fibonacci

fibonacci.ipynb

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   "cell_type": "code",
   "execution_count": 21,
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   "source": [
    "def fib(n):\n",
    "    if n < 2:\n",
    "        return n\n",
    "    else: return fib(n-1)+fib(n-2) # O(2^n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib(n) for n in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "326 ms ± 14.9 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%timeit fib(30)  # unbelievably slow, since it repeats a huge amount of work"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def helper(a, b, n):\n",
    "    if n < 2:\n",
    "        return b\n",
    "    else:\n",
    "        return helper(b, a+b, n-1)\n",
    "def fib_rec(n):\n",
    "    return helper(0,1,n) # O(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_rec(n) for n in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "50.5 µs ± 2 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit fib_rec(300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fib_dyn(n):\n",
    "    a,b = 1,1\n",
    "    for _ in range(2, n):\n",
    "        a, b = b, a+b\n",
    "    return b  # O(n), but not recursive, so it doesn't hit pythons recursion limit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_dyn(n) for n in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "17.7 µs ± 434 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit fib_dyn(300) # significantly faster than recursive because of the lack of overhead of python function calls"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "def modified_helper(a,b,n):\n",
    "    if n < 2:\n",
    "        return (a,b,n)\n",
    "    else:\n",
    "        return modified_helper(b, a+b, n-1)\n",
    "\n",
    "def fib_rec_trampolined(n):\n",
    " '''\n",
    "     Conceptually this is basically the same as the recursive version.\n",
    "     However, because it 'bounces' the state back up into the parent function, it avoids the recursion limit.\n",
    "     While this is still slower than the dynamic method because of function call overhead, it presents a straightforward way to convert tail recursive functions into iterative ones.\n",
    " '''\n",
    "    a,b = 0,1\n",
    "    while n >=2:\n",
    "        (a, b, n) = modified_helper(a,b,n)\n",
    "    return b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1, 2, 3, 5, 8, 13, 21, 34]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[fib_rec_trampolined(n) for n in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "55.4 µs ± 1.6 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit fib_rec_trampolined(300)"
   ]
  }
 ],
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