mirror of
https://github.com/donnemartin/interactive-coding-challenges.git
synced 2024-03-22 13:11:13 +08:00
412 lines
11 KiB
Python
412 lines
11 KiB
Python
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"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)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Solution Notebook"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Problem: Implement a min heap with extract_min and insert methods.\n",
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"\n",
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"* [Constraints](#Constraints)\n",
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"* [Test Cases](#Test-Cases)\n",
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"* [Algorithm](#Algorithm)\n",
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"* [Code](#Code)\n",
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"* [Unit Test](#Unit-Test)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Constraints\n",
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"\n",
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"* Can we assume the inputs are ints?\n",
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" * Yes\n",
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"* Can we assume this fits memory?\n",
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" * Yes"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Test Cases\n",
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"\n",
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"* Extract min of an empty tree\n",
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"* Extract min general case\n",
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"* Insert into an empty tree\n",
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"* Insert general case (left and right insert)\n",
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"\n",
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"<pre>\n",
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" _5_\n",
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" / \\\n",
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" 20 15\n",
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" / \\ / \\\n",
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" 22 40 25\n",
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" \n",
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"extract_min(): 5\n",
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"\n",
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" _15_\n",
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" / \\\n",
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" 20 25\n",
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" / \\ / \\\n",
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" 22 40 \n",
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"\n",
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"insert(2):\n",
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"\n",
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" _2_\n",
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" / \\\n",
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" 20 5\n",
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" / \\ / \\\n",
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" 22 40 25 15\n",
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"</pre>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Algorithm\n",
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"\n",
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"A heap is a complete binary tree where each node is smaller than its children.\n",
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"\n",
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"### extract_min\n",
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"\n",
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"<pre>\n",
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" _5_\n",
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" / \\\n",
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" 20 15\n",
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" / \\ / \\\n",
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" 22 40 25\n",
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"\n",
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"Save the root as the value to be returned: 5\n",
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"Move the right most element to the root: 25\n",
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"\n",
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" _25_\n",
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" / \\\n",
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" 20 15\n",
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" / \\ / \\\n",
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" 22 40 \n",
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"\n",
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"Bubble down 25: Swap 25 and 15 (the smaller child)\n",
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"\n",
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" _15_\n",
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" / \\\n",
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" 20 25\n",
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" / \\ / \\\n",
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" 22 40 \n",
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"\n",
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"Return 5\n",
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"</pre>\n",
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"\n",
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"We'll use an array to represent the tree, here are the indices:\n",
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"\n",
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"<pre>\n",
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" _0_\n",
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" / \\\n",
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" 1 2\n",
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" / \\ / \\\n",
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" 3 4 \n",
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"</pre>\n",
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"\n",
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"To get to a child, we take 2 * index + 1 (left child) or 2 * index + 2 (right child).\n",
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"\n",
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"For example, the right child of index 1 is 2 * 1 + 2 = 4.\n",
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"\n",
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"Complexity:\n",
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"* Time: O(lg(n))\n",
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"* Space: O(lg(n)) for the recursion depth (tree height), or O(1) if using an iterative approach\n",
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"\n",
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"### insert\n",
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"\n",
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"<pre>\n",
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" _5_\n",
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" / \\\n",
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" 20 15\n",
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" / \\ / \\\n",
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" 22 40 25\n",
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"\n",
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"insert(2):\n",
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"Insert at the right-most spot to maintain the heap property.\n",
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"\n",
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" _5_\n",
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" / \\\n",
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" 20 15\n",
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" / \\ / \\\n",
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" 22 40 25 2\n",
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"\n",
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"Bubble up 2: Swap 2 and 15\n",
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"\n",
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" _5_\n",
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" / \\\n",
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" 20 2\n",
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" / \\ / \\\n",
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" 22 40 25 15\n",
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"\n",
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"Bubble up 2: Swap 2 and 5\n",
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"\n",
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" _2_\n",
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" / \\\n",
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" 20 5\n",
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" / \\ / \\\n",
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" 22 40 25 15\n",
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"</pre>\n",
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"\n",
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"We'll use an array to represent the tree, here are the indices:\n",
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"\n",
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"<pre>\n",
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" _0_\n",
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" / \\\n",
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" 1 2\n",
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" / \\ / \\\n",
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" 3 4 5 6\n",
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"</pre>\n",
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"\n",
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"To get to a parent, we take (index - 1) // 2. \n",
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"\n",
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"For example, the parent of index 6 is (6 - 1) // 2 = 2.\n",
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"\n",
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"Complexity:\n",
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"* Time: O(lg(n))\n",
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"* Space: O(lg(n)) for the recursion depth (tree height), or O(1) if using an iterative approach"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Code"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Overwriting min_heap.py\n"
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]
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}
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],
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"source": [
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"%%writefile min_heap.py\n",
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"from __future__ import division\n",
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"\n",
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"import sys\n",
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"\n",
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"\n",
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"class MinHeap(object):\n",
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"\n",
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" def __init__(self):\n",
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" self.array = []\n",
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"\n",
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" def __len__(self):\n",
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" return len(self.array)\n",
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"\n",
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" def extract_min(self):\n",
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" if not self.array:\n",
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" return None\n",
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" if len(self.array) == 1:\n",
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" return self.array.pop(0)\n",
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" minimum = self.array[0]\n",
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" # Move the last element to the root\n",
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" self.array[0] = self.array.pop(-1)\n",
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" self._bubble_down(index=0)\n",
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" return minimum\n",
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"\n",
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" def peek_min(self):\n",
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" return self.array[0] if self.array else None\n",
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"\n",
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" def insert(self, key):\n",
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" if key is None:\n",
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" raise TypeError('key cannot be None')\n",
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" self.array.append(key)\n",
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" self._bubble_up(index=len(self.array) - 1)\n",
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"\n",
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" def _bubble_up(self, index):\n",
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" if index == 0:\n",
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" return\n",
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" index_parent = (index - 1) // 2\n",
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" if self.array[index] < self.array[index_parent]:\n",
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" # Swap the indices and recurse\n",
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" self.array[index], self.array[index_parent] = \\\n",
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" self.array[index_parent], self.array[index]\n",
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" self._bubble_up(index_parent)\n",
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"\n",
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" def _bubble_down(self, index):\n",
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" min_child_index = self._find_smaller_child(index)\n",
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" if min_child_index == -1:\n",
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" return\n",
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" if self.array[index] > self.array[min_child_index]:\n",
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" # Swap the indices and recurse\n",
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" self.array[index], self.array[min_child_index] = \\\n",
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" self.array[min_child_index], self.array[index]\n",
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" self._bubble_down(min_child_index)\n",
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"\n",
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" def _find_smaller_child(self, index):\n",
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" left_child_index = 2 * index + 1\n",
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" right_child_index = 2 * index + 2\n",
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" # No right child\n",
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" if right_child_index >= len(self.array):\n",
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" # No left child\n",
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" if left_child_index >= len(self.array):\n",
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" return -1\n",
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" # Left child only\n",
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" else:\n",
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" return left_child_index\n",
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" else:\n",
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" # Compare left and right children\n",
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" if self.array[left_child_index] < self.array[right_child_index]:\n",
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" return left_child_index\n",
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" else:\n",
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" return right_child_index"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%run min_heap.py"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Unit Test"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Overwriting test_min_heap.py\n"
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]
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}
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],
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"source": [
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"%%writefile test_min_heap.py\n",
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"from nose.tools import assert_equal\n",
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"\n",
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"\n",
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"class TestMinHeap(object):\n",
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"\n",
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" def test_min_heap(self):\n",
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" heap = MinHeap()\n",
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" assert_equal(heap.peek_min(), None)\n",
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" assert_equal(heap.extract_min(), None)\n",
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" heap.insert(20)\n",
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" assert_equal(heap.array[0], 20)\n",
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" heap.insert(5)\n",
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" assert_equal(heap.array[0], 5)\n",
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" assert_equal(heap.array[1], 20)\n",
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" heap.insert(15)\n",
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" assert_equal(heap.array[0], 5)\n",
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" assert_equal(heap.array[1], 20)\n",
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" assert_equal(heap.array[2], 15)\n",
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" heap.insert(22)\n",
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" assert_equal(heap.array[0], 5)\n",
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" assert_equal(heap.array[1], 20)\n",
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" assert_equal(heap.array[2], 15)\n",
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" assert_equal(heap.array[3], 22)\n",
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" heap.insert(40)\n",
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" assert_equal(heap.array[0], 5)\n",
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" assert_equal(heap.array[1], 20)\n",
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" assert_equal(heap.array[2], 15)\n",
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" assert_equal(heap.array[3], 22)\n",
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" assert_equal(heap.array[4], 40)\n",
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" heap.insert(3)\n",
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" assert_equal(heap.array[0], 3)\n",
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" assert_equal(heap.array[1], 20)\n",
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" assert_equal(heap.array[2], 5)\n",
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" assert_equal(heap.array[3], 22)\n",
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" assert_equal(heap.array[4], 40)\n",
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" assert_equal(heap.array[5], 15)\n",
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" mins = []\n",
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" while heap:\n",
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" mins.append(heap.extract_min())\n",
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" assert_equal(mins, [3, 5, 15, 20, 22, 40])\n",
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" print('Success: test_min_heap')\n",
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"\n",
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" \n",
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"def main():\n",
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" test = TestMinHeap()\n",
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" test.test_min_heap()\n",
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"\n",
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" \n",
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"if __name__ == '__main__':\n",
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" main()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Success: test_min_heap\n"
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]
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}
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],
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"source": [
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"%run -i test_min_heap.py"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.4.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 0
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}
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