{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sebastian Raschka \n",
      "last updated: 2017-08-14 \n",
      "\n",
      "CPython 3.6.1\n",
      "IPython 6.1.0\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a 'Sebastian Raschka' -u -d -v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Binary Search"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An implementation of the binary search algorithm. For details will follow. A good summary can be found on Wikipedia: https://en.wikipedia.org/wiki/Binary_search_algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The figures below provide a short illustration of how the implementation works on a toy example:\n",
    "\n",
    "![](images/binary_search/ex-1-1.png)\n",
    "\n",
    "![](images/binary_search/ex-1-2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binary Search Implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def binary_search(array, value):\n",
    "    ary = array\n",
    "    min_idx = 0\n",
    "    max_idx = len(array)\n",
    "    \n",
    "    while min_idx < max_idx:\n",
    "        middle_idx = (min_idx + max_idx) // 2\n",
    "\n",
    "        if array[middle_idx] == value:\n",
    "            return middle_idx\n",
    "        elif array[middle_idx] < value:\n",
    "            min_idx = middle_idx + 1\n",
    "        else:\n",
    "            max_idx = middle_idx\n",
    "            \n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "binary_search(array=[],\n",
    "              value=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "              value=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "              value=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                value=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                value=11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                value=99)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binary Search using Recursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this implementation of recursive binary search deliberately avoid slicing the `array` (e.g., `array[:middle_idx]`), because slicing Python lists is expensive due to the random memory access. E.g., slicing a Python list with as `a_list[:k]` is an O(k) operation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def recursive_binary_search(array, value, start_idx=None, end_idx=None):\n",
    "    \n",
    "    len_ary = len(array)\n",
    "    \n",
    "    if start_idx is None:\n",
    "        start_idx = 0\n",
    "    if end_idx is None:\n",
    "        end_idx = len(array) - 1\n",
    "    \n",
    "    if not len_ary or start_idx >= end_idx:\n",
    "        return None\n",
    "    \n",
    "    middle_idx = (start_idx + end_idx) // 2\n",
    "    if array[middle_idx] == value:\n",
    "        return middle_idx\n",
    "\n",
    "    elif array[middle_idx] > value:\n",
    "        return recursive_binary_search(array, \n",
    "                                       value, \n",
    "                                       start_idx=start_idx,\n",
    "                                       end_idx=middle_idx)\n",
    "    else:\n",
    "        return recursive_binary_search(array,\n",
    "                                       value,\n",
    "                                       start_idx=middle_idx + 1,\n",
    "                                       end_idx=len_ary)\n",
    "    return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "recursive_binary_search(array=[],\n",
    "                        value=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "recursive_binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                        value=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "recursive_binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                        value=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "recursive_binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                        value=11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "recursive_binary_search(array=[1, 2, 4, 7, 8, 10, 11],\n",
    "                        value=99)"
   ]
  }
 ],
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