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9.1. 图（Graph）
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</a>
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</div>
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
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0. 写在前面
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</label>
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0. 写在前面
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<li class="md-nav__item">
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<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
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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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1. 引言
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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1. 引言
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
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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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<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
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1.2. 算法是什么
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</a>
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<a href="../../chapter_introduction/summary/" class="md-nav__link">
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1.3. 小结
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 复杂度分析
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</label>
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<span class="md-nav__icon md-icon"></span>
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2. 复杂度分析
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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_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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<li class="md-nav__item">
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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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<li class="md-nav__item">
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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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</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_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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<li class="md-nav__item">
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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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</li>
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<li class="md-nav__item">
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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>
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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（Linked 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/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（Hash Map）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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6.2. 哈希冲突处理
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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6.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
|
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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">
|
|
|
7.3. 二叉搜索树
|
|
|
</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
|
|
|
</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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|
|
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|
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|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
|
|
8. 堆
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
8. 堆
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
|
8.1. 堆（Heap）
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
8.2. 建堆操作 *
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
|
8.3. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--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">
|
|
|
9. 图
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
|
|
|
<label class="md-nav__title" for="__nav_10">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
9. 图
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
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|
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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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|
|
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|
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|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
9.1. 图（Graph）
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
9.1. 图（Graph）
|
|
|
</a>
|
|
|
|
|
|
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|
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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">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
目录
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#911" class="md-nav__link">
|
|
|
9.1.1. 图常见类型
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#912" class="md-nav__link">
|
|
|
9.1.2. 图常用术语
|
|
|
</a>
|
|
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|
|
</li>
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#913" class="md-nav__link">
|
|
|
9.1.3. 图的表示
|
|
|
</a>
|
|
|
|
|
|
<nav class="md-nav" aria-label="9.1.3. 图的表示">
|
|
|
<ul class="md-nav__list">
|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
邻接矩阵
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
邻接表
|
|
|
</a>
|
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|
|
</li>
|
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</ul>
|
|
|
</nav>
|
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|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#914" class="md-nav__link">
|
|
|
9.1.4. 图常见应用
|
|
|
</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">
|
|
|
<a href="../graph_operations/" class="md-nav__link">
|
|
|
9.2. 图基础操作
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../graph_traversal/" class="md-nav__link">
|
|
|
9.3. 图的遍历
|
|
|
</a>
|
|
|
</li>
|
|
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|
<li class="md-nav__item">
|
|
|
<a href="../summary/" class="md-nav__link">
|
|
|
9.4. 小结
|
|
|
</a>
|
|
|
</li>
|
|
|
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|
|
</ul>
|
|
|
</nav>
|
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|
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|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
|
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
|
|
|
10. 查找算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_11">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
10. 查找算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|
|
10.1. 线性查找
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|
|
10.2. 二分查找
|
|
|
</a>
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
|
|
10.3. 哈希查找
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|
|
10.4. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
|
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
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<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
|
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11.1. 排序简介
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11.2. 冒泡排序
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11.3. 插入排序
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11.4. 快速排序
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11.5. 归并排序
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11.6. 小结
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12. 附录
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12.2. 一起参与创作
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参考文献
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目录
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9.1.1. 图常见类型
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9.1.2. 图常用术语
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9.1.3. 图的表示
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<a href="#_1" class="md-nav__link">
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邻接矩阵
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</a>
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<li class="md-nav__item">
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邻接表
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<a href="#914" class="md-nav__link">
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9.1.4. 图常见应用
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_graph/graph.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="91-graph">9.1. 图（Graph）<a class="headerlink" href="#91-graph" title="Permanent link">¶</a></h1>
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<p>「图 Graph」是一种非线性数据结构，由「顶点 Vertex」和「边 Edge」组成。我们可将图 <span class="arithmatex">\(G\)</span> 抽象地表示为一组顶点 <span class="arithmatex">\(V\)</span> 和一组边 <span class="arithmatex">\(E\)</span> 的集合。例如，以下表示一个包含 5 个顶点和 7 条边的图</p>
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<div class="arithmatex">\[
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\begin{aligned}
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V & = \{ 1, 2, 3, 4, 5 \} \newline
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E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
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G & = \{ V, E \} \newline
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\end{aligned}
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\]</div>
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<p><img alt="链表、树、图之间的关系" src="../graph.assets/linkedlist_tree_graph.png" /></p>
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<p align="center"> Fig. 链表、树、图之间的关系 </p>
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<p>那么，图与其他数据结构的关系是什么？如果我们把「顶点」看作结点，把「边」看作连接各个结点的指针，则可将「图」看成一种从「链表」拓展而来的数据结构。<strong>相比线性关系（链表）和分治关系（树），网络关系（图）的自由度更高，也从而更为复杂</strong>。</p>
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<h2 id="911">9.1.1. 图常见类型<a class="headerlink" href="#911" title="Permanent link">¶</a></h2>
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<p>根据边是否有方向，分为「无向图 Undirected Graph」和「有向图 Directed Graph」。</p>
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<ul>
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<li>在无向图中，边表示两顶点之间“双向”的连接关系，例如微信或 QQ 中的“好友关系”；</li>
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<li>在有向图中，边是有方向的，即 <span class="arithmatex">\(A \rightarrow B\)</span> 和 <span class="arithmatex">\(A \leftarrow B\)</span> 两个方向的边是相互独立的，例如微博或抖音上的“关注”与“被关注”关系；</li>
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</ul>
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<p><img alt="有向图与无向图" src="../graph.assets/directed_graph.png" /></p>
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<p align="center"> Fig. 有向图与无向图 </p>
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<p>根据所有顶点是否连通，分为「连通图 Connected Graph」和「非连通图 Disconnected Graph」。</p>
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<ul>
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<li>对于连通图，从某个顶点出发，可以到达其余任意顶点；</li>
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<li>对于非连通图，从某个顶点出发，至少有一个顶点无法到达；</li>
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</ul>
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<p><img alt="连通图与非连通图" src="../graph.assets/connected_graph.png" /></p>
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<p align="center"> Fig. 连通图与非连通图 </p>
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<p>我们可以给边添加“权重”变量，得到「有权图 Weighted Graph」。例如，在王者荣耀等游戏中，系统会根据共同游戏时间来计算玩家之间的“亲密度”，这种亲密度网络就可以使用有权图来表示。</p>
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<p><img alt="有权图与无权图" src="../graph.assets/weighted_graph.png" /></p>
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<p align="center"> Fig. 有权图与无权图 </p>
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<h2 id="912">9.1.2. 图常用术语<a class="headerlink" href="#912" title="Permanent link">¶</a></h2>
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<ul>
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<li>「邻接 Adjacency」：当两顶点之间有边相连时，称此两顶点“邻接”。例如，上图中顶点 1 的邻接顶点为顶点 2, 3, 5 。</li>
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<li>「路径 Path」：从顶点 A 到顶点 B 走过的边构成的序列，被称为从 A 到 B 的“路径”。例如，上图中 1, 5, 2, 4 是顶点 1 到顶点 4 的一个路径。</li>
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<li>「度 Degree」表示一个顶点具有多少条边。对于有向图，「入度 In-Degree」表示有多少条边指向该顶点，「出度 Out-Degree」表示有多少条边从该顶点指出。</li>
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</ul>
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<h2 id="913">9.1.3. 图的表示<a class="headerlink" href="#913" title="Permanent link">¶</a></h2>
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<p>图的常用表示方法有「邻接矩阵」和「邻接表」。以下使用「无向图」来举例。</p>
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<h3 id="_1">邻接矩阵<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
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<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，「邻接矩阵 Adjacency Matrix」使用一个 <span class="arithmatex">\(n \times n\)</span> 大小的矩阵来表示图，每一行（列）代表一个顶点，矩阵元素代表边，使用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 来表示两个顶点之间有边或无边。</p>
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<p>如下图所示，记邻接矩阵为 <span class="arithmatex">\(M\)</span> 、顶点列表为 <span class="arithmatex">\(V\)</span> ，则矩阵元素 <span class="arithmatex">\(M[i][j] = 1\)</span> 代表着顶点 <span class="arithmatex">\(V[i]\)</span> 到顶点 <span class="arithmatex">\(V[j]\)</span> 之间有边，相反地 <span class="arithmatex">\(M[i][j] = 0\)</span> 代表两顶点之间无边。</p>
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<p><img alt="图的邻接矩阵表示" src="../graph.assets/adjacency_matrix.png" /></p>
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<p align="center"> Fig. 图的邻接矩阵表示 </p>
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<p>邻接矩阵具有以下性质：</p>
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<ul>
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<li>顶点不能与自身相连，因而邻接矩阵主对角线元素没有意义。</li>
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<li>「无向图」两个方向的边等价，此时邻接矩阵关于主对角线对称。</li>
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<li>将邻接矩阵的元素从 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(0\)</span> 替换为权重，则能够表示「有权图」。</li>
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</ul>
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<p>使用邻接矩阵表示图时，我们可以直接通过访问矩阵元素来获取边，因此增删查操作的效率很高，时间复杂度均为 <span class="arithmatex">\(O(1)\)</span> 。然而，矩阵的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，内存占用较大。</p>
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<h3 id="_2">邻接表<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
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<p>「邻接表 Adjacency List」使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表结点表示顶点。第 <span class="arithmatex">\(i\)</span> 条链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（即与该顶点相连的顶点）。</p>
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<p><img alt="图的邻接表表示" src="../graph.assets/adjacency_list.png" /></p>
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<p align="center"> Fig. 图的邻接表表示 </p>
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<p>邻接表仅存储存在的边，而边的总数往往远小于 <span class="arithmatex">\(n^2\)</span> ，因此更加节省空间。但是，因为在邻接表中需要通过遍历链表来查找边，所以其时间效率不如邻接矩阵。</p>
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<p>观察上图发现，<strong>邻接表结构与哈希表「链地址法」非常相似，因此我们也可以用类似方法来优化效率</strong>。比如，当链表较长时，可以把链表转化为 AVL 树或红黑树，从而将时间效率从 <span class="arithmatex">\(O(n)\)</span> 优化至 <span class="arithmatex">\(O(\log n)\)</span> ，还可以通过中序遍历获取有序序列；还可以将链表转化为哈希表，将时间复杂度降低至 <span class="arithmatex">\(O(1)\)</span> 。</p>
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<h2 id="914">9.1.4. 图常见应用<a class="headerlink" href="#914" title="Permanent link">¶</a></h2>
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<p>现实中的许多系统都可以使用图来建模，对应的待求解问题也可以被约化为图计算问题。</p>
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<table>
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<thead>
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<th></th>
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<th>顶点</th>
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<th>边</th>
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<th>图计算问题</th>
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<tr>
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<td>星体</td>
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<td>星体间的万有引力作用</td>
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<td>行星轨道计算</td>
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