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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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2.4. 小结
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3.1. 数据结构分类
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3.1. 数据结构分类
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3.1.1. 逻辑结构：线性与非线性
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3.1.2. 物理结构：连续与离散
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3.3. 数字编码 *
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3.5. 小结
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<label for="__nav_5">
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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_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
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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
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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
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</a>
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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 class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__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--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（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/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/hash_algorithm/" class="md-nav__link">
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6.3. 哈希算法（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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6.4. 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--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--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--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_graph/">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="false">
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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="../../chapter_graph/graph/" class="md-nav__link">
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9.1. 图
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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_graph/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="../../chapter_graph/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="../../chapter_graph/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--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
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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
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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--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 ">
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<a href="../../chapter_sorting/">11. 排序</a>
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<label for="__nav_12">
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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_12_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
|
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11. 排序
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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/sorting_algorithm/" class="md-nav__link">
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11.1. 排序算法
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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_sorting/selection_sort/" class="md-nav__link">
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11.2. 选择排序
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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/bubble_sort/" class="md-nav__link">
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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">
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11.5. 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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11.6. 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/heap_sort/" class="md-nav__link">
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11.7. 堆排序
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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/bucket_sort/" class="md-nav__link">
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11.8. 桶排序
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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/counting_sort/" class="md-nav__link">
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11.9. 计数排序
|
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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/radix_sort/" class="md-nav__link">
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11.10. 基数排序
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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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11.11. 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--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_backtracking/">12. 回溯</a>
|
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|
<label for="__nav_13">
|
|
|
<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_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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12. 回溯
|
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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_backtracking/backtracking_algorithm/" class="md-nav__link">
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|
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12.1. 回溯算法
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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_backtracking/permutations_problem/" class="md-nav__link">
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12.2. 全排列问题
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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_backtracking/subset_sum_problem/" class="md-nav__link">
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12.3. 子集和问题（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_backtracking/n_queens_problem/" class="md-nav__link">
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12.4. N 皇后问题
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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_backtracking/summary/" class="md-nav__link">
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12.5. 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--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_dynamic_programming/">13. 动态规划</a>
|
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|
<label for="__nav_14">
|
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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_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
|
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<span class="md-nav__icon md-icon"></span>
|
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13. 动态规划
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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_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
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13.1. 初探动态规划（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_dynamic_programming/dp_problem_features/" class="md-nav__link">
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13.2. DP 问题特性（New）
|
|
|
</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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<h1 id="31">3.1. 数据结构分类<a class="headerlink" href="#31" title="Permanent link">¶</a></h1>
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<p>数据结构可以从逻辑结构和物理结构两个维度进行分类。</p>
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<h2 id="311">3.1.1. 逻辑结构：线性与非线性<a class="headerlink" href="#311" title="Permanent link">¶</a></h2>
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<p><strong>「逻辑结构」揭示了数据元素之间的逻辑关系</strong>。在数组和链表中，数据按照顺序依次排列，体现了数据之间的线性关系；而在树中，数据从顶部向下按层次排列，表现出祖先与后代之间的派生关系；图则由节点和边构成，反映了复杂的网络关系。</p>
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<p>逻辑结构通常分为「线性」和「非线性」两类。线性结构比较直观，指数据在逻辑关系上呈线性排列；非线性结构则相反，呈非线性排列，例如网状或树状结构。</p>
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<ul>
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<li><strong>线性数据结构</strong>：数组、链表、栈、队列、哈希表；</li>
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<li><strong>非线性数据结构</strong>：树、图、堆、哈希表；</li>
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</ul>
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<p><img alt="线性与非线性数据结构" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></p>
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<p align="center"> Fig. 线性与非线性数据结构 </p>
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<h2 id="312">3.1.2. 物理结构：连续与离散<a class="headerlink" href="#312" title="Permanent link">¶</a></h2>
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<p>在计算机中，内存和硬盘是两种主要的存储硬件设备。「硬盘」主要用于长期存储数据，容量较大（通常可达到 TB 级别）、速度较慢。「内存」用于运行程序时暂存数据，速度较快，但容量较小（通常为 GB 级别）。</p>
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<p><strong>在算法运行过程中，相关数据都存储在内存中</strong>。下图展示了一个计算机内存条，其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格，其中每个单元格都可以存储 1 byte 的数据，在算法运行时，所有数据都被存储在这些单元格中。</p>
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<p><strong>系统通过「内存地址 Memory Location」来访问目标内存位置的数据</strong>。计算机根据特定规则为表格中的每个单元格分配编号，确保每个内存空间都有唯一的内存地址。有了这些地址，程序便可以访问内存中的数据。</p>
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<p><img alt="内存条、内存空间、内存地址" src="../classification_of_data_structure.assets/computer_memory_location.png" /></p>
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<p align="center"> Fig. 内存条、内存空间、内存地址 </p>
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<p>内存是所有程序的共享资源，当内存被某个程序占用时，其他程序无法同时使用。<strong>因此，在数据结构与算法的设计中，内存资源是一个重要的考虑因素</strong>。例如，算法所占用的内存峰值不应超过系统剩余空闲内存；如果运行的程序很多并且缺少大量连续的内存空间，那么所选用的数据结构必须能够存储在离散的内存空间内。</p>
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<p><strong>「物理结构」反映了数据在计算机内存中的存储方式</strong>，可分为数组的连续空间存储和链表的离散空间存储。物理结构从底层决定了数据的访问、更新、增删等操作方法，同时在时间效率和空间效率方面呈现出互补的特点。</p>
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<p><img alt="连续空间存储与离散空间存储" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></p>
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<p align="center"> Fig. 连续空间存储与离散空间存储 </p>
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<p><strong>所有数据结构都是基于数组、链表或二者的组合实现的</strong>。例如，栈和队列既可以使用数组实现，也可以使用链表实现；而哈希表的实现可能同时包含数组和链表。</p>
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<ul>
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<li><strong>基于数组可实现</strong>：栈、队列、哈希表、树、堆、图、矩阵、张量（维度 <span class="arithmatex">\(\geq 3\)</span> 的数组）等；</li>
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<li><strong>基于链表可实现</strong>：栈、队列、哈希表、树、堆、图等；</li>
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</ul>
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<p>基于数组实现的数据结构也被称为「静态数据结构」，这意味着此类数据结构在初始化后长度不可变。相对应地，基于链表实现的数据结构被称为「动态数据结构」，这类数据结构在初始化后，仍可以在程序运行过程中对其长度进行调整。</p>
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<p class="admonition-title">Tip</p>
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<p>如若感觉理解物理结构有困难，建议先阅读下一章“数组与链表”，然后再回头理解物理结构的含义。数组与链表是其他所有数据结构的基石，建议你投入更多时间深入了解这两种基本数据结构。</p>
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