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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
|
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|
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<span class="md-ellipsis">
|
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第 0 章 前言
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0.1 关于本书
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|
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</a>
|
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0.2 如何使用本书
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</a>
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0.3 小结
|
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|
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第 1 章 初识算法
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第 1 章 初识算法
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1.1 算法无处不在
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|
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</a>
|
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1.2 算法是什么
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</a>
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1.3 小结
|
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</span>
|
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|
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|
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<span class="md-ellipsis">
|
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第 2 章 时空复杂度
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第 2 章 时空复杂度
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2.1 算法效率评估
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</span>
|
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|
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|
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</a>
|
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</li>
|
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2.2 时间复杂度
|
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|
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|
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|
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</a>
|
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</li>
|
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2.3 空间复杂度
|
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</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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|
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2.4 小结
|
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</span>
|
||
|
||
|
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</a>
|
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|
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|
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|
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<span class="md-ellipsis">
|
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第 3 章 数据结构
|
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|
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|
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<label class="md-nav__link " for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
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第 3 章 数据结构
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<a href="../classification_of_data_structure/" class="md-nav__link">
|
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|
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3.1 数据结构分类
|
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|
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|
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|
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</a>
|
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</li>
|
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3.2 基本数据类型
|
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|
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|
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|
||
</a>
|
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</li>
|
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|
||
|
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3.3 数字编码 *
|
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|
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|
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|
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<span class="md-ellipsis">
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3.3 数字编码 *
|
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|
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</a>
|
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|
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|
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|
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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目录
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3.3.1 原码、反码和补码
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|
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|
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|
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<a href="#332" class="md-nav__link">
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3.3.2 浮点数编码
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|
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|
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|
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|
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<span class="md-ellipsis">
|
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3.4 字符编码 *
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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|
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|
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<a href="../summary/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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3.5 小结
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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第 4 章 数组与链表
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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|
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4.1 数组
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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4.2 链表
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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<li class="md-nav__item">
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|
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4.3 列表
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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4.4 小结
|
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</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
||
|
||
</ul>
|
||
</nav>
|
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|
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</li>
|
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|
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|
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|
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|
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|
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|
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||
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|
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|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__container">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
|
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|
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<span class="md-ellipsis">
|
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第 5 章 栈与队列
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<label class="md-nav__link " for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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第 5 章 栈与队列
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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|
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5.1 栈
|
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</span>
|
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|
||
|
||
</a>
|
||
</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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|
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5.2 队列
|
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</span>
|
||
|
||
|
||
</a>
|
||
</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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<span class="md-ellipsis">
|
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5.3 双向队列
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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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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<span class="md-ellipsis">
|
||
5.4 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
|
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</nav>
|
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</li>
|
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|
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|
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||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_hashing/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
第 6 章 哈希表
|
||
</span>
|
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|
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|
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</a>
|
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|
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<label class="md-nav__link " for="__nav_7">
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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第 6 章 哈希表
|
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<ul class="md-nav__list" data-md-scrollfix>
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|
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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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|
||
|
||
<span class="md-ellipsis">
|
||
6.1 哈希表
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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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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|
||
|
||
<span class="md-ellipsis">
|
||
6.2 哈希冲突
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
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|
||
|
||
<span class="md-ellipsis">
|
||
6.3 哈希算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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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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|
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|
||
<span class="md-ellipsis">
|
||
6.4 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
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</ul>
|
||
</nav>
|
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|
||
</li>
|
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|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_tree/" class="md-nav__link ">
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|
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|
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
第 7 章 树
|
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</span>
|
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|
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|
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</a>
|
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|
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<label class="md-nav__link " for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
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</div>
|
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
|
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第 7 章 树
|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.1 二叉树
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
||
7.2 二叉树遍历
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
||
7.3 二叉树数组表示
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.4 二叉搜索树
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
7.5 AVL 树 *
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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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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|
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|
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<span class="md-ellipsis">
|
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7.6 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
|
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</nav>
|
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||
</li>
|
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|
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|
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|
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|
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|
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|
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||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_heap/" class="md-nav__link ">
|
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|
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
第 8 章 堆
|
||
</span>
|
||
|
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|
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</a>
|
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|
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<label class="md-nav__link " for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
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第 8 章 堆
|
||
</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_heap/heap/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
||
8.1 堆
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
||
8.2 建堆操作
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
||
<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.3 Top-K 问题
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.4 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_graph/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 9 章 图
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " for="__nav_10">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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>
|
||
第 9 章 图
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.1 图
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.2 图基础操作
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.3 图的遍历
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.4 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_searching/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 10 章 搜索
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " for="__nav_11">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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>
|
||
第 10 章 搜索
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.1 二分查找
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.2 二分查找插入点
|
||
</span>
|
||
|
||
|
||
|
||
|
||
<span class="md-status md-status--new" title="最近添加">
|
||
</span>
|
||
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.3 二分查找边界
|
||
</span>
|
||
|
||
|
||
|
||
|
||
<span class="md-status md-status--new" title="最近添加">
|
||
</span>
|
||
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.4 哈希优化策略
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.5 重识搜索算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.6 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_sorting/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 11 章 排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " for="__nav_12">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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>
|
||
第 11 章 排序
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.1 排序算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.2 选择排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.3 冒泡排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.4 插入排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.5 快速排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.6 归并排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.7 堆排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.8 桶排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.9 计数排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.10 基数排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.11 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 12 章 分治
|
||
</span>
|
||
|
||
|
||
|
||
|
||
<span class="md-status md-status--new" title="最近添加">
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第 12 章 分治
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12.1 分治算法
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12.2 分治搜索策略
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
|
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|
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第 13 章 回溯
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第 13 章 回溯
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13.1 回溯算法
|
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|
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|
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|
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</a>
|
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13.2 全排列问题
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|
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|
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|
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</a>
|
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|
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13.3 子集和问题
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|
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|
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|
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</a>
|
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13.4 N 皇后问题
|
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|
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13.5 小结
|
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|
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|
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第 14 章 动态规划
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第 14 章 动态规划
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14.1 初探动态规划
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14.2 DP 问题特性
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14.3 DP 解题思路
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14.5 完全背包问题
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14.6 编辑距离问题
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14.7 小结
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第 15 章 贪心
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15.1 贪心算法
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15.2 分数背包问题
|
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15.3 最大容量问题
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15.4 最大切分乘积问题
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15.5 小结
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第 16 章 附录
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3.3.1 原码、反码和补码
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3.3.2 浮点数编码
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|
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<h1 id="33">3.3 数字编码 *<a class="headerlink" href="#33" title="Permanent link">¶</a></h1>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>在本书中，标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难，可以先跳过，等学完必读章节后再单独攻克。</p>
|
||
</div>
|
||
<h2 id="331">3.3.1 原码、反码和补码<a class="headerlink" href="#331" title="Permanent link">¶</a></h2>
|
||
<p>在上一节的表格中我们发现，所有整数类型能够表示的负数都比正数多一个，例如 <code>byte</code> 的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> 。这个现象比较反直觉，它的内在原因涉及到原码、反码、补码的相关知识。</p>
|
||
<p>首先需要指出，<strong>数字是以“补码”的形式存储在计算机中的</strong>。在分析这样做的原因之前，我们首先给出三者的定义：</p>
|
||
<ul>
|
||
<li><strong>原码</strong>：我们将数字的二进制表示的最高位视为符号位，其中 <span class="arithmatex">\(0\)</span> 表示正数，<span class="arithmatex">\(1\)</span> 表示负数，其余位表示数字的值。</li>
|
||
<li><strong>反码</strong>：正数的反码与其原码相同，负数的反码是对其原码除符号位外的所有位取反。</li>
|
||
<li><strong>补码</strong>：正数的补码与其原码相同，负数的补码是在其反码的基础上加 <span class="arithmatex">\(1\)</span> 。</li>
|
||
</ul>
|
||
<p>下图展示了原吗、反码和补码之间的转换方法。</p>
|
||
<p><img alt="原码、反码与补码之间的相互转换" src="../number_encoding.assets/1s_2s_complement.png" /></p>
|
||
<p align="center"> 图：原码、反码与补码之间的相互转换 </p>
|
||
|
||
<p>「原码 true form」虽然最直观，但存在一些局限性。一方面，<strong>负数的原码不能直接用于运算</strong>。例如在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& 1 + (-2) \newline
|
||
& \rightarrow 0000 \space 0001 + 1000 \space 0010 \newline
|
||
& = 1000 \space 0011 \newline
|
||
& \rightarrow -3
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>为了解决此问题，计算机引入了「反码 1's complement code」。如果我们先将原码转换为反码，并在反码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，最后将结果从反码转化回原码，则可得到正确结果 <span class="arithmatex">\(-1\)</span> 。</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& 1 + (-2) \newline
|
||
& \rightarrow 0000 \space 0001 \space \text{(原码)} + 1000 \space 0010 \space \text{(原码)} \newline
|
||
& = 0000 \space 0001 \space \text{(反码)} + 1111 \space 1101 \space \text{(反码)} \newline
|
||
& = 1111 \space 1110 \space \text{(反码)} \newline
|
||
& = 1000 \space 0001 \space \text{(原码)} \newline
|
||
& \rightarrow -1
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>另一方面，<strong>数字零的原码有 <span class="arithmatex">\(+0\)</span> 和 <span class="arithmatex">\(-0\)</span> 两种表示方式</strong>。这意味着数字零对应着两个不同的二进制编码，其可能会带来歧义。比如在条件判断中，如果没有区分正零和负零，则可能会导致判断结果出错。而如果我们想要处理正零和负零歧义，则需要引入额外的判断操作，其可能会降低计算机的运算效率。</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
+0 & \rightarrow 0000 \space 0000 \newline
|
||
-0 & \rightarrow 1000 \space 0000
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了「补码 2's complement code」。我们先来观察一下负零的原码、反码、补码的转换过程：</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
-0 \rightarrow \space & 1000 \space 0000 \space \text{(原码)} \newline
|
||
= \space & 1111 \space 1111 \space \text{(反码)} \newline
|
||
= 1 \space & 0000 \space 0000 \space \text{(补码)} \newline
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>在负零的反码基础上加 <span class="arithmatex">\(1\)</span> 会产生进位，但 <code>byte</code> 类型的长度只有 8 位，因此溢出到第 9 位的 <span class="arithmatex">\(1\)</span> 会被舍弃。也就是说，<strong>负零的补码为 <span class="arithmatex">\(0000 \space 0000\)</span> ，与正零的补码相同</strong>。这意味着在补码表示中只存在一个零，正负零歧义从而得到解决。</p>
|
||
<p>还剩余最后一个疑惑：<code>byte</code> 类型的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> ，多出来的一个负数 <span class="arithmatex">\(-128\)</span> 是如何得到的呢？我们注意到，区间 <span class="arithmatex">\([-127, +127]\)</span> 内的所有整数都有对应的原码、反码和补码，并且原码和补码之间是可以互相转换的。</p>
|
||
<p>然而，<strong>补码 <span class="arithmatex">\(1000 \space 0000\)</span> 是一个例外，它并没有对应的原码</strong>。根据转换方法，我们得到该补码的原码为 <span class="arithmatex">\(0000 \space 0000\)</span> 。这显然是矛盾的，因为该原码表示数字 <span class="arithmatex">\(0\)</span> ，它的补码应该是自身。计算机规定这个特殊的补码 <span class="arithmatex">\(1000 \space 0000\)</span> 代表 <span class="arithmatex">\(-128\)</span> 。实际上，<span class="arithmatex">\((-1) + (-127)\)</span> 在补码下的计算结果就是 <span class="arithmatex">\(-128\)</span> 。</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& (-127) + (-1) \newline
|
||
& \rightarrow 1111 \space 1111 \space \text{(原码)} + 1000 \space 0001 \space \text{(原码)} \newline
|
||
& = 1000 \space 0000 \space \text{(反码)} + 1111 \space 1110 \space \text{(反码)} \newline
|
||
& = 1000 \space 0001 \space \text{(补码)} + 1111 \space 1111 \space \text{(补码)} \newline
|
||
& = 1000 \space 0000 \space \text{(补码)} \newline
|
||
& \rightarrow -128
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>你可能已经发现，上述的所有计算都是加法运算。这暗示着一个重要事实：<strong>计算机内部的硬件电路主要是基于加法运算设计的</strong>。这是因为加法运算相对于其他运算（比如乘法、除法和减法）来说，硬件实现起来更简单，更容易进行并行化处理，运算速度更快。</p>
|
||
<p>请注意，这并不意味着计算机只能做加法。<strong>通过将加法与一些基本逻辑运算结合，计算机能够实现各种其他的数学运算</strong>。例如，计算减法 <span class="arithmatex">\(a - b\)</span> 可以转换为计算加法 <span class="arithmatex">\(a + (-b)\)</span> ；计算乘法和除法可以转换为计算多次加法或减法。</p>
|
||
<p>现在我们可以总结出计算机使用补码的原因：基于补码表示，计算机可以用同样的电路和操作来处理正数和负数的加法，不需要设计特殊的硬件电路来处理减法，并且无须特别处理正负零的歧义问题。这大大简化了硬件设计，提高了运算效率。</p>
|
||
<p>补码的设计非常精妙，因篇幅关系我们就先介绍到这里，建议有兴趣的读者进一步深度了解。</p>
|
||
<h2 id="332">3.3.2 浮点数编码<a class="headerlink" href="#332" title="Permanent link">¶</a></h2>
|
||
<p>细心的你可能会发现：<code>int</code> 和 <code>float</code> 长度相同，都是 4 bytes，但为什么 <code>float</code> 的取值范围远大于 <code>int</code> ？这非常反直觉，因为按理说 <code>float</code> 需要表示小数，取值范围应该变小才对。</p>
|
||
<p>实际上，<strong>这是因为浮点数 <code>float</code> 采用了不同的表示方式</strong>。记一个 32-bit 长度的二进制数为：</p>
|
||
<div class="arithmatex">\[
|
||
b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
|
||
\]</div>
|
||
<p>根据 IEEE 754 标准，32-bit 长度的 <code>float</code> 由以下部分构成：</p>
|
||
<ul>
|
||
<li>符号位 <span class="arithmatex">\(\mathrm{S}\)</span> ：占 1 bit ，对应 <span class="arithmatex">\(b_{31}\)</span> 。</li>
|
||
<li>指数位 <span class="arithmatex">\(\mathrm{E}\)</span> ：占 8 bits ，对应 <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span> 。</li>
|
||
<li>分数位 <span class="arithmatex">\(\mathrm{N}\)</span> ：占 23 bits ，对应 <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span> 。</li>
|
||
</ul>
|
||
<p>二进制数 <code>float</code> 对应的值的计算方法：</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：</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：</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 的计算示例" src="../number_encoding.assets/ieee_754_float.png" /></p>
|
||
<p align="center"> 图：IEEE 754 标准下的 float 的计算示例 </p>
|
||
|
||
<p>观察上图，给定一个示例数据 <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> ，则有：</p>
|
||
<div class="arithmatex">\[
|
||
\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
|
||
\]</div>
|
||
<p>现在我们可以回答最初的问题：<strong><code>float</code> 的表示方式包含指数位，导致其取值范围远大于 <code>int</code></strong> 。根据以上计算，<code>float</code> 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>尽管浮点数 <code>float</code> 扩展了取值范围，但其副作用是牺牲了精度</strong>。整数类型 <code>int</code> 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 <code>float</code> 的数值越大，相邻两个数字之间的差值就会趋向越大。</p>
|
||
<p>如下表所示，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>
|
||
<p align="center"> 表：指数位含义 </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浮点数的精度，这是因为：</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 <code>double</code> 也采用类似 <code>float</code> 的表示方法，此处不再详述。</p>
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