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<title>9.3. 图的遍历 - Hello 算法</title>
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9.3. 图的遍历
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</label>
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<span class="md-nav__icon md-icon"></span>
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0. 写在前面
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0.1. 关于本书
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0.2. 如何使用本书
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0.3. 编程环境安装
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1. 引言
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<span class="md-nav__icon md-icon"></span>
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1. 引言
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1.1. 算法无处不在
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</a>
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</li>
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1.2. 算法是什么
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</nav>
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 计算复杂度
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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_3_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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2. 计算复杂度
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
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2.1. 算法效率评估
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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_computational_complexity/time_complexity/" class="md-nav__link">
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2.2. 时间复杂度
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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_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>
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</li>
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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>
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</ul>
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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_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
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3. 数据结构简介
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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_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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3. 数据结构简介
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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_data_structure/data_and_memory/" class="md-nav__link">
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3.1. 数据与内存
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</a>
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</li>
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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3.2. 数据结构分类
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</a>
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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3.3. 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 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_5" >
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
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4. 数组与链表
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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_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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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_array_and_linkedlist/array/" class="md-nav__link">
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4.1. 数组（Array）
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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_array_and_linkedlist/linked_list/" class="md-nav__link">
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4.2. 链表（LinkedList）
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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_array_and_linkedlist/list/" class="md-nav__link">
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4.3. 列表（List）
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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_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 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_6" >
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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">
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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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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（HashMap）
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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">
|
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7. 二叉树
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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_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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7. 二叉树
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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_tree/binary_tree/" class="md-nav__link">
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7.1. 二叉树（Binary Tree）
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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_tree/binary_tree_traversal/" class="md-nav__link">
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7.2. 二叉树遍历
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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_tree/binary_search_tree/" class="md-nav__link">
|
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7.3. 二叉搜索树
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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_tree/avl_tree/" class="md-nav__link">
|
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7.4. AVL 树 *
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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_tree/summary/" class="md-nav__link">
|
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7.5. 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_9" >
|
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
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8. 堆
|
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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_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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8. 堆
|
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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_heap/heap/" class="md-nav__link">
|
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8.1. 堆（Heap）
|
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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--active md-nav__item--section md-nav__item--nested">
|
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|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
|
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|
<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
|
|
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|
|
|
9. 图
|
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|
|
<span class="md-nav__icon md-icon"></span>
|
|
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|
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</label>
|
|
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|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
|
|
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|
|
<label class="md-nav__title" for="__nav_10">
|
|
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|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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|
|
|
9. 图
|
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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="../graph/" class="md-nav__link">
|
|
|
|
|
|
9.1. 图（Graph）
|
|
|
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
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|
|
<a href="../graph_operations/" class="md-nav__link">
|
|
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|
|
|
9.2. 图基础操作
|
|
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|
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</a>
|
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</li>
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<li class="md-nav__item md-nav__item--active">
|
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|
|
<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
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|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
|
|
|
9.3. 图的遍历
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
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|
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|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
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|
|
|
9.3. 图的遍历
|
|
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|
|
</a>
|
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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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|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
目录
|
|
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|
|
</label>
|
|
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|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
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|
|
<a href="#931" class="md-nav__link">
|
|
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|
|
9.3.1. 广度优先遍历
|
|
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|
|
|
</a>
|
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|
|
<nav class="md-nav" aria-label="9.3.1. 广度优先遍历">
|
|
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|
|
<ul class="md-nav__list">
|
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|
<li class="md-nav__item">
|
|
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|
<a href="#_1" class="md-nav__link">
|
|
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|
|
|
算法实现
|
|
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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="#_2" class="md-nav__link">
|
|
|
|
|
|
复杂度分析
|
|
|
|
|
|
</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">
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<a href="#932" class="md-nav__link">
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9.3.2. 深度优先遍历
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</a>
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<nav class="md-nav" aria-label="9.3.2. 深度优先遍历">
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#_3" class="md-nav__link">
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算法实现
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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="#_4" class="md-nav__link">
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复杂度分析
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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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</ul>
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</nav>
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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_11" >
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
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10. 查找算法
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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_11_label" aria-expanded="false">
|
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|
<label class="md-nav__title" for="__nav_11">
|
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|
<span class="md-nav__icon md-icon"></span>
|
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10. 查找算法
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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_searching/linear_search/" class="md-nav__link">
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10.1. 线性查找
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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_searching/binary_search/" class="md-nav__link">
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10.2. 二分查找
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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_searching/hashing_search/" class="md-nav__link">
|
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10.3. 哈希查找
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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_searching/summary/" class="md-nav__link">
|
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10.4. 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_12" >
|
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|
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
|
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|
|
11. 排序算法
|
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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_12_label" aria-expanded="false">
|
|
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|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
11. 排序算法
|
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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_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>
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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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|
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11.6. 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_13" >
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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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|
</div>
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
|
|
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|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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|
|
|
参考文献
|
|
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|
|
</label>
|
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|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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</ul>
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</nav>
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</li>
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</ul>
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</nav>
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</div>
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</div>
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</div>
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|
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
|
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|
|
<div class="md-sidebar__scrollwrap">
|
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|
<div class="md-sidebar__inner">
|
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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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|
|
<span class="md-nav__icon md-icon"></span>
|
|
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|
|
|
目录
|
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|
</label>
|
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|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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9.3.1. 广度优先遍历
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算法实现
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复杂度分析
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9.3.2. 深度优先遍历
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复杂度分析
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_graph/graph_traversal.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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<h1 id="93">9.3. 图的遍历<a class="headerlink" href="#93" title="Permanent link">¶</a></h1>
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<div class="admonition note">
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<p class="admonition-title">图与树的关系</p>
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<p>树代表的是“一对多”的关系，而图则自由度更高，可以代表任意“多对多”关系。本质上，<strong>可以把树看作是图的一类特例</strong>。那么显然，树遍历操作也是图遍历操作的一个特例，两者的方法是非常类似的，建议你在学习本章节的过程中将两者融会贯通。</p>
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</div>
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<p>「图」与「树」都是非线性数据结构，都需要使用「搜索算法」来实现遍历操作。</p>
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<p>类似地，图的遍历方式也分为两种，即「广度优先遍历 Breadth-First Traversal」和「深度优先遍历 Depth-First Travsersal」，也称「广度优先搜索 Breadth-First Search」和「深度优先搜索 Depth-First Search」，简称为 BFS 和 DFS 。</p>
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<h2 id="931">9.3.1. 广度优先遍历<a class="headerlink" href="#931" title="Permanent link">¶</a></h2>
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<p><strong>广度优先遍历优是一种由近及远的遍历方式，从距离最近的顶点开始访问，并一层层向外扩张</strong>。具体地，从某个顶点出发，先遍历该顶点的所有邻接顶点，随后遍历下个顶点的所有邻接顶点，以此类推……</p>
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<p><img alt="graph_bfs" src="../graph_traversal.assets/graph_bfs.png" /></p>
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<h3 id="_1">算法实现<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
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<p>BFS 常借助「队列」来实现。队列具有“先入先出”的性质，这与 BFS “由近及远”的思想是异曲同工的。</p>
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<ol>
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<li>将遍历起始顶点 <code>startVet</code> 加入队列，并开启循环；</li>
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<li>在循环的每轮迭代中，弹出队首顶点弹出并记录访问，并将该顶点的所有邻接顶点加入到队列尾部；</li>
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<li>循环 <code>2.</code> ，直到所有顶点访问完成后结束。</li>
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</ol>
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<p>为了防止重复遍历顶点，我们需要借助一个哈希表 <code>visited</code> 来记录哪些结点已被访问。</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="highlight"><span class="filename">graph_bfs.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 广度优先遍历 BFS */</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1">// 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">List</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="nf">graphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </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="w"> </span><span class="c1">// 顶点遍历序列</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="c1">// 哈希表，用于记录已被访问过的顶点</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="n">Set</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o"><></span><span class="p">()</span><span class="w"> </span><span class="p">{{</span><span class="w"> </span><span class="n">add</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span><span class="w"> </span><span class="p">}};</span>
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<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="c1">// 队列用于实现 BFS</span>
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<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="n">Queue</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">LinkedList</span><span class="o"><></span><span class="p">()</span><span class="w"> </span><span class="p">{{</span><span class="w"> </span><span class="n">offer</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span><span class="w"> </span><span class="p">}};</span>
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<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="c1">// 以顶点 vet 为起点，循环直至访问完所有顶点</span>
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">que</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 队首顶点出队</span>
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<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录访问顶点</span>
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<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="c1">// 遍历该顶点的所有邻接顶点</span>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
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<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="w"> </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// 跳过已被访问过的顶点</span>
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<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// 只入队未访问的顶点</span>
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<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// 标记该顶点已被访问</span>
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<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="w"> </span><span class="c1">// 返回顶点遍历序列</span>
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<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><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"><span class="filename">graph_bfs.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">graph_bfs.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">graph_bfs.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">graph_bfs.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">graph_bfs.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">graph_bfs.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_bfs.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_bfs.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_bfs.zig</span><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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<p>代码相对抽象，建议对照以下动画图示来加深理解。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:11"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Step 1</label><label for="__tabbed_2_2">Step 2</label><label for="__tabbed_2_3">Step 3</label><label for="__tabbed_2_4">Step 4</label><label for="__tabbed_2_5">Step 5</label><label for="__tabbed_2_6">Step 6</label><label for="__tabbed_2_7">Step 7</label><label for="__tabbed_2_8">Step 8</label><label for="__tabbed_2_9">Step 9</label><label for="__tabbed_2_10">Step 10</label><label for="__tabbed_2_11">Step 11</label></div>
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<div class="tabbed-content">
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<p><img alt="graph_bfs_step1" src="../graph_traversal.assets/graph_bfs_step1.png" /></p>
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</div>
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<p><img alt="graph_bfs_step2" src="../graph_traversal.assets/graph_bfs_step2.png" /></p>
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</div>
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<p><img alt="graph_bfs_step3" src="../graph_traversal.assets/graph_bfs_step3.png" /></p>
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</div>
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<p><img alt="graph_bfs_step4" src="../graph_traversal.assets/graph_bfs_step4.png" /></p>
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</div>
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<p><img alt="graph_bfs_step5" src="../graph_traversal.assets/graph_bfs_step5.png" /></p>
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<p><img alt="graph_bfs_step6" src="../graph_traversal.assets/graph_bfs_step6.png" /></p>
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</div>
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<p><img alt="graph_bfs_step7" src="../graph_traversal.assets/graph_bfs_step7.png" /></p>
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</div>
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<p><img alt="graph_bfs_step8" src="../graph_traversal.assets/graph_bfs_step8.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_bfs_step9" src="../graph_traversal.assets/graph_bfs_step9.png" /></p>
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</div>
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<p><img alt="graph_bfs_step10" src="../graph_traversal.assets/graph_bfs_step10.png" /></p>
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<p><img alt="graph_bfs_step11" src="../graph_traversal.assets/graph_bfs_step11.png" /></p>
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</div>
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</div>
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</div>
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<div class="admonition question">
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<p class="admonition-title">广度优先遍历的序列是否唯一？</p>
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<p>不唯一。广度优先遍历只要求“由近及远”，而相同距离的多个顶点的遍历顺序允许任意被打乱。以上图为例，顶点 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(3\)</span> 的访问顺序可以交换、顶点 <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(4\)</span> , <span class="arithmatex">\(6\)</span> 的访问顺序也可以任意交换、以此类推……</p>
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</div>
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<h3 id="_2">复杂度分析<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
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<p><strong>时间复杂度：</strong> 所有顶点都会入队、出队一次，使用 <span class="arithmatex">\(O(|V|)\)</span> 时间；在遍历邻接顶点的过程中，由于是无向图，因此所有边都会被访问 <span class="arithmatex">\(2\)</span> 次，使用 <span class="arithmatex">\(O(2|E|)\)</span> 时间；总体使用 <span class="arithmatex">\(O(|V| + |E|)\)</span> 时间。</p>
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<p><strong>空间复杂度：</strong> 列表 <code>res</code> ，哈希表 <code>visited</code> ，队列 <code>que</code> 中的顶点数量最多为 <span class="arithmatex">\(|V|\)</span> ，使用 <span class="arithmatex">\(O(|V|)\)</span> 空间。</p>
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<h2 id="932">9.3.2. 深度优先遍历<a class="headerlink" href="#932" title="Permanent link">¶</a></h2>
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<p><strong>深度优先遍历是一种优先走到底、无路可走再回头的遍历方式</strong>。具体地，从某个顶点出发，不断地访问当前结点的某个邻接顶点，直到走到尽头时回溯，再继续走到底 + 回溯，以此类推……直至所有顶点遍历完成时结束。</p>
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<p><img alt="graph_dfs" src="../graph_traversal.assets/graph_dfs.png" /></p>
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<h3 id="_3">算法实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
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<p>这种“走到头 + 回溯”的算法形式一般基于递归来实现。与 BFS 类似，在 DFS 中我们也需要借助一个哈希表 <code>visited</code> 来记录已被访问的顶点，以避免重复访问顶点。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:10"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_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"><span class="filename">graph_dfs.java</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 深度优先遍历 DFS 辅助函数 */</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Set</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录访问顶点</span>
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<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// 标记该顶点已被访问</span>
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<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w"> </span><span class="c1">// 遍历该顶点的所有邻接顶点</span>
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<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
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<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// 跳过已被访问过的顶点</span>
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<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="c1">// 递归访问邻接顶点</span>
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<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">);</span>
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<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="p">}</span>
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<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a>
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<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="cm">/* 深度优先遍历 DFS */</span>
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<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="c1">// 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点</span>
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<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="n">List</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="nf">graphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w"> </span><span class="c1">// 顶点遍历序列</span>
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<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w"> </span><span class="c1">// 哈希表，用于记录已被访问过的顶点</span>
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<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w"> </span><span class="n">Set</span><span class="o"><</span><span class="n">Vertex</span><span class="o">></span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
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<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_dfs.cpp</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_dfs.py</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_dfs.go</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_dfs.js</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>
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</code></pre></div>
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<div class="highlight"><span class="filename">graph_dfs.ts</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>
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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"><span class="filename">graph_dfs.c</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a>
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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"><span class="filename">graph_dfs.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a>
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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"><span class="filename">graph_dfs.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a>
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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"><span class="filename">graph_dfs.zig</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></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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<p>深度优先遍历的算法流程如下图所示，其中</p>
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<ul>
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<li><strong>直虚线代表向下递推</strong>，代表开启了一个新的递归方法来访问新顶点；</li>
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<li><strong>曲虚线代表向上回溯</strong>，代表此递归方法已经返回，回溯到了开启此递归方法的位置；</li>
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</ul>
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<p>为了加深理解，请你将图示与代码结合起来，在脑中（或者用笔画下来）模拟整个 DFS 过程，包括每个递归方法何时开启、何时返回。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:11"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Step 1</label><label for="__tabbed_4_2">Step 2</label><label for="__tabbed_4_3">Step 3</label><label for="__tabbed_4_4">Step 4</label><label for="__tabbed_4_5">Step 5</label><label for="__tabbed_4_6">Step 6</label><label for="__tabbed_4_7">Step 7</label><label for="__tabbed_4_8">Step 8</label><label for="__tabbed_4_9">Step 9</label><label for="__tabbed_4_10">Step 10</label><label for="__tabbed_4_11">Step 11</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step1" src="../graph_traversal.assets/graph_dfs_step1.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step2" src="../graph_traversal.assets/graph_dfs_step2.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step3" src="../graph_traversal.assets/graph_dfs_step3.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step4" src="../graph_traversal.assets/graph_dfs_step4.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step5" src="../graph_traversal.assets/graph_dfs_step5.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step6" src="../graph_traversal.assets/graph_dfs_step6.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step7" src="../graph_traversal.assets/graph_dfs_step7.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step8" src="../graph_traversal.assets/graph_dfs_step8.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step9" src="../graph_traversal.assets/graph_dfs_step9.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step10" src="../graph_traversal.assets/graph_dfs_step10.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="graph_dfs_step11" src="../graph_traversal.assets/graph_dfs_step11.png" /></p>
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</div>
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</div>
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</div>
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<div class="admonition question">
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<p class="admonition-title">深度优先遍历的序列是否唯一？</p>
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<p>与广度优先遍历类似，深度优先遍历序列的顺序也不是唯一的。给定某顶点，先往哪个方向探索都行，都是深度优先遍历。</p>
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<p>以树的遍历为例，“根 <span class="arithmatex">\(\rightarrow\)</span> 左 <span class="arithmatex">\(\rightarrow\)</span> 右”、“左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右”、“左 <span class="arithmatex">\(\rightarrow\)</span> 右 <span class="arithmatex">\(\rightarrow\)</span> 根”分别对应前序、中序、后序遍历，体现三种不同的遍历优先级，而三者都属于深度优先遍历。</p>
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</div>
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<h3 id="_4">复杂度分析<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
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<p><strong>时间复杂度：</strong> 所有顶点都被访问一次；所有边都被访问了 <span class="arithmatex">\(2\)</span> 次，使用 <span class="arithmatex">\(O(2|E|)\)</span> 时间；总体使用 <span class="arithmatex">\(O(|V| + |E|)\)</span> 时间。</p>
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<p><strong>空间复杂度：</strong> 列表 <code>res</code> ，哈希表 <code>visited</code> 顶点数量最多为 <span class="arithmatex">\(|V|\)</span> ，递归深度最大为 <span class="arithmatex">\(|V|\)</span> ，因此使用 <span class="arithmatex">\(O(|V|)\)</span> 空间。</p>
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<h2 id="__comments">评论</h2>
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