《算法图解》读书笔记

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第一章 算法简介

1.1 引言

算法是一组完成任务的指令。任何代码片段都可视为算法

1.2 二分查找

二分查找是一种算法，其输入是一个有序的元素列表（必须有序）。如果要查找的元素在列表中，返回其所在位置，否则返回null。

试想如果你要从电话簿中查找名字开头为K的联系人，是会从头开始查找，还是从中间开始查找。

二分查找原理：让你从1——100之间猜一个数字，你每次猜，我都会说‘大了’还是‘小了’。简单查找从1开始，如果数字是99，需要查找99次；二分查找，从50（取得初始中间下标为=Math.floor（0+99）/2=54）开始，说‘小了’，排除1——50，又猜75，‘小了’，又排除51——75，也就是说，每次都能排除一半的数据。一般而言，对于包含n个元素的列表，用二分查找最多需要log₂n步。

对数运算是幂运算的逆运算，log₂4相当于问多少个2相乘等于4，因此log₂4=2

1.3 大O表示法

大O表示法是一种特殊的表示法，指出了算法的速度有多快。就是说要知道运行时间如何随列表增长而增加。大o表示法让你能够比较操作数，它指出了算法运行时间的增速

例子：为检查长度为n的列表，二分查找的运行时间用大O表示法为O(log₂n)

一些常见的大O运行时间： O(logn),也叫对数时间，这样的算法包括二分查找 O(n)，也叫线性时间，这样的算法包括简单查找 O(nlogn)，这样的算法包括快速排序——一种速度较快的排序算法 O(n²)，这样的算法包括选择排序——一种速度较慢的排序算法 O(n!)，这样的算法包括旅行商问题的解决方案，一种非常慢的算法

启示：

  • 算法的速度指的并非是时间，而是操作数的增速
  • 谈论算法说的是随着输入的增加，其运行时间将以什么样的速度增加
  • 算法的运行时间用大O表示法表示
  • O(logn)比O(n)快，元素越多差距越明显

第二章 选择排序

2.2 数组和链表

数组：所有数据在内存中都是相连的，一旦这一块连续位置存储不下后续数据，又得请求内存重新开辟一块能够存储数组全部数据的连续内存位置。在同一个数组中，所有元素类型必须相同。 链表：链表中的元素可存储在内存中的任何地方，链表中的每一个元素都存储了下一个元素的地址，从而使一系列的随机地址串在一起。

数组和链表的优缺点： 数组:缺点是新增元素时，如果这一块连续内存不够用，得移动元素到另一块可以放在全部数据的连续内存，新增元素速度慢；优点是支持随机访问，效率高，只要知道起始位置，做个简单的运算就能知道要查找的元素位置； 链表：缺点是要读取链表最后一个元素时，不能直接读取，只能从第一个元素开始，读到最后一个，需要读取所有元素，链表速度高，需要跳跃读取，链表效率低；内存不连续，新增元素不用移动，新增效率高；

2.3 选择排序

例如要将一组含有n个数据的列表进行从大到小排序，可以遍历列表，每次遍历将最大的数据添加到新列表，需要遍历n次，其速度为O(n²)。

第三章 递归

3.1 递归

递归函数调用自己，每个递归函数都有两部分：基线条件和递归条件。满足递归条件则调用自己，满足基线条件则不再继续往下递归。

3.3 栈

先进后出的数据结构，在计算机内部执行中称为调用栈。栈包含未完成的函数调用，后调用的函数压在栈顶，栈顶函数调用结束弹出。如果栈很长，将占用大量内存，并可能导致栈溢出。

第四章 快速排序

4.1 分而治之

找出最简单的基线条件 确定如何缩小问题的规模，使其符合基线条件

欧几里得算法：两个整数的最大公约数等于较小的那个数与两数相除余数的最大公约数

4.2 快速排序

归纳证明：分为基线条件和归纳条件 基线条件中，证明算法对空数组或者包含一个元素的数组有用 归纳条件中，证明如果这种算法对一个元素数组有用，那么对两个元素的数组也有用，如果对两个元素的数组有用，那么对三个元素的数组也有用，以此类推。

快速排序原理： 选一个基准值，将小于基准值的放入一个数组，大于基准值的放入另一个数组，再对这两个数组做同样重复的操作，直到两边数组都为空，将数组拼接起来就得到一个排序后的数组

快速排序的速度取决于选择的基准值。在最糟糕的情况下，其速度为O(n²)，和选择排序一样；平均情况下，其速度为O(nlogn)；最佳情况下，总是将中间大的元素作为基准值，速度为O(nlogn)；最佳情况和平均情况一样；合并排序速度也为O(nlogn)，大O表达式法的常量在比较快速排序和合并排序时作用很大

本人实践：

// 对以下数组进行排序
const array = 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// 运行时间 45~75ms
const getSort = (arr) => {
    if (!arr.length) return []
    let arrPrev = []
    const arrEqual = []
    let arrNext = []
    for (let i = 0; i < arr.length; i++) {
        if (arr[i] < arr[0]) {
            arrPrev.push(arr[i])
        } else if (arr[i] === arr[0]) {
            arrEqual.push(arr[i])
        } else {
            arrNext.push(arr[i])
        }
    }
    arrPrev = getSort(arrPrev)
    arrNext = getSort(arrNext)
    return [...arrPrev, ...arrEqual, ...arrNext]
}
const arrSort = getSort(nums)
console.log(arrSort)

第五章 散列表

5.1 散列函数

散列表也称为散列映射、映射、字典和关联数组，是一种功能强大的数据结构，由键和值组成，散列表将键映射到值，其操作速度快。散列表通过散列函数获取数据存储位置，散列表也使用数组来存储数据。

  • 散列表被用于大海捞针式的查找
  • 模拟映射关系
  • 防止重复
  • 缓存/记住数据

image.png

使用散列表，避开最糟糕的情况至关重要，因此需要：

  • 较低的填装因子
  • 良好的散列函数

第六章 广度优先搜索

6.1 图简介

一个例子，你要从双子峰前往金门大桥，希望换乘最少。需要两个步骤： 1）使用图来建立问题模型 2）使用广度优先搜索解决问题

6.2 图是什么

图模拟一组连接，图由节点和边组成。一个节点可能与众多其他节点相连，这些节点被称为邻居。

6.3 广度优先搜索

广度优先搜索是一种用于图的查找算法，可帮助回答两类问题： 第一类：从节点A出发，由前往节点B的路径吗 第二类：从节点A出发，前往节点B的哪条路径最短

队列：队列类似于栈，不能随机访问队列中的元素，队列只有两个操作：入队和出队，遵循先进先出原则。

6.4 实现图

可以使用散列表来实现图，将节点映射到所有邻居

有向图：关系是单向的；无向图：没有箭头,直接相连的互为邻居

image.png

image.png

算法原理：

  1. 创建一个队列，用于存储要检查的节点
  2. 从队头弹出一个节点
  3. 检查是否符合要求
  4. 符合结束查找；不符合，将这个节点的所有邻居加入队列
  5. 回第2步
  6. 如果队列为空，说明没有符合要求的节点 算法将执行直到：找到符合要求的节点或者队列为空

第七章 狄克斯特拉算法

7.1 使用狄克斯特拉算法

  1. 找出最便宜的节点，即可在最短时间内到达的节点
  2. 对于该节点的邻居，检查是否有前往它们更短的路径，如果有，就更新其开销
  3. 重复这个过程，直到对图中每个节点都这样做了
  4. 计算最终路径

广度优先搜索找的是最少段数路径 狄克斯特拉算法对每条边加了权重，找的是权重最小的路径 带权重的图为加权图，使用狄克斯特拉算法计算；不带权重的图为非加权图，使用广度优先搜索算法计算 狄克斯特拉算法只适用于有向无环图 对节点记录其最少开销是的父节点，以便回溯 如果有负权边，会导致狄克斯特拉算法不管用 负权边请用贝尔曼-福德算法

本人实践

/**
 * 获取每个节点到其所有下一个邻居的开销的散列表
 * @param {*} arr
 * @returns
 */
const getPriceMap = (arr) => {
  const priceMap = {};
  for (let price of arr) {
    let key = price.start;
    if (priceMap[key]) {
      priceMap[key][price.end] = price.price;
    } else {
      let lin = {};
      lin[price.end] = price.price;
      priceMap[key] = lin;
    }
  }
  return priceMap;
};

/**
 * 获取从起点到终点的最小开销
 * @param {*} map
 * @param {*} path
 */
const getAmount = (map, path) => {
  // 起点到每个点的最小开销
  const amounts = {};
  // 待处理的节点
  const queen = [];
  // 起点
  let start = path.start;
  // 终点
  let end = path.end;
  queen.push(start);
  // 起点的初始开销为0
  amounts[start] = 0;
  // 处理过的节点
  const done = [];
  // 当所有路径处理完成
  while (queen.length) {
    let cur = queen[0];
    done.push(cur);
    let curPrice = map[cur];
    if (!curPrice) {
      queen.shift(0);
      continue;
    }
    for (let key of Object.keys(curPrice)) {
      // 将当前节点的所有下一个邻居压入待处理队列
      if (!done.includes(key)) {
        queen.push(key);
      }
      let newAmount = amounts[cur] + curPrice[key];
      if (amounts[key]) {
        // 取最小开销
        amounts[key] = Math.min(newAmount, amounts[key]);
      } else {
        amounts[cur] + curPrice[key];
        amounts[key] = newAmount;
      }
    }
    // 处理过的节点弹出待处理队列
    queen.shift(0);
  }
  console.log(amounts);
  // 如果没有终点开销数据，意味着从起点开始没有达到终点的路劲
  if (amounts[end]) {
    console.log(amounts[end]);
  } else {
    console.log(-1);
  }
};

const fn = (arg1, arg2) => {
  const priceMap = getPriceMap(arg1);
  console.log(priceMap);
  // {
  //   A: { C: 10, D: 5 }, A可以直接到达C和D，A到C花费10，A到D花费5
  //   B: { D: 3 },
  //   C: { D: 2, E: 5 },
  //   D: { E: 7, F: 8 },
  //   E: { F: 4 }
  // }
  for (let path of arg2) {
    getAmount(priceMap, path);
  }
};
/**
 * 每段路开销列表
 * { start: "A", end: "C", price: 10 }
 * 表示从A到C需要花费10
 * { start: "B", end: "D", price: 3 },
 * 表示从B到D需要华为3
 */
let arr1 = [
  { start: "A", end: "C", price: 1 },
  { start: "A", end: "D", price: 5 },
  { start: "B", end: "D", price: 3 },
  { start: "C", end: "D", price: 5 },
  { start: "C", end: "E", price: 1 },
  { start: "D", end: "E", price: 7 },
  { start: "E", end: "D", price: 2 },
  { start: "D", end: "F", price: 8 },
  { start: "E", end: "F", price: 4 },
];
/**
 * 起始地点和目的地
 * { start: "A", end: "E" }
 * 表示求A到E最小开销
 */
let arr2 = [
  { start: "A", end: "D" },
  { start: "B", end: "F" },
];
fn(arr1, arr2);

第八章 贪婪算法

8.1 教室调度问题

每一步选择局部最优解，最后的到全局最优解 贪婪算法寻找局部最优解，企图以这种方式获得全局最优解

8.2 背包问题

贪婪算法并不总是能得到全局最优解，但是有时候，你只需要找到一个能够大致解决问题的算法，得到结果又与正确结果相当接近

例子：一个小偷背一个包偷东西，包只能装10kg东西，他要拿背包装得下的最贵的东西，会先从最贵的且能装得下的东西开始。虽然结果不一定准确，但确实简单快速且接近正确答案的解

8.3 集合覆盖问题

判断近似算法优劣标准： 速度有多快 得到的近似解与最优解的接近程度

近似算法解广播台覆盖所有州的问题：

  1. 选出覆盖最多未覆盖州的广播台
  2. 重复第一步，直到覆盖所有州

8.4 NP完全问题

旅行商问题和集合覆盖问题都有一些共同之处：你需要计算所有的解，并从中选出最小/最短的那个。这两个问题都属于NP完全问题，目前还没有编写出壳快速解决这些问题的算法，最佳的做法是使用近似算法。贪婪算法易于实现，运行速度快，是不错的近似算法

如何识别NP完全问题：

  • 元素较少时算法运行速度非常快，随着元素的增加，速度会变得非常慢
  • 涉及“所有组合”的问题通常是NP完全问题
  • 不能将问题分为小问题，必须考虑所有可能情况
  • 如果问题涉及序列且难以解决，可能是NP完全问题
  • 如果问题涉及集合，如广播台集合且难以解决
  • 如果问题可以转换为旅行商问题或集合覆盖问题，那肯定是NP完全问题

第九章 动态规划

动态规划将问题分为小问题，先解决小问题，并用小问题答案来解决大问题。但仅当子问题都是离散的，即不依赖于其它子问题时，动态规划才有用 动态规划解决背包问题： 在这个问题中，行表示可偷商品，包括上一行可偷商品，列表示背包容量。第一行第一列格子表示，背包容量为1磅并且只能拿吉他时，最大价值；第一行第二列为背包容量为2磅并且只能偷吉他时，最大价值；第二行第一列表示背包容量为1磅，并且只能偷吉他和音响时，最大价值；以此类推 image.png

新增商品iphone： image.png

9.3 最长公共子串

image.png

最长公共子序列

image.png

动态规划实际应用：

  • 生物学家根据最长公共序列来确定DNA链的相似性
  • Git diff比较两个文件的差异
  • 编辑距离算法，拼写检查和判断盗版
  • word断字

9.4 小结

  • 需要在给定约束条件下优化某种指标时，动态规划很有用
  • 问题可分解为离散子问题时，可使用动态规划来解决
  • 每种动态规划解决方案都涉及网格
  • 单元格中的值通常就是你要优化的值
  • 每个单元格都是一个子问题，因此你需要考虑如何将问题分解为子问题
  • 没有放之四海而皆准的计算动态规划解决方案的公式

第十章 K最近邻算法

你将使用KNN来做两项基本工作：分类和回归

  • 分类就是编组；
  • 回归就是预测结果
  • 特征抽取意味着将物品转化为一系列可比较的数字
  • 能否挑选合适的特征事关KNN算法的成败

10.3 机器学习简介

机器学习的例子：

OCR：光学字符识别（optical character recognition），这意味你可拍摄印刷页面的照片，计算机将自动识别出其中的文字。 一般而言，OCR算法提取线段、点和曲线等特征 OCR的第一步是查看大量的数字图像并提取特征，这被称为训练 KNN的理念也可以用于语音识别和人脸识别

垃圾邮件过滤器-朴素贝叶斯分类器

第十一章 接下来如何做

11.1 树

二叉查找树：B树，红黑树，堆，伸展树

11.2 反向索引

11.3 傅里叶变换

11.4 并行算法

11.5 MapReduce

11.6 布隆过滤器和HyperLogLog

...

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