    "        timestamp = time.time() - start_time\n",
    "        method_time.append(end - start)\n",
    "\n",
    "    \n",
    "    def func(x, a, b, c):\n",
    "        return a * np.exp(b * x) + c\n",
    "\n",
    "    popt, pcov = curve_fit(func, input_range, method_time)\n",
    "\n",
    "    plt.figure().canvas.manager.set_window_title(\"Time Complexity\")\n",
    "    plt.plot(input_range, method_time, 'ko', label=\"Original Noised Data\")\n",
    "    plt.plot(input_range, func(input_range, *popt), 'r-', label=\"Fitted Curve\")\n",
    "    plt.xlabel(\"Input Length\")\n",
    "    plt.ylabel(\"Execution Time (s)\")\n",
    "    plt.legend()\n",
    "    plt.show()\n",
    "\n",
    "    a = popt[0]\n",
    "    b = popt[1]\n",
    "    c = popt[2]\n",
    "\n",
    "    print(\"The regression curve is\")\n",
    "    print(f\"{a} * exp({b} x) + {c}\")\n",
    "    base = math.exp(b)\n",
    "    print(f\"The regression extimates the constant c as {base}\")\n",
    "\n",
    "time_plot(powerset, 1, 20)"
   ]
  },
  {
   "source": [
    "We see it the \"powerset\" example has an exponential behaviour. The base constant c in $O(c^n)$ is estimated to around 2, but the value fluctuates a bit each time the data is generated.\n",
    "\n",
    "The base value is obtained as $e^{b}$ from the regression formula $T_x = a e^{b x} + {c}$ by using $ e^{b x}=\\left( e^{b}\\right)^x$.\n",
    "\n",
    "Now for the moment of truth. What will happen when we try to find the base value,$c$ for our \"fibonnachi\" method?"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
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     },
     "metadata": {}
    },
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The regression curve is\n1.3651692620976048e-07 * exp(0.47859837037103253 x) + -0.00017374246112787685\nThe regression extimates the constant c as 1.6138108511288576\n"
     ]
    }
   ],
   "source": [
    "time_plot(fibonnachi, 1, 35)"
   ]
  },
  {
   "source": [
    "The value is thus estimated as 1.6176029818338031 (the value on your screen may be different, sice the code can be rerun). This gives an estimate which is correct on the first 2 decimals.\n",
    "\n",
    "Attention should be drawn to the fact that is a terrible way of estimating the golden ratio. Indeed even the method \"fibonnachi\" is a very slow and therefore poor implementation.\n",
    "\n",
    "Let us thus end with a good implementation of a function computing the fibonacchi numbers."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fib_fast(n):\n",
    "    ''' Returns the n'th term in the fibonnachi sequence '''\n",
    "    if n <= 2:\n",
    "        return 1\n",
    "    prev, cur = 1, 1\n",
    "    i = 2\n",
    "    while i < n:\n",
    "        prev, cur = cur, prev + cur\n",
    "        i += 1\n",
    "    return cur\n",
    "    \n",
    "def test_fib_fast():\n",
    "    assert fib_fast(2) == 1\n",
    "    assert fib_fast(3) == 2\n",
    "    assert fib_fast(4) == 3\n",
    "    assert fib_fast(5) == 5\n",
    "    assert fib_fast(6) == 8\n",
    "\n",
    "test_fib_fast()"
   ]
  },
  {
   "source": [
    "Let us compare this implementation to the slow one from earlier."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The regression for the slow curve is\n1.2227731687893812e-06 * exp(0.3632289063372369 x) + -9.470939864951051e-06\nThe regression for the fast curve is\n7.709431087317102e-08 * x + 4.40948050389317e-07\n"
     ]
    },
    {
     "output_type": "display_data",
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ZY0xRNWIE7N0Lf/tbsV9zLJoCE4yqtveeK8cnHGOMKQZWrIDnn4c77oC0tERH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\nMcYE4v8D5WkIumVKq3oAAAAASUVORK5CYII=\n"
     },
     "metadata": {}
    }
   ],
   "source": [
    "def exponential(x, a, b, c):\n",
    "        return a * np.exp(b * x) + c\n",
    "\n",
    "def linear(x,a,b):\n",
    "    return a * x + b\n",
    "\n",
    "def time_method(method, start, end):\n",
    "    method_time = []\n",
    "    input_range = range(start,end)\n",
    "\n",
    "    for n in input_range:\n",
    "        start = time.perf_counter()\n",
    "        method(n)\n",
    "        end = time.perf_counter()\n",
    "        method_time.append(end - start)\n",
    "\n",
    "    return method_time\n",
    "\n",
    "def fit_and_plot(fit_func_assumption, input_range, method_time, method_name, colour,single_figure = True):\n",
    "    popt, pcov = curve_fit(fit_func_assumption, input_range, method_time)\n",
    "    if single_figure:\n",
    "        plt.figure().canvas.manager.set_window_title(\"Time Complexity\")\n",
    "    plt.plot(input_range, method_time, f'{colour}o')\n",
    "    plt.plot(input_range, fit_func_assumption(input_range, *popt), f'{colour}-', label=f\"{method_name} Curve\")\n",
    "    plt.xlabel(\"Input Length\")\n",
    "    plt.ylabel(\"Execution Time (s)\")\n",
    "    plt.legend()\n",
    "    if single_figure:\n",
    "        plt.show()\n",
    "    return popt\n",
    "\n",
    "# Plotting the 2 Fibonacchi methods agains each other\n",
    "plt.figure().canvas.manager.set_window_title(\"Time Complexity\")\n",
    "method_time = time_method(fibonnachi, 1, 20)\n",
    "popt = fit_and_plot(exponential, range(1, 20), method_time,\"Fibonnachi slow\",'r', False)\n",
    "a = popt[0]\n",
    "b = popt[1]\n",
    "c = popt[2]\n",
    "print(\"The regression for the slow curve is\")\n",
    "print(f\"{a} * exp({b} x) + {c}\")\n",
    "method_time = time_method(fib_fast, 1, 20)\n",
    "popt = fit_and_plot(linear, range(1, 20), method_time,\"Fibonnachi fast\",'b', False)\n",
    "a = popt[0]\n",
    "b = popt[1]\n",
    "print(\"The regression for the fast curve is\")\n",
    "print(f\"{a} * x + {b}\")\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "We can make an even faster method, by using the closed form of the fibonnachi sequence:"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fib_maybe_very_fast(n):\n",
    "    '''Returns the n'th term in the fibonnachi sequence'''\n",
    "    return round(1.6180339887 ** n /math.sqrt(5))\n",
    "    \n",
    "def test_fib_maybe_very_fast():\n",
    "    assert fib_maybe_very_fast(2) == 1\n",
    "    assert fib_maybe_very_fast(3) == 2\n",
    "    assert fib_maybe_very_fast(4) == 3\n",
    "    assert fib_maybe_very_fast(5) == 5\n",
    "    assert fib_maybe_very_fast(6) == 8\n",
    "\n",
    "test_fib_maybe_very_fast()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The regression for the very fast curve is\n2.14308468601709e-10 * x + 1.220224595503382e-06\nThe regression for the fast curve is\n1.218446062311882e-07 * x + 1.0571116448334428e-06\n"
     ]
    },
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MMXz77bc0bdqUcePG0bhxY26//faj6l6zZk1M02yrVq1izZo1tG/fnvfff59bb72V//zP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\no3biVXw8xfOpjVg8n/gSLE5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     },
     "metadata": {}
    }
   ],
   "source": [
    "input_range = range(1,1000)\n",
    "\n",
    "plt.figure().canvas.manager.set_window_title(\"Time Complexity\")\n",
    "method_time = time_method(fib_maybe_very_fast, input_range[0], input_range[-1]+1)\n",
    "popt = fit_and_plot(linear, input_range, method_time,\"fib_maybe_very_fast\",'r', False)\n",
    "a = popt[0]\n",
    "b = popt[1]\n",
    "print(\"The regression for the very fast curve is\")\n",
    "print(f\"{a} * x + {b}\")\n",
    "method_time = time_method(fib_fast, input_range[0], input_range[-1]+1)\n",
    "popt = fit_and_plot(linear, input_range, method_time,\"Fibonnachi fast\",'b', False)\n",
    "a = popt[0]\n",
    "b = popt[1]\n",
    "print(\"The regression for the fast curve is\")\n",
    "print(f\"{a} * x + {b}\")\n",
    "plt.show()"
   ]
  },
  {
   "source": [
    "We see that it is indeed faster to use the closed form rather than sum a lot."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "Below I try to see if the ratio between the actual compation time and the fib value is indeed constant. If this was the case (as it should be in theory), then one should be able to implement a very very slow method that computes the n'th fibonnachi number by first computing it using the slow method and then measuring the time it took and adjusting that time.\n",
    "\n",
    "It seems to work a bit different in practice than it does on the purely theoretic algorithmic level and I was not able to find a perfect relationship between the computation time and the value of the fibonnachi number. Oh well..."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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d0TUYcabVXg/8EviSaLU3MbY3j8YlxgBz3P3WKi4rAY4kTNXdjbDWYz6wDOhiZnsREsUI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\n"
     },
     "metadata": {}
    },
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The regression for the ratio curve is\n2.8163453251807272e-09 * x + 2.444448156474604e-07\n"
     ]
    }
   ],
   "source": [
    "input_range = range(5,15)\n",
    "num_iterations = 100000\n",
    "\n",
    "average_method_time = []\n",
    "for n in input_range:\n",
    "    start = time.perf_counter()\n",
    "    for i in range(num_iterations):\n",
    "        fibonnachi(n)\n",
    "    end = time.perf_counter()\n",
    "    average_method_time.append((end - start)/num_iterations)\n",
    "\n",
    "time_to_output_ratios = [(average_method_time[i-input_range[0]])/fib_maybe_very_fast(i) for i in input_range]\n",
    "\n",
    "popt, pcov = curve_fit(linear, input_range, time_to_output_ratios)\n",
    "    \n",
    "plt.figure().canvas.manager.set_window_title(\"Time Complexity\")\n",
    "plt.plot(input_range, time_to_output_ratios, f'ko')\n",
    "plt.plot(input_range, linear(input_range, *popt), 'r-', label=f\"Ratio Curve\")\n",
    "plt.xlabel(\"Input Length\")\n",
    "plt.ylabel(\"Ratios of Time and Fib number\")\n",
    "plt.legend()\n",
    "plt.show()\n",
    "a = popt[0]\n",
    "b = popt[1]\n",
    "print(\"The regression for the ratio curve is\")\n",
    "print(f\"{a} * x + {b}\")"
   ]
  },
  {
   "source": [
    "Examples of regression lines with different ways of iterating.\n",
    "\n",
    "-3.4914444034112626e-10 * x + 2.968440845792717e-07\n",
    "\n",
    "6.6372040304205115e-09 * x + 2.079098234992264e-07\n",
    "\n",
    ".910374018775641e-09 * x + 1.9283661090008395e-07\n",
    "\n",
    "7.05452024647927e-09 * x + 1.918605670597993e-07\n",
    "\n",
    "5.372013001210546e-09 * x + 2.072016519928606e-07"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [],
   "source": [
    "fib_approx = [time/2.7e-07 for time in average_method_time]\n",
    "fib_appox_diff = [fib_approx[i]-fib_maybe_very_fast(i+5) for i in range(len(fib_approx))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[-0.2877142598110911,\n",
       " -0.653625665870667,\n",
       " -0.10177992621471255,\n",
       " 0.10983366664084926,\n",
       " 0.6680965923480002,\n",
       " 0.9546608893907873,\n",
       " 5.871105925887136,\n",
       " 8.508011000463739,\n",
       " 11.80197674172274,\n",
       " -9.959361221529207]"
      ]
     },
     "metadata": {},
     "execution_count": 208
    }
   ],
   "source": [
    "fib_appox_diff"
   ]
  },
  {
   "source": [
    "Thanks for reading"
   ],
   "cell_type": "markdown",
   "metadata": {}
  }
 ]
}