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Hello 算法
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3.1. 数据与内存
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0. 写在前面
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1. 引言
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1.1. 算法无处不在
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1.2. 算法是什么
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1.3. 小结
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2. 复杂度分析
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2. 复杂度分析
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< label class = "md-nav__link" for = "__nav_4" id = "__nav_4_label" tabindex = "0" >
3. 数据结构简介
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_4_label" aria-expanded = "true" >
< label class = "md-nav__title" for = "__nav_4" >
< span class = "md-nav__icon md-icon" > < / span >
3. 数据结构简介
< / label >
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< li class = "md-nav__item md-nav__item--active" >
< input class = "md-nav__toggle md-toggle" type = "checkbox" id = "__toc" >
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3.1. 数据与内存
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< a href = "./" class = "md-nav__link md-nav__link--active" >
3.1. 数据与内存
< / a >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#311" class = "md-nav__link" >
3.1.1. 基本数据类型
< / a >
< nav class = "md-nav" aria-label = "3.1.1. 基本数据类型" >
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#_1" class = "md-nav__link" >
整数表示方式
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
浮点数表示方式 *
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_3" class = "md-nav__link" >
基本数据类型与数据结构的关系
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item" >
< a href = "#312" class = "md-nav__link" >
3.1.2. 计算机内存
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item" >
< a href = "../classification_of_data_structure/" class = "md-nav__link" >
3.2. 数据结构分类
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../summary/" class = "md-nav__link" >
3.3. 小结
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
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< label class = "md-nav__link" for = "__nav_5" id = "__nav_5_label" tabindex = "0" >
4. 数组与链表
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_5_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_5" >
< span class = "md-nav__icon md-icon" > < / span >
4. 数组与链表
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/array/" class = "md-nav__link" >
4.1. 数组
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/linked_list/" class = "md-nav__link" >
4.2. 链表
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
4.3. 列表
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/summary/" class = "md-nav__link" >
4.4. 小结
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
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< label class = "md-nav__link" for = "__nav_6" id = "__nav_6_label" tabindex = "0" >
5. 栈与队列
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_6_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_6" >
< span class = "md-nav__icon md-icon" > < / span >
5. 栈与队列
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
5.1. 栈
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
5.2. 队列
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
5.3. 双向队列
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/summary/" class = "md-nav__link" >
5.4. 小结
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_7" >
< label class = "md-nav__link" for = "__nav_7" id = "__nav_7_label" tabindex = "0" >
6. 散列表
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_7_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_7" >
< span class = "md-nav__icon md-icon" > < / span >
6. 散列表
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_map/" class = "md-nav__link" >
6.1. 哈希表
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
6.2. 哈希冲突处理
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_hashing/summary/" class = "md-nav__link" >
6.3. 小结
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_8" >
< label class = "md-nav__link" for = "__nav_8" id = "__nav_8_label" tabindex = "0" >
7. 树
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
7. 树
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree/" class = "md-nav__link" >
7.1. 二叉树
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
7.2. 二叉树遍历
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
7.3. 二叉搜索树
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
7.4. AVL 树 *
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
7.5. 小结
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
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< label class = "md-nav__link" for = "__nav_9" id = "__nav_9_label" tabindex = "0" >
8. 堆
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
8.1. 堆
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
8.2. 建堆操作 *
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
8.3. 小结
< / a >
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< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
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< label class = "md-nav__link" for = "__nav_10" id = "__nav_10_label" tabindex = "0" >
9. 图
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_10" >
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9. 图
< / label >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
9.1. 图
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
9.2. 图基础操作
< / a >
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< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
9.3. 图的遍历（ ）
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
9.4. 小结
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< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
< label class = "md-nav__link" for = "__nav_11" id = "__nav_11_label" tabindex = "0" >
10. 查找算法
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
10. 查找算法
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/linear_search/" class = "md-nav__link" >
10.1. 线性查找
< / a >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
10.2. 二分查找
< / a >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/hashing_search/" class = "md-nav__link" >
10.3. 哈希查找
< / a >
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< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
10.4. 小结
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< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
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< label class = "md-nav__link" for = "__nav_12" id = "__nav_12_label" tabindex = "0" >
11. 排序算法
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< / label >
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< label class = "md-nav__title" for = "__nav_12" >
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11. 排序算法
< / label >
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< a href = "../../chapter_sorting/intro_to_sort/" class = "md-nav__link" >
11.1. 排序简介
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
11.2. 冒泡排序
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
11.3. 插入排序
< / a >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
11.4. 快速排序
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
11.5. 归并排序
< / a >
< / li >
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< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
11.6. 计数排序（ ）
< / a >
< / li >
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< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
11.7. 小结
< / a >
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< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
< label class = "md-nav__link" for = "__nav_13" id = "__nav_13_label" tabindex = "0" >
12. 附录
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_13_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_13" >
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12. 附录
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< a href = "../../chapter_appendix/installation/" class = "md-nav__link" >
12.1. 编程环境安装
< / a >
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< a href = "../../chapter_appendix/contribution/" class = "md-nav__link" >
12.2. 一起参与创作
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< / ul >
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< / li >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_14" >
< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_reference/" > 参考文献< / a >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_14_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_14" >
< span class = "md-nav__icon md-icon" > < / span >
参考文献
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< / li >
< / ul >
< / nav >
< / div >
< / div >
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< div class = "md-sidebar md-sidebar--secondary" data-md-component = "sidebar" data-md-type = "toc" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#311" class = "md-nav__link" >
3.1.1. 基本数据类型
< / a >
< nav class = "md-nav" aria-label = "3.1.1. 基本数据类型" >
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#_1" class = "md-nav__link" >
整数表示方式
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
浮点数表示方式 *
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_3" class = "md-nav__link" >
基本数据类型与数据结构的关系
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< li class = "md-nav__item" >
< a href = "#312" class = "md-nav__link" >
3.1.2. 计算机内存
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< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
< / a >
< h1 id = "31" > 3.1. 数据与内存< a class = "headerlink" href = "#31" title = "Permanent link" > ¶ < / a > < / h1 >
< h2 id = "311" > 3.1.1. 基本数据类型< a class = "headerlink" href = "#311" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 谈到计算机中的数据， < / p >
< p > < strong > 「基本数据类型」是 CPU 可以直接进行运算的类型，在算法中直接被使用< / strong > 。< / p >
< ul >
< li > 「整数」根据不同的长度分为 byte, short, int, long ，根据算法需求选用，即在满足取值范围的情况下尽量减小内存空间占用；< / li >
< li > 「浮点数」代表小数，根据长度分为 float, double ，同样根据算法的实际需求选用；< / li >
< li > 「字符」在计算机中是以字符集的形式保存的， ； < / li >
< li > 「布尔」代表逻辑中的“是”与“否”，其占用空间需根据编程语言确定，通常为 1 byte 或 1 bit ； < / li >
< / ul >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > 类别< / th >
< th > 符号< / th >
< th > 占用空间< / th >
< th > 取值范围< / th >
< th > 默认值< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > 整数< / td >
< td > byte< / td >
< td > 1 byte< / td >
< td > < span class = "arithmatex" > \(-2^7\)< / span > ~ < span class = "arithmatex" > \(2^7 - 1\)< / span > ( < span class = "arithmatex" > \(-128\)< / span > ~ < span class = "arithmatex" > \(127\)< / span > )< / td >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< / tr >
< tr >
< td > < / td >
< td > short< / td >
< td > 2 bytes< / td >
< td > < span class = "arithmatex" > \(-2^{15}\)< / span > ~ < span class = "arithmatex" > \(2^{15} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< / tr >
< tr >
< td > < / td >
< td > < strong > int< / strong > < / td >
< td > 4 bytes< / td >
< td > < span class = "arithmatex" > \(-2^{31}\)< / span > ~ < span class = "arithmatex" > \(2^{31} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< / tr >
< tr >
< td > < / td >
< td > long< / td >
< td > 8 bytes< / td >
< td > < span class = "arithmatex" > \(-2^{63}\)< / span > ~ < span class = "arithmatex" > \(2^{63} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< / tr >
< tr >
< td > 浮点数< / td >
< td > < strong > float< / strong > < / td >
< td > 4 bytes< / td >
< td > < span class = "arithmatex" > \(-3.4 \times 10^{38}\)< / span > ~ < span class = "arithmatex" > \(3.4 \times 10^{38}\)< / span > < / td >
< td > < span class = "arithmatex" > \(0.0\)< / span > f< / td >
< / tr >
< tr >
< td > < / td >
< td > double< / td >
< td > 8 bytes< / td >
< td > < span class = "arithmatex" > \(-1.7 \times 10^{308}\)< / span > ~ < span class = "arithmatex" > \(1.7 \times 10^{308}\)< / span > < / td >
< td > < span class = "arithmatex" > \(0.0\)< / span > < / td >
< / tr >
< tr >
< td > 字符< / td >
< td > < strong > char< / strong > < / td >
< td > 2 bytes / 1 byte< / td >
< td > < span class = "arithmatex" > \(0\)< / span > ~ < span class = "arithmatex" > \(2^{16} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< / tr >
< tr >
< td > 布尔< / td >
< td > < strong > bool< / strong > < / td >
< td > 1 byte / 1 bit< / td >
< td > < span class = "arithmatex" > \(\text{true}\)< / span > 或 < span class = "arithmatex" > \(\text{false}\)< / span > < / td >
< td > < span class = "arithmatex" > \(\text{false}\)< / span > < / td >
< / tr >
< / tbody >
< / table >
< / div >
< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
< p > 以上表格中，加粗项在「算法题」中最为常用。此表格无需硬背，大致理解即可，需要时可以通过查表来回忆。< / p >
< / div >
< h3 id = "_1" > 整数表示方式< a class = "headerlink" href = "#_1" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 整数的取值范围取决于变量使用的内存长度，即字节（或比特）数。在计算机中， 1 字节 (byte) = 8 比特 (bit) ， < / p >
< ol >
< li > 整数类型 int 占用 4 bytes = 32 bits ，因此可以表示 < span class = "arithmatex" > \(2^{32}\)< / span > 个不同的数字；< / li >
< li > 将最高位看作符号位，< span class = "arithmatex" > \(0\)< / span > 代表正数，< span class = "arithmatex" > \(1\)< / span > 代表负数，从而可以表示 < span class = "arithmatex" > \(2^{31}\)< / span > 个正数和 < span class = "arithmatex" > \(2^{31}\)< / span > 个负数；< / li >
< li > 当所有 bits 为 0 时代表数字 < span class = "arithmatex" > \(0\)< / span > ，从零开始增大，可得最大正数为 < span class = "arithmatex" > \(2^{31} - 1\)< / span > ； < / li >
< li > 剩余 < span class = "arithmatex" > \(2^{31}\)< / span > 个数字全部用来表示负数，因此最小负数为 < span class = "arithmatex" > \(-2^{31}\)< / span > ；具体细节涉及到到“源码、反码、补码”知识，有兴趣的同学可以查阅学习； < / li >
< / ol >
< p > 其它整数类型 byte, short, long 取值范围的计算方法与 int 类似，在此不再赘述。< / p >
< h3 id = "_2" > 浮点数表示方式 *< a class = "headerlink" href = "#_2" title = "Permanent link" > ¶ < / a > < / h3 >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 在本书中，标题后的 < code > *< / code > 符号代表选读章节，如果你觉得理解困难，建议先跳过，等学完必读章节后续再单独攻克。 < / p >
< / div >
< p > 细心的你可能会疑惑： int 和 float 长度相同，都是 4 bytes ， < strong > 但为什么 float 的取值范围远大于 int< / strong > ？按说 float 需要表示小数，取值范围应该变小才对。< / p >
< p > 其实，这是因为浮点数 float 采用了不同的表示方式。IEEE 754 标准规定， < / p >
< ul >
< li > 符号位 < span class = "arithmatex" > \(\mathrm{S}\)< / span > ：占 1 bit ； < / li >
< li > 指数位 < span class = "arithmatex" > \(\mathrm{E}\)< / span > ：占 8 bits ； < / li >
< li > 分数位 < span class = "arithmatex" > \(\mathrm{N}\)< / span > ：占 24 bits ，其中 23 位显式存储；< / li >
< / ul >
< p > 设 32-bit 二进制数的第 < span class = "arithmatex" > \(i\)< / span > 位为 < span class = "arithmatex" > \(b_i\)< / span > ，则 float 值的计算方法定义为< / p >
< div class = "arithmatex" > \[
\text { val } = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
\]< / div >
< p > 转化到十进制下的计算公式为< / p >
< div class = "arithmatex" > \[
\text { val }=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
\]< / div >
< p > 其中各项的取值范围为< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\mathrm{S} \in & \{ 0, 1\} , \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]< / div >
< p > < img alt = "IEEE 754 标准下的 float 表示方式" src = "../data_and_memory.assets/ieee_754_float.png" / > < / p >
< p align = "center" > Fig. IEEE 754 标准下的 float 表示方式 < / p >
< p > 以上图为例，< span class = "arithmatex" > \(\mathrm{S} = 0\)< / span > ， < span class = "arithmatex" > \(\mathrm{E} = 124\)< / span > ， < span class = "arithmatex" > \(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)< / span > ，易得< / p >
< div class = "arithmatex" > \[
\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
\]< / div >
< p > 现在我们可以回答开始的问题：< strong > float 的表示方式包含指数位，导致其取值范围远大于 int< / strong > 。根据以上计算， float 可表示的最大正数为 < span class = "arithmatex" > \(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)< / span > ，切换符号位便可得到最小负数。< / p >
< p > < strong > 浮点数 float 虽然拓展了取值范围，但副作用是牺牲了精度< / strong > 。整数类型 int 将全部 32 位用于表示数字，数字是均匀分布的；而由于指数位的存在，浮点数 float 的数值越大，相邻两个数字之间的差值就会趋向越大。< / p >
< p > 进一步地，指数位 < span class = "arithmatex" > \(E = 0\)< / span > 和 < span class = "arithmatex" > \(E = 255\)< / span > 具有特殊含义，< strong > 用于表示零、无穷大、< span class = "arithmatex" > \(\mathrm{NaN}\)< / span > 等< / strong > 。< / p >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > 指数位 E< / th >
< th > 分数位 < span class = "arithmatex" > \(\mathrm{N} = 0\)< / span > < / th >
< th > 分数位 < span class = "arithmatex" > \(\mathrm{N} \ne 0\)< / span > < / th >
< th > 计算公式< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< td > < span class = "arithmatex" > \(\pm 0\)< / span > < / td >
< td > 次正规数< / td >
< td > < span class = "arithmatex" > \((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)< / span > < / td >
< / tr >
< tr >
< td > < span class = "arithmatex" > \(1, 2, \dots, 254\)< / span > < / td >
< td > 正规数< / td >
< td > 正规数< / td >
< td > < span class = "arithmatex" > \((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)< / span > < / td >
< / tr >
< tr >
< td > < span class = "arithmatex" > \(255\)< / span > < / td >
< td > < span class = "arithmatex" > \(\pm \infty\)< / span > < / td >
< td > < span class = "arithmatex" > \(\mathrm{NaN}\)< / span > < / td >
< td > < / td >
< / tr >
< / tbody >
< / table >
< / div >
< p > 特别地，次正规数显著提升了小数精度：< / p >
< ul >
< li > 最小正正规数为 < span class = "arithmatex" > \(2^{-126} \approx 1.18 \times 10^{-38}\)< / span > ； < / li >
< li > 最小正次正规数为 < span class = "arithmatex" > \(2^{-126} \times 2^{-23} \approx 1.4 \times 10^{-45}\)< / span > ； < / li >
< / ul >
< p > 双精度 double 也采用类似 float 的表示方法，在此不再赘述。< / p >
< h3 id = "_3" > 基本数据类型与数据结构的关系< a class = "headerlink" href = "#_3" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 我们知道，< strong > 数据结构是在计算机中组织与存储数据的方式< / strong > ，它的主语是“结构”，而不是“数据”。如果我们想要表示“一排数字”，自然想到使用「数组」数据结构。数组的存储方式可以表示数字的相邻关系、顺序关系，但至于其中存储的是整数 int ，还是小数 float ，或是字符 char ， < strong > 则与所谓的数据的结构无关了< / strong > 。< / p >
< p > 换言之，基本数据类型提供了数据的“内容类型”，而数据结构提供数据的“组织方式”。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:10" > < input checked = "checked" id = "__tabbed_1_1" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_2" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_3" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_4" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_5" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_6" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_7" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_8" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_9" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_10" name = "__tabbed_1" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JavaScript< / label > < label for = "__tabbed_1_6" > TypeScript< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > numbers< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [< / span > < span class = "mi" > 5< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "kt" > float< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > decimals< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > float< / span > < span class = "o" > [< / span > < span class = "mi" > 5< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "kt" > char< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > characters< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > char< / span > < span class = "o" > [< / span > < span class = "mi" > 5< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "kt" > boolean< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > booleans< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > boolean< / span > < span class = "o" > [< / span > < span class = "mi" > 5< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > numbers< / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "kt" > float< / span > < span class = "w" > < / span > < span class = "n" > decimals< / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "kt" > char< / span > < span class = "w" > < / span > < span class = "n" > characters< / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "kt" > bool< / span > < span class = "w" > < / span > < span class = "n" > booleans< / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "sd" > " " " Python 的 list 可以自由存储各种基本数据类型和对象 " " " < / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "nb" > list< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span > < span class = "s1" > ' a' < / span > < span class = "p" > ,< / span > < span class = "kc" > False< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "c1" > // 使用多种「基本数据类型」来初始化「数组」< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "kd" > var< / span > < span class = "w" > < / span > < span class = "nx" > numbers< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ]< / span > < span class = "kt" > int< / span > < span class = "p" > {}< / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "kd" > var< / span > < span class = "w" > < / span > < span class = "nx" > decimals< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ]< / span > < span class = "kt" > float64< / span > < span class = "p" > {}< / span >
< a id = "__codelineno-3-4" name = "__codelineno-3-4" href = "#__codelineno-3-4" > < / a > < span class = "kd" > var< / span > < span class = "w" > < / span > < span class = "nx" > characters< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ]< / span > < span class = "kt" > byte< / span > < span class = "p" > {}< / span >
< a id = "__codelineno-3-5" name = "__codelineno-3-5" href = "#__codelineno-3-5" > < / a > < span class = "kd" > var< / span > < span class = "w" > < / span > < span class = "nx" > booleans< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mi" > 5< / span > < span class = "p" > ]< / span > < span class = "kt" > bool< / span > < span class = "p" > {}< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "cm" > /* JavaScript 的数组可以自由存储各种基本数据类型和对象 */< / span >
< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > array< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mf" > 0< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "s1" > ' a' < / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > false< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > numbers< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > number< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [];< / span >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > characters< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > string< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [];< / span >
< a id = "__codelineno-5-4" name = "__codelineno-5-4" href = "#__codelineno-5-4" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > booleans< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > boolean< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-6-2" name = "__codelineno-6-2" href = "#__codelineno-6-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > numbers< / span > < span class = "p" > [< / span > < span class = "mi" > 10< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-6-3" name = "__codelineno-6-3" href = "#__codelineno-6-3" > < / a > < span class = "kt" > float< / span > < span class = "w" > < / span > < span class = "n" > decimals< / span > < span class = "p" > [< / span > < span class = "mi" > 10< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-6-4" name = "__codelineno-6-4" href = "#__codelineno-6-4" > < / a > < span class = "kt" > char< / span > < span class = "w" > < / span > < span class = "n" > characters< / span > < span class = "p" > [< / span > < span class = "mi" > 10< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-6-5" name = "__codelineno-6-5" href = "#__codelineno-6-5" > < / a > < span class = "kt" > bool< / span > < span class = "w" > < / span > < span class = "n" > booleans< / span > < span class = "p" > [< / span > < span class = "mi" > 10< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > numbers< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "m" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "kt" > float< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > decimals< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > float< / span > < span class = "p" > [< / span > < span class = "m" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "kt" > char< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > characters< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > char< / span > < span class = "p" > [< / span > < span class = "m" > 5< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "kt" > bool< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > booleans< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > bool< / span > < span class = "p" > [< / span > < span class = "m" > 5< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "cm" > /* 使用多种「基本数据类型」来初始化「数组」 */< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a > < span class = "kd" > let< / span > < span class = "nv" > numbers< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "p" > (),< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > decimals< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "nb" > Double< / span > < span class = "p" > (),< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-8-4" name = "__codelineno-8-4" href = "#__codelineno-8-4" > < / a > < span class = "kd" > let< / span > < span class = "nv" > characters< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "nb" > Character< / span > < span class = "p" > (< / span > < span class = "s" > " a" < / span > < span class = "p" > ),< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-8-5" name = "__codelineno-8-5" href = "#__codelineno-8-5" > < / a > < span class = "kd" > let< / span > < span class = "nv" > booleans< / span > < span class = "p" > =< / span > < span class = "nb" > Array< / span > < span class = "p" > (< / span > < span class = "n" > repeating< / span > < span class = "p" > :< / span > < span class = "nb" > Bool< / span > < span class = "p" > (),< / span > < span class = "bp" > count< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< h2 id = "312" > 3.1.2. 计算机内存< a class = "headerlink" href = "#312" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 在计算机中，内存和硬盘是两种主要的存储硬件设备。「硬盘」主要用于长期存储数据，容量较大（通常可达到 TB 级别）、速度较慢。「内存」用于运行程序时暂存数据，速度较快，但容量较小（通常为 GB 级别）。< / p >
< p > < strong > 算法运行中，相关数据都被存储在内存中< / strong > 。下图展示了一个计算机内存条，其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格，其中每个单元格都可以存储 1 byte 的数据，在算法运行时，所有数据都被存储在这些单元格中。< / p >
< p > < strong > 系统通过「内存地址 Memory Location」来访问目标内存位置的数据< / strong > 。计算机根据特定规则给表格中每个单元格编号，保证每块内存空间都有独立的内存地址。自此，程序便通过这些地址，访问内存中的数据。< / p >
< p > < img alt = "内存条、内存空间、内存地址" src = "../data_and_memory.assets/computer_memory_location.png" / > < / p >
< p align = "center" > Fig. 内存条、内存空间、内存地址 < / p >
< p > < strong > 内存资源是设计数据结构与算法的重要考虑因素< / strong > 。内存是所有程序的公共资源，当内存被某程序占用时，不能被其它程序同时使用。我们需要根据剩余内存资源的情况来设计算法。例如，若剩余内存空间有限，则要求算法占用的峰值内存不能超过系统剩余内存；若运行的程序很多、缺少大块连续的内存空间，则要求选取的数据结构必须能够存储在离散的内存空间内。< / p >
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