{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This notebook was prepared by [Donne Martin](https://github.com/donnemartin). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solution Notebook"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem: Implement a min heap with extract_min and insert methods.\n",
"\n",
"* [Constraints](#Constraints)\n",
"* [Test Cases](#Test-Cases)\n",
"* [Algorithm](#Algorithm)\n",
"* [Code](#Code)\n",
"* [Unit Test](#Unit-Test)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Constraints\n",
"\n",
"* Can we assume the inputs are ints?\n",
" * Yes\n",
"* Can we assume this fits memory?\n",
" * Yes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Test Cases\n",
"\n",
"* Extract min of an empty tree\n",
"* Extract min general case\n",
"* Insert into an empty tree\n",
"* Insert general case (left and right insert)\n",
"\n",
"\n",
" _5_\n",
" / \\\n",
" 20 15\n",
" / \\ / \\\n",
" 22 40 25\n",
" \n",
"extract_min(): 5\n",
"\n",
" _15_\n",
" / \\\n",
" 20 25\n",
" / \\ / \\\n",
" 22 40 \n",
"\n",
"insert(2):\n",
"\n",
" _2_\n",
" / \\\n",
" 20 5\n",
" / \\ / \\\n",
" 22 40 25 15\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algorithm\n",
"\n",
"A heap is a complete binary tree where each node is smaller than its children.\n",
"\n",
"### extract_min\n",
"\n",
"\n",
" _5_\n",
" / \\\n",
" 20 15\n",
" / \\ / \\\n",
" 22 40 25\n",
"\n",
"Save the root as the value to be returned: 5\n",
"Move the right most element to the root: 25\n",
"\n",
" _25_\n",
" / \\\n",
" 20 15\n",
" / \\ / \\\n",
" 22 40 \n",
"\n",
"Bubble down 25: Swap 25 and 15 (the smaller child)\n",
"\n",
" _15_\n",
" / \\\n",
" 20 25\n",
" / \\ / \\\n",
" 22 40 \n",
"\n",
"Return 5\n",
"\n",
"\n",
"We'll use an array to represent the tree, here are the indices:\n",
"\n",
"\n",
" _0_\n",
" / \\\n",
" 1 2\n",
" / \\ / \\\n",
" 3 4 \n",
"\n",
"\n",
"To get to a child, we take 2 * index + 1 (left child) or 2 * index + 2 (right child).\n",
"\n",
"For example, the right child of index 1 is 2 * 1 + 2 = 4.\n",
"\n",
"Complexity:\n",
"* Time: O(lg(n))\n",
"* Space: O(lg(n)) for the recursion depth (tree height), or O(1) if using an iterative approach\n",
"\n",
"### insert\n",
"\n",
"\n",
" _5_\n",
" / \\\n",
" 20 15\n",
" / \\ / \\\n",
" 22 40 25\n",
"\n",
"insert(2):\n",
"Insert at the right-most spot to maintain the heap property.\n",
"\n",
" _5_\n",
" / \\\n",
" 20 15\n",
" / \\ / \\\n",
" 22 40 25 2\n",
"\n",
"Bubble up 2: Swap 2 and 15\n",
"\n",
" _5_\n",
" / \\\n",
" 20 2\n",
" / \\ / \\\n",
" 22 40 25 15\n",
"\n",
"Bubble up 2: Swap 2 and 5\n",
"\n",
" _2_\n",
" / \\\n",
" 20 5\n",
" / \\ / \\\n",
" 22 40 25 15\n",
"\n",
"\n",
"We'll use an array to represent the tree, here are the indices:\n",
"\n",
"\n",
" _0_\n",
" / \\\n",
" 1 2\n",
" / \\ / \\\n",
" 3 4 5 6\n",
"\n",
"\n",
"To get to a parent, we take (index - 1) // 2. \n",
"\n",
"For example, the parent of index 6 is (6 - 1) // 2 = 2.\n",
"\n",
"Complexity:\n",
"* Time: O(lg(n))\n",
"* Space: O(lg(n)) for the recursion depth (tree height), or O(1) if using an iterative approach"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Code"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Overwriting min_heap.py\n"
]
}
],
"source": [
"%%writefile min_heap.py\n",
"from __future__ import division\n",
"\n",
"import sys\n",
"\n",
"\n",
"class MinHeap(object):\n",
"\n",
" def __init__(self):\n",
" self.array = []\n",
"\n",
" def __len__(self):\n",
" return len(self.array)\n",
"\n",
" def extract_min(self):\n",
" if not self.array:\n",
" return None\n",
" if len(self.array) == 1:\n",
" return self.array.pop(0)\n",
" minimum = self.array[0]\n",
" # Move the last element to the root\n",
" self.array[0] = self.array.pop(-1)\n",
" self._bubble_down(index=0)\n",
" return minimum\n",
"\n",
" def peek_min(self):\n",
" return self.array[0] if self.array else None\n",
"\n",
" def insert(self, key):\n",
" if key is None:\n",
" raise TypeError('key cannot be None')\n",
" self.array.append(key)\n",
" self._bubble_up(index=len(self.array) - 1)\n",
"\n",
" def _bubble_up(self, index):\n",
" if index == 0:\n",
" return\n",
" index_parent = (index - 1) // 2\n",
" if self.array[index] < self.array[index_parent]:\n",
" # Swap the indices and recurse\n",
" self.array[index], self.array[index_parent] = \\\n",
" self.array[index_parent], self.array[index]\n",
" self._bubble_up(index_parent)\n",
"\n",
" def _bubble_down(self, index):\n",
" min_child_index = self._find_smaller_child(index)\n",
" if min_child_index == -1:\n",
" return\n",
" if self.array[index] > self.array[min_child_index]:\n",
" # Swap the indices and recurse\n",
" self.array[index], self.array[min_child_index] = \\\n",
" self.array[min_child_index], self.array[index]\n",
" self._bubble_down(min_child_index)\n",
"\n",
" def _find_smaller_child(self, index):\n",
" left_child_index = 2 * index + 1\n",
" right_child_index = 2 * index + 2\n",
" # No right child\n",
" if right_child_index >= len(self.array):\n",
" # No left child\n",
" if left_child_index >= len(self.array):\n",
" return -1\n",
" # Left child only\n",
" else:\n",
" return left_child_index\n",
" else:\n",
" # Compare left and right children\n",
" if self.array[left_child_index] < self.array[right_child_index]:\n",
" return left_child_index\n",
" else:\n",
" return right_child_index"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%run min_heap.py"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Unit Test"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Overwriting test_min_heap.py\n"
]
}
],
"source": [
"%%writefile test_min_heap.py\n",
"from nose.tools import assert_equal\n",
"\n",
"\n",
"class TestMinHeap(object):\n",
"\n",
" def test_min_heap(self):\n",
" heap = MinHeap()\n",
" assert_equal(heap.peek_min(), None)\n",
" assert_equal(heap.extract_min(), None)\n",
" heap.insert(20)\n",
" assert_equal(heap.array[0], 20)\n",
" heap.insert(5)\n",
" assert_equal(heap.array[0], 5)\n",
" assert_equal(heap.array[1], 20)\n",
" heap.insert(15)\n",
" assert_equal(heap.array[0], 5)\n",
" assert_equal(heap.array[1], 20)\n",
" assert_equal(heap.array[2], 15)\n",
" heap.insert(22)\n",
" assert_equal(heap.array[0], 5)\n",
" assert_equal(heap.array[1], 20)\n",
" assert_equal(heap.array[2], 15)\n",
" assert_equal(heap.array[3], 22)\n",
" heap.insert(40)\n",
" assert_equal(heap.array[0], 5)\n",
" assert_equal(heap.array[1], 20)\n",
" assert_equal(heap.array[2], 15)\n",
" assert_equal(heap.array[3], 22)\n",
" assert_equal(heap.array[4], 40)\n",
" heap.insert(3)\n",
" assert_equal(heap.array[0], 3)\n",
" assert_equal(heap.array[1], 20)\n",
" assert_equal(heap.array[2], 5)\n",
" assert_equal(heap.array[3], 22)\n",
" assert_equal(heap.array[4], 40)\n",
" assert_equal(heap.array[5], 15)\n",
" mins = []\n",
" while heap:\n",
" mins.append(heap.extract_min())\n",
" assert_equal(mins, [3, 5, 15, 20, 22, 40])\n",
" print('Success: test_min_heap')\n",
"\n",
" \n",
"def main():\n",
" test = TestMinHeap()\n",
" test.test_min_heap()\n",
"\n",
" \n",
"if __name__ == '__main__':\n",
" main()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Success: test_min_heap\n"
]
}
],
"source": [
"%run -i test_min_heap.py"
]
}
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