算法笔记_什么是数据结构_向量vector

深藏阁楼爱情的钟 2023-06-30 02:57 27阅读 0赞

P34 [https://www.bilibili.com/video/av75509584?p=34][https_www.bilibili.com_video_av75509584_p_34]

# 基本概念 #

## 抽象数据类型(ADT) vs 数据结构 ##

汽车使用手册，在某种意义上，就是一种ADT，用户只需要关系如何使用它，不必关系内部的实现，内部的实现由生产者决定。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70][]

算法中，将向量的元素对应的索引，称之为“**秩**”。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 1][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 2][]

**disordered():** 返回**逆序对**的个数；若返回0，说明逆序对为0，该序列为有序序列。

**search(X):** 返回不超过X的最大的元素的秩（索引）。若X小于vector中最小的值，则返回-1。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 3][]

# 自定义一个 vector 数据结构 #

向量源自，并且是脱胎于数组。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 4][]

统一的采用模板式的方法，可以使数据结构的定义非常的规范，更重要的是，此后，他们可以互相的融合、组合，便捷的搭建更为复杂的数据结构。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 5][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 6][]

## 存储空间扩容 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 7][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 8][]

### 采用加倍扩容 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 9][]

### 为什么不采用固定值扩容？ ###

因为固定大小扩容的时间成本较大，**平均分摊到每次，**需要的时间复杂度为**O(n)**；

而倍增式的扩容，平均每次需要的时间复杂度为**O(1)**。详细分析如下。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 10][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 11][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 12][]

**总结:** 倍增策略，是在空间上做出了一些牺牲（使用率仍大于50%），来换取时间复杂度上的巨大收益。

# 平均复杂度 vs 分摊复杂度 #

整体考虑连续操作。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 13][]

# 无序向量 #

**比对**：能够比较元素之间是否相等。

## 元素 \[ \] 访问 ##

采用 \[ \] 的方式访问向量元素时，至关重要的方式是 “秩” 的概念；这种方式也称之为 **“循秩访问”**。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 14][]

## 元素插入 ##

插入注意点：移动需要移动的元素时，是**从后往前**依次的移动顺序。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 15][]

## 元素删除 ##

区间删除注意点：移动需要移动的元素时，是**从前往后**依次的移动顺序。原因：避免因为移动区间有重叠部分，导致无意间元素被覆盖。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 16][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 17][]

## 遍历 ##

通过函数对象的方式，对向量元素进行遍历的方法，通用性更强。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 18][]

重载一个结构体的 “ ( ) ” 操作符，使之看上去或者说使用上去，类似于一个函数。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 19][]

# 有序向量 #

**比较：**元素之间谁大谁小。

## 找出向量中的逆序对 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 20][]

## 去除有序向量中的重复元素 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 21][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 22][]

复制元素：直接复制到位，而不是每发现一个重复的，删除一个。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 23][]

## 二分查找 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 24][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 25][]

### 二分查找：减而治之版 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 26][]

**建议小技巧**：以下算法，**在比较时，用的都是 “<” 小于号**，这样更加直观。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 27][]

时间复杂度只有 O( log(n) )。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 28][]

参考： [https://www.bilibili.com/video/av75509584?p=55][https_www.bilibili.com_video_av75509584_p_55]

[https_www.bilibili.com_video_av75509584_p_34]: https://www.bilibili.com/video/av75509584?p=34
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[https_www.bilibili.com_video_av75509584_p_55]: https://www.bilibili.com/video/av75509584?p=55