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krahets/hello-algo
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2. 复杂度
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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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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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2.4. 小结
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</a>
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3. 数据结构
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3.1. 数据结构分类
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</a>
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
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3.2. 基本数据类型
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</a>
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3.3. 数字编码 *
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3.4. 字符编码 *
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3.5. 小结
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<div class="md-nav__link md-nav__link--index ">
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<a href="../">4. 数组与链表</a>
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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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4.3. 列表
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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4.4. 小结
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<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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4.4. 小结
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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="#441-q-a" class="md-nav__link">
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4.4.1. Q & A
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</a>
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</li>
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</ul>
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</nav>
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</li>
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</ul>
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</nav>
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</li>
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<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 ">
|
|
|
<a href="../../chapter_graph/">9. 图</a>
|
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|
|
<label for="__nav_10">
|
|
|
<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>
|
|
|
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">
|
|
|
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
|
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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_15" >
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
|
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14. 附录
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<span class="md-nav__icon md-icon"></span>
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</label>
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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="44">4.4. 小结<a class="headerlink" href="#44" title="Permanent link">¶</a></h1>
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<ul>
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<li>数组和链表是两种基本数据结构，分别代表数据在计算机内存中的连续空间存储和离散空间存储方式。两者的优缺点呈现出互补的特性。</li>
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<li>数组支持随机访问、占用内存较少；但插入和删除元素效率低，且初始化后长度不可变。</li>
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<li>链表通过更改指针实现高效的节点插入与删除，且可以灵活调整长度；但节点访问效率低、占用内存较多。常见的链表类型包括单向链表、循环链表、双向链表。</li>
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<li>动态数组，又称列表，是基于数组实现的一种数据结构。它保留了数组的优势，同时可以灵活调整长度。列表的出现极大地提高了数组的易用性，但可能导致部分内存空间浪费。</li>
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<li>下表总结并对比了数组与链表的各项特性与操作效率。</li>
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</ul>
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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>数组</th>
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<th>链表</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>存储方式</td>
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<td>连续内存空间</td>
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<td>离散内存空间</td>
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</tr>
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<tr>
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<td>数据结构长度</td>
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<td>长度不可变</td>
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<td>长度可变</td>
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</tr>
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<tr>
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<td>内存使用率</td>
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<td>占用内存少、缓存局部性好</td>
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<td>占用内存多</td>
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</tr>
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<tr>
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<td>优势操作</td>
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<td>随机访问</td>
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<td>插入、删除</td>
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</tr>
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<tr>
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<td>访问元素</td>
|
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<td><span class="arithmatex">\(O(1)\)</span></td>
|
|
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<td><span class="arithmatex">\(O(N)\)</span></td>
|
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</tr>
|
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<tr>
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<td>添加元素</td>
|
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<td><span class="arithmatex">\(O(N)\)</span></td>
|
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<td><span class="arithmatex">\(O(1)\)</span></td>
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</tr>
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<tr>
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<td>删除元素</td>
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<td><span class="arithmatex">\(O(N)\)</span></td>
|
|
|
<td><span class="arithmatex">\(O(1)\)</span></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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<div class="admonition note">
|
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<p class="admonition-title">缓存局部性</p>
|
|
|
<p>在计算机中，数据读写速度排序是“硬盘 < 内存 < CPU 缓存”。当我们访问数组元素时，计算机不仅会加载它，还会缓存其周围的其他数据，从而借助高速缓存来提升后续操作的执行速度。链表则不然，计算机只能挨个地缓存各个节点，这样的多次“搬运”降低了整体效率。</p>
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</div>
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<h2 id="441-q-a">4.4.1. Q & A<a class="headerlink" href="#441-q-a" title="Permanent link">¶</a></h2>
|
|
|
<div class="admonition question">
|
|
|
<p class="admonition-title">数组存储在栈上和存储在堆上，对时间效率和空间效率是否有影响？</p>
|
|
|
<p>栈内存分配由编译器自动完成，而堆内存由程序员在代码中分配（注意，这里的栈和堆和数据结构中的栈和堆不是同一概念）。</p>
|
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<ol>
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|
<li>栈不灵活，分配的内存大小不可更改；堆相对灵活，可以动态分配内存；</li>
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|
|
<li>栈是一块比较小的内存，容易出现内存不足；堆内存很大，但是由于是动态分配，容易碎片化，管理堆内存的难度更大、成本更高；</li>
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<li>访问栈比访问堆更快，因为栈内存较小、对缓存友好，堆帧分散在很大的空间内，会出现更多的缓存未命中；</li>
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</ol>
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</div>
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<div class="admonition question">
|
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<p class="admonition-title">为什么数组会强调要求相同类型的元素，而在链表中却没有强调同类型呢？</p>
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|
|
<p>链表由结点组成，结点之间由指针连接，各个结点可以存储不同类型的数据，例如 int, double, string, object 等。</p>
|
|
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<p>相对地，数组元素则必须是相同类型的，这样才能通过计算偏移量来获取对应元素位置。例如，如果数组同时包含 int 和 long 两种类型，单个元素分别占用 4 bytes 和 8 bytes ，那么此时就不能用以下公式计算偏移量了，因为数组中包含了两种 <code>elementLength</code> 。</p>
|
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a>// 元素内存地址 = 数组内存地址 + 元素长度 * 元素索引
|
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|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>elementAddr = firtstElementAddr + elementLength * elementIndex
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</code></pre></div>
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</div>
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<div class="admonition question">
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<p class="admonition-title">删除节点后，是否需要把 <code>P.next</code> 设为 <span class="arithmatex">\(\text{None}\)</span> 呢？</p>
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<p>不修改 <code>P.next</code> 也可以。从该链表的角度看，从头结点遍历到尾结点已经遇不到 <code>P</code> 了。这意味着结点 <code>P</code> 已经从链表中删除了，此时结点 <code>P</code> 指向哪里都不会对这条链表产生影响了。</p>
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<p>从垃圾回收的角度看，对于 Java, Python, Go 等拥有自动垃圾回收的语言来说，节点 <code>P</code> 是否被回收取决于是否有仍存在指向它的引用，而不是 <code>P.next</code> 的值。在 C, C++ 等语言中，我们需要手动释放节点内存。</p>
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</div>
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<div class="admonition question">
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<p class="admonition-title">在链表中插入和删除操作的时间复杂度是 <span class="arithmatex">\(O(1)\)</span> 。但是增删之前都需要 <span class="arithmatex">\(O(n)\)</span> 查找元素，那为什么时间复杂度不是 <span class="arithmatex">\(O(n)\)</span> 呢？</p>
|
|
|
<p>如果是先查找元素、再删除元素，确实是 <span class="arithmatex">\(O(n)\)</span> 。然而，链表的 <span class="arithmatex">\(O(1)\)</span> 增删的优势可以在其他应用上得到体现。例如，双向队列适合使用链表实现，我们维护一个指针变量始终指向头结点、尾结点，每次插入与删除操作都是 <span class="arithmatex">\(O(1)\)</span> 。</p>
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</div>
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|
|
<div class="admonition question">
|
|
|
<p class="admonition-title">图片“链表定义与存储方式”中，浅蓝色的存储结点指针是占用一块内存地址吗？还是和结点值各占一半呢？</p>
|
|
|
<p>文中只是一个示意图，只是定性表示。定量的话需要根据具体情况分析：</p>
|
|
|
<ul>
|
|
|
<li>不同类型的结点值占用的空间是不同的，比如 int, long, double, 或者是类的实例等等。</li>
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|
|
<li>指针变量占用的内存空间大小根据所使用的操作系统及编译环境而定，大多为 8 字节或 4 字节。</li>
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|
</ul>
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|
|
</div>
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|
|
<div class="admonition question">
|
|
|
<p class="admonition-title">在列表末尾添加元素是否时时刻刻都为 <span class="arithmatex">\(O(1)\)</span> ？</p>
|
|
|
<p>如果添加元素时超出列表长度，则需要先扩容列表再添加。系统会申请一块新的内存，并将原列表的所有元素搬运过去，这时候时间复杂度就会是 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
|
|
</div>
|
|
|
<div class="admonition question">
|
|
|
<p class="admonition-title">“列表的出现大大提升了数组的实用性，但副作用是会造成部分内存空间浪费”，这里的空间浪费是指额外增加的变量如容量、长度、扩容倍数所占的内存吗？</p>
|
|
|
<p>这里的空间浪费主要有两方面含义：一方面，列表都会设定一个初始长度，我们不一定需要用这么多。另一方面，为了防止频繁扩容，扩容一般都会乘以一个系数，比如 <span class="arithmatex">\(\times 1.5\)</span> 。这样一来，也会出现很多空位，我们通常不能完全填满它们。</p>
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</div>
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