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2. 复杂度分析
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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3. 数据结构简介
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3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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3.5. 小结
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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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
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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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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_stack_and_queue/">5. 栈与队列</a>
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<label for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<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
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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
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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
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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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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_hashing/">6. 散列表</a>
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<label for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<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
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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（New）
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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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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_tree/">7. 树</a>
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<label for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<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. 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
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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/array_representation_of_tree/" class="md-nav__link">
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7.3. 二叉树数组表示
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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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7.4. 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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7.5. 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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7.6. 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_9" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_heap/">8. 堆</a>
|
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<label for="__nav_9">
|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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<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
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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_heap/build_heap/" class="md-nav__link">
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8.2. 建堆操作
|
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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_heap/top_k/" class="md-nav__link">
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8.3. Top-K 问题（New）
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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_heap/summary/" class="md-nav__link">
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8.4. 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--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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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../">9. 图</a>
|
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<label for="__nav_10">
|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
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</div>
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<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 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">
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9.1. 图
|
|
|
<span class="md-nav__icon md-icon"></span>
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</label>
|
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<a href="./" class="md-nav__link md-nav__link--active">
|
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9.1. 图
|
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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>
|
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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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<li class="md-nav__item">
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<a href="#911" class="md-nav__link">
|
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9.1.1. 图常见类型
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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="#912" class="md-nav__link">
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9.1.2. 图常用术语
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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="#913" class="md-nav__link">
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9.1.3. 图的表示
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</a>
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<nav class="md-nav" aria-label="9.1.3. 图的表示">
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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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邻接矩阵
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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">
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邻接表
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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">
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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>
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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="../graph_operations/" class="md-nav__link">
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9.2. 图基础操作
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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="../graph_traversal/" class="md-nav__link">
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9.3. 图的遍历
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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="../summary/" class="md-nav__link">
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9.4. 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_11" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_searching/">10. 搜索</a>
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<label for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<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. 搜索
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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/binary_search/" class="md-nav__link">
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10.1. 二分查找（New）
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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_edge/" class="md-nav__link">
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10.2. 二分查找边界（New）
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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/replace_linear_by_hashing/" class="md-nav__link">
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10.3. 哈希优化策略
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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/searching_algorithm_revisited/" class="md-nav__link">
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10.4. 重识搜索算法
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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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10.5. 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_12" >
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<div class="md-nav__link md-nav__link--index ">
|
|
|
<a href="../../chapter_sorting/">11. 排序</a>
|
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|
<label for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
11. 排序
|
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</label>
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|
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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/sorting_algorithm/" class="md-nav__link">
|
|
|
11.1. 排序算法
|
|
|
</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
|
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11.2. 选择排序（New）
|
|
|
</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">
|
|
|
11.3. 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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11.4. 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">
|
|
|
11.5. 快速排序
|
|
|
</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">
|
|
|
11.6. 归并排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
|
|
11.7. 堆排序（New）
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
|
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|
|
11.8. 桶排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|
|
11.9. 计数排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|
|
11.10. 基数排序
|
|
|
</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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11.11. 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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<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 ">
|
|
|
<a href="../../chapter_backtracking/">12. 回溯</a>
|
|
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|
|
|
<label for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
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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">
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
12. 回溯
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
|
|
|
12.1. 回溯算法
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
|
|
|
12.2. 全排列问题
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
|
|
|
12.3. N 皇后问题
|
|
|
</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_14" >
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
|
|
|
13. 附录
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_14">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
13. 附录
|
|
|
</label>
|
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_appendix/installation/" class="md-nav__link">
|
|
|
13.1. 编程环境安装
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
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13.2. 一起参与创作
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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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邻接矩阵
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邻接表
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9.1.4. 图常见应用
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<h1 id="91">9.1. 图<a class="headerlink" href="#91" 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。我们可以将图 <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。以下示例展示了一个包含 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」看作节点，把「边」看作连接各个节点的指针，则可将「图」看作是一种从「链表」拓展而来的数据结构。<strong>相较于线性关系（链表）和分治关系（树），网络关系（图）的自由度更高，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>根据边是否具有方向，可分为「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，边表示两顶点之间的“双向”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，边具有方向性，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，可分为「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>我们还可以为边添加“权重”变量，从而得到「有权图 Weighted Graph」。例如，在王者荣耀等手游中，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」：当两顶点之间存在边相连时，称这两顶点“邻接”。在上图中，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 经过的边构成的序列被称为从 A 到 B 的“路径”。在上图中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
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|
<li>「度 Degree」表示一个顶点拥有的边数。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>图的常用表示方法包括「邻接矩阵」和「邻接表」。以下使用无向图进行举例。</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，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示两个顶点之间是否存在边。</p>
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<p>如下图所示，设邻接矩阵为 <span class="arithmatex">\(M\)</span> 、顶点列表为 <span class="arithmatex">\(V\)</span> ，那么矩阵元素 <span class="arithmatex">\(M[i][j] = 1\)</span> 表示顶点 <span class="arithmatex">\(V[i]\)</span> 到顶点 <span class="arithmatex">\(V[j]\)</span> 之间存在边，反之 <span class="arithmatex">\(M[i][j] = 0\)</span> 表示两顶点之间无边。</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>邻接矩阵具有以下特性：</p>
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<ul>
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<li>顶点不能与自身相连，因此邻接矩阵主对角线元素没有意义。</li>
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<li>对于无向图，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，则可表示有权图。</li>
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</ul>
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<p>使用邻接矩阵表示图时，我们可以直接访问矩阵元素以获取边，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> ，内存占用较多。</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，链表节点表示顶点。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>邻接表仅存储实际存在的边，而边的总数通常远小于 <span class="arithmatex">\(n^2\)</span> ，因此它更加节省空间。然而，在邻接表中需要通过遍历链表来查找边，因此其时间效率不如邻接矩阵。</p>
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<p>观察上图可发现，<strong>邻接表结构与哈希表中的「链地址法」非常相似，因此我们也可以采用类似方法来优化效率</strong>。例如，当链表较长时，可以将链表转化为 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；此外，还可以将链表转换为哈希表，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>实际应用中，许多系统都可以用图来建模，相应的待求解问题也可以约化为图计算问题。</p>
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<div class="center-table">
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<table>
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<thead>
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<tr>
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<th></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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</thead>
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<tbody>
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<tr>
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<td>社交网络</td>
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<td>用户</td>
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<td>好友关系</td>
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<td>潜在好友推荐</td>
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</tr>
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<tr>
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<td>地铁线路</td>
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<td>站点</td>
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<td>站点间的连通性</td>
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<td>最短路线推荐</td>
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<td>太阳系</td>
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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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</tr>
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</tbody>
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</table>
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</div>
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