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2. 计算复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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2.3. 空间复杂度
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</a>
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</li>
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<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
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2.4. 权衡时间与空间
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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2.5. 小结
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</a>
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<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
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3. 数据结构简介
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3.1. 数据与内存
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<label class="md-nav__title" for="__toc">
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目录
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整数表示方式
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<li class="md-nav__item">
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<a href="#_2" class="md-nav__link">
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浮点数表示方式 *
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基本数据类型与数据结构的关系
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3.1.2. 计算机内存
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4.1. 数组（Array）
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</li>
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<li class="md-nav__item">
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4.2. 链表（Linked List）
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</a>
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<li class="md-nav__item">
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4.4. 小结
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</nav>
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
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5. 栈与队列
|
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_6">
|
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<span class="md-nav__icon md-icon"></span>
|
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5. 栈与队列
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈（Stack）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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5.2. 队列（Queue）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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5.3. 双向队列（Deque）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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5.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
|
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6. 散列表
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_7">
|
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|
<span class="md-nav__icon md-icon"></span>
|
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6. 散列表
|
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</label>
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|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
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6.1. 哈希表（Hash Map）
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
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6.2. 哈希冲突处理
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
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6.3. 小结
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
|
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
|
|
|
7. 二叉树
|
|
|
<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. 二叉树
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
7.1. 二叉树（Binary Tree）
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
|
|
7.2. 二叉树遍历
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|
|
7.3. 二叉搜索树
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|
|
7.4. AVL 树 *
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/summary/" class="md-nav__link">
|
|
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7.5. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
|
|
8. 堆
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</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>
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
|
8.1. 堆（Heap）
|
|
|
</a>
|
|
|
</li>
|
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
8.2. 建堆操作 *
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|
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|
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|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
|
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
|
|
|
9. 图
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</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">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
9. 图
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
|
|
9.1. 图（Graph）
|
|
|
</a>
|
|
|
</li>
|
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|
<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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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|
|
9.3. 图的遍历
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
|
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
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<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>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|
|
10.1. 线性查找
|
|
|
</a>
|
|
|
</li>
|
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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>
|
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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>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|
|
10.4. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
|
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
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<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
|
|
|
11. 排序算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
11. 排序算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
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<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
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11.1. 排序简介
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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11.2. 冒泡排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
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11.3. 插入排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
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11.4. 快速排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
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11.5. 归并排序
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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11.6. 小结
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</a>
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</li>
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_reference/">参考文献</a>
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<a href="#311" class="md-nav__link">
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3.1.1. 基本数据类型
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整数表示方式
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<li class="md-nav__item">
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<a href="#_2" class="md-nav__link">
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浮点数表示方式 *
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<li class="md-nav__item">
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<a href="#_3" class="md-nav__link">
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基本数据类型与数据结构的关系
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<a href="#312" class="md-nav__link">
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3.1.2. 计算机内存
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</a>
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</li>
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</ul>
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<div class="md-content" data-md-component="content">
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<article class="md-content__inner md-typeset">
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_data_structure/data_and_memory.md" title="编辑此页" class="md-content__button md-icon">
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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>
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</a>
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<h1 id="31">3.1. 数据与内存<a class="headerlink" href="#31" title="Permanent link">¶</a></h1>
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<h2 id="311">3.1.1. 基本数据类型<a class="headerlink" href="#311" title="Permanent link">¶</a></h2>
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<p>谈到计算机中的数据，我们能够想到文本、图片、视频、语音、3D 模型等等，这些数据虽然组织形式不同，但都是由各种基本数据类型构成的。</p>
|
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<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>
|
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|
<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>
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<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>
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<li>「字符」在计算机中是以字符集的形式保存的，char 的值实际上是数字，代表字符集中的编号，计算机通过字符集查表来完成编号到字符的转换。占用空间与具体编程语言有关，通常为 2 bytes 或 1 byte ；</li>
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<li>「布尔」代表逻辑中的“是”与“否”，其占用空间需要具体根据编程语言确定，通常为 1 byte 或 1 bit ；</li>
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</ul>
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<div class="center-table">
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<table>
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<thead>
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<tr>
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<th>类别</th>
|
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<th>符号</th>
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<th>占用空间</th>
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<th>取值范围</th>
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<th>默认值</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>整数</td>
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<td>byte</td>
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<td>1 byte</td>
|
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<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>
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<td><span class="arithmatex">\(0\)</span></td>
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</tr>
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<tr>
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<td></td>
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<td>short</td>
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<td>2 bytes</td>
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<td><span class="arithmatex">\(-2^{15}\)</span> ~ <span class="arithmatex">\(2^{15} - 1\)</span></td>
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<td><span class="arithmatex">\(0\)</span></td>
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</tr>
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<tr>
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<td></td>
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<td><strong>int</strong></td>
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<td>4 bytes</td>
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<td><span class="arithmatex">\(-2^{31}\)</span> ~ <span class="arithmatex">\(2^{31} - 1\)</span></td>
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<td><span class="arithmatex">\(0\)</span></td>
|
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</tr>
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<tr>
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<td></td>
|
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<td>long</td>
|
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<td>8 bytes</td>
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<td><span class="arithmatex">\(-2^{63}\)</span> ~ <span class="arithmatex">\(2^{63} - 1\)</span></td>
|
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<td><span class="arithmatex">\(0\)</span></td>
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</tr>
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<tr>
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<td>浮点数</td>
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<td><strong>float</strong></td>
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<td>4 bytes</td>
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<td><span class="arithmatex">\(-3.4 \times 10^{38}\)</span> ~ <span class="arithmatex">\(3.4 \times 10^{38}\)</span></td>
|
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<td><span class="arithmatex">\(0.0\)</span> f</td>
|
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</tr>
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<tr>
|
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<td></td>
|
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<td>double</td>
|
|
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<td>8 bytes</td>
|
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|
<td><span class="arithmatex">\(-1.7 \times 10^{308}\)</span> ~ <span class="arithmatex">\(1.7 \times 10^{308}\)</span></td>
|
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<td><span class="arithmatex">\(0.0\)</span></td>
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</tr>
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<tr>
|
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<td>字符</td>
|
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<td><strong>char</strong></td>
|
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<td>2 bytes / 1 byte</td>
|
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<td><span class="arithmatex">\(0\)</span> ~ <span class="arithmatex">\(2^{16} - 1\)</span></td>
|
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<td><span class="arithmatex">\(0\)</span></td>
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</tr>
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<tr>
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<td>布尔</td>
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<td><strong>bool</strong></td>
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<td>1 byte / 1 bit</td>
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<td><span class="arithmatex">\(\text{true}\)</span> 或 <span class="arithmatex">\(\text{false}\)</span></td>
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<td><span class="arithmatex">\(\text{false}\)</span></td>
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</tr>
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</tbody>
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</table>
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</div>
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<div class="admonition tip">
|
|
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<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>
|
|
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</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) ， 1 比特即 1 个二进制位。以 int 类型为例：</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>
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|
|
</div>
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|
|
<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，32-bit 长度的 float 由以下部分构成：</p>
|
|
|
<ul>
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|
<li>符号位 <span class="arithmatex">\(\mathrm{S}\)</span> ：占 1 bit ；</li>
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|
|
<li>指数位 <span class="arithmatex">\(\mathrm{E}\)</span> ：占 8 bits ；</li>
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|
|
<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>
|
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<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>
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<table>
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<thead>
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<tr>
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<th>指数位 E</th>
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<th>分数位 <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
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<th>分数位 <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
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<th>计算公式</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td><span class="arithmatex">\(0\)</span></td>
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<td><span class="arithmatex">\(\pm 0\)</span></td>
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<td>次正规数（subnormal number）</td>
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<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)</span></td>
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</tr>
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<tr>
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<td><span class="arithmatex">\(1, 2, \dots, 254\)</span></td>
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<td>正规数</td>
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<td>正规数</td>
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<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)</span></td>
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</tr>
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<tr>
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<td><span class="arithmatex">\(255\)</span></td>
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<td><span class="arithmatex">\(\pm \infty\)</span></td>
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<td><span class="arithmatex">\(\mathrm{NaN}\)</span></td>
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<td></td>
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</tr>
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</tbody>
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</table>
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<p>特别地，次正规数显著提升了小数精度：</p>
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<ul>
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<li>最小正正规数为 <span class="arithmatex">\(2^{-126} \approx 1.18 \times 10^{-38}\)</span> ；</li>
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<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>
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</ul>
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<p>双精度 double 也采用类似 float 的表示方法，在此不再赘述。</p>
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<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>
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<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>
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<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>
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<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>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
|
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<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>
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<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>
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<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>
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<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>
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<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>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h2 id="312">3.1.2. 计算机内存<a class="headerlink" href="#312" title="Permanent link">¶</a></h2>
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<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>
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<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>
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<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>
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<p><img alt="内存条、内存空间、内存地址" src="../data_and_memory.assets/computer_memory_location.png" /></p>
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<p align="center"> Fig. 内存条、内存空间、内存地址 </p>
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<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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