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<div class="md-source__repository">
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krahets/hello-algo
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</div>
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</a>
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</div>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
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0. 写在前面
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_1">
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<span class="md-nav__icon md-icon"></span>
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0. 写在前面
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</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_preface/about_the_book/" class="md-nav__link">
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0.1. 关于本书
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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_preface/suggestions/" class="md-nav__link">
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0.2. 如何使用本书
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_preface/summary/" class="md-nav__link">
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0.3. 小结
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</a>
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</nav>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
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1. 引言
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_2">
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<span class="md-nav__icon md-icon"></span>
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1. 引言
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
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1.1. 算法无处不在
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
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1.2. 算法是什么
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/summary/" class="md-nav__link">
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1.3. 小结
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</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_3" >
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 复杂度分析
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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2. 复杂度分析
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</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_computational_complexity/performance_evaluation/" class="md-nav__link">
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2.1. 算法效率评估
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
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2.2. 时间复杂度
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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2.3. 空间复杂度
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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2.4. 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_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
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3. 数据结构与类型
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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3. 数据结构与类型
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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3.1. 数据结构分类
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/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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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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3.3. 数字编码 *
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
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3.4. 字符编码 *
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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3.5. 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_5" >
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
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4. 数组与链表
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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4.1. 数组
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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>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
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5. 栈与队列
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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5. 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈
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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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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
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6. 二分查找
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
|
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6. 二分查找
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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_binary_search/binary_search/" class="md-nav__link">
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6.1. 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_binary_search/binary_search_edge/" class="md-nav__link">
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6.2. 二分查找边界
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
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7. 散列表
|
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_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. 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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|
7.1. 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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7.2. 哈希冲突处理
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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7.3. 小结
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|
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
|
|
8. 树
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
8. 树
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
8.1. 二叉树
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
|
|
8.2. 二叉树遍历
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
|
|
8.3. 二叉树数组表示
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|
|
8.4. 二叉搜索树
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
|
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|
|
8.5. AVL 树 *
|
|
|
</a>
|
|
|
</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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8.6. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
|
|
|
9. 堆
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_10">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
9. 堆
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
|
9.1. 堆
|
|
|
</a>
|
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
9.2. 建堆操作 *
|
|
|
</a>
|
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
|
9.3. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
|
|
|
10. 图
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_11">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
10. 图
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
|
|
10.1. 图
|
|
|
</a>
|
|
|
</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
|
|
10.2. 图基础操作
|
|
|
</a>
|
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|
|
10.3. 图的遍历
|
|
|
</a>
|
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|
|
10.4. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
|
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
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<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
|
|
|
11. 排序算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<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. 排序算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
|
|
11.1. 排序算法
|
|
|
</a>
|
|
|
</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|
|
11.2. 冒泡排序
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
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|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|
|
11.3. 插入排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|
|
11.4. 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">
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11.5. 归并排序
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
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11.6. 桶排序
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
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11.7. 计数排序
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
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11.8. 基数排序
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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11.9. 小结
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
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12. 搜索算法
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_13">
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<span class="md-nav__icon md-icon"></span>
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12. 搜索算法
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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12.1. 搜索算法（New）
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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12.2. 哈希优化策略
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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12.3. 小结
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13. 回溯算法
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
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<span class="md-nav__icon md-icon"></span>
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13. 回溯算法
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<a href="../backtracking_algorithm/" class="md-nav__link">
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13.1. 回溯算法（New）
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<a href="../permutations_problem/" class="md-nav__link">
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13.2. 全排列问题（New）
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13.3. N 皇后问题（New）
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<span class="md-nav__icon md-icon"></span>
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<a href="./" class="md-nav__link md-nav__link--active">
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13.3. N 皇后问题（New）
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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目录
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<a href="#1331" class="md-nav__link">
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13.3.1. 复杂度分析
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</a>
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_15">
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<span class="md-nav__icon md-icon"></span>
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14. 附录
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</label>
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<a href="../../chapter_appendix/installation/" class="md-nav__link">
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14.1. 编程环境安装
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</a>
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
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14.2. 一起参与创作
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</a>
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_reference/">参考文献</a>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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参考文献
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<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
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<div class="md-sidebar__scrollwrap">
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<div class="md-sidebar__inner">
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目录
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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="#1331" class="md-nav__link">
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13.3.1. 复杂度分析
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</a>
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<div class="md-content" data-md-component="content">
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_backtracking/n_queens_problem.md" title="编辑此页" class="md-content__button md-icon">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
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</a>
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<h1 id="133-n">13.3. N 皇后问题<a class="headerlink" href="#133-n" title="Permanent link">¶</a></h1>
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<div class="admonition question">
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<p class="admonition-title">根据国际象棋的规则，皇后可以攻击与之处在同一行或同一列或同一斜线上的棋子。给定 <span class="arithmatex">\(n\)</span> 个皇后和一个 <span class="arithmatex">\(n \times n\)</span> 大小的棋盘，寻找使得所有皇后之间无法相互攻击的摆放方案。</p>
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</div>
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<p>如下图所示，当 <span class="arithmatex">\(n = 4\)</span> 时，共可以找到两个解。从回溯算法的角度看，<span class="arithmatex">\(n \times n\)</span> 大小的棋盘共有 <span class="arithmatex">\(n^2\)</span> 个格子，给出了所有的选择 <code>choices</code> 。在逐个放置皇后的过程中，棋盘状态在不断地变化，每个时刻的棋盘就是状态 <code>state</code> 。</p>
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<p><img alt="4 皇后问题的解" src="../n_queens_problem.assets/solution_4_queens.png" /></p>
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<p align="center"> Fig. 4 皇后问题的解 </p>
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<p>本题共有三个约束条件：<strong>多个皇后不能在同一行、同一列和同一对角线</strong>。值得注意的是，对角线分为主对角线 <code>\</code> 和副对角线 <code>/</code> 两种。</p>
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<p><img alt="n 皇后问题的约束条件" src="../n_queens_problem.assets/n_queens_constraints.png" /></p>
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<p align="center"> Fig. n 皇后问题的约束条件 </p>
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<p>皇后的数量和棋盘的行数都为 <span class="arithmatex">\(n\)</span> ，因此我们容易得到第一个推论：<strong>棋盘每行都允许且只允许放置一个皇后</strong>。这意味着，我们可以采取逐行放置策略：从第一行开始，在每行放置一个皇后，直至最后一行结束。<strong>此策略起到了剪枝的作用</strong>，它避免了同一行出现多个皇后的所有搜索分支。</p>
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<p>下图展示了 <span class="arithmatex">\(4\)</span> 皇后问题的逐行放置过程。受篇幅限制，下图仅展开了第一行的一个搜索分支。在搜索过程中，我们将不满足列约束和对角线约束的方案都剪枝了。</p>
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<p><img alt="逐行放置策略" src="../n_queens_problem.assets/n_queens_placing.png" /></p>
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<p align="center"> Fig. 逐行放置策略 </p>
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<p>为了实现根据列约束剪枝，我们可以利用一个长度为 <span class="arithmatex">\(n\)</span> 的布尔型数组 <code>cols</code> 记录每一列是否有皇后。在每次决定放置前，我们通过 <code>cols</code> 将已有皇后的列剪枝，并在回溯中动态更新 <code>cols</code> 的状态。</p>
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<p>那么，如何处理对角线约束呢？设棋盘中某个格子的行列索引为 <code>(row, col)</code> ，观察矩阵的某条主对角线，<strong>我们发现该对角线上所有格子的行索引减列索引相等</strong>，即 <code>row - col</code> 为恒定值。换句话说，若两个格子满足 <code>row1 - col1 == row2 - col2</code> ，则这两个格子一定处在一条主对角线上。</p>
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<p>利用该性质，我们可以借助一个数组 <code>diag1</code> 来记录每条主对角线上是否有皇后。注意，<span class="arithmatex">\(n\)</span> 维方阵 <code>row - col</code> 的范围是 <span class="arithmatex">\([-n + 1, n - 1]\)</span> ，因此共有 <span class="arithmatex">\(2n - 1\)</span> 条主对角线。</p>
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<p><img alt="处理列约束和对角线约束" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></p>
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<p align="center"> Fig. 处理列约束和对角线约束 </p>
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<p>同理，<strong>次对角线上的所有格子的 <code>row + col</code> 是恒定值</strong>。我们可以使用同样的方法，借助数组 <code>diag2</code> 来处理次对角线约束。</p>
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<p>根据以上分析，我们便可以写出 <span class="arithmatex">\(n\)</span> 皇后的解题代码。</p>
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<div class="highlight"><span class="filename">n_queens.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="n">copyState</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">(</span><span class="n">sRow</span><span class="p">));</span>
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<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
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<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"Q"</span><span class="p">);</span>
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<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="w"> </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
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<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">);</span>
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<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="w"> </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
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<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="p">}</span>
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<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>
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<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="cm">/* 求解 N 皇后 */</span>
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<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="s">"#"</span><span class="p">);</span>
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<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
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<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
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<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a>
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<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a>
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<a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>></span><span class="w"> </span><span class="o">&</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="o">&</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">cols</span><span class="p">,</span>
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<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
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<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
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<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">;</span>
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<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
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<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">;</span>
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<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
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<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
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<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
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<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* 求解 N 皇后 */</span>
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<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">></span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">));</span>
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<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">cols</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">diags1</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">diags2</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a>
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<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a>
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<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a> <span class="n">row</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
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<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]],</span>
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<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]],</span>
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<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="n">cols</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
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<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="n">diags1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
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<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">diags2</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
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<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="p">):</span>
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<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="sd">"""回溯算法：N 皇后"""</span>
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<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="c1"># 当放置完所有行时，记录解</span>
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<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">if</span> <span class="n">row</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
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<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">state</span><span class="p">])</span>
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<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="k">return</span>
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<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="c1"># 遍历所有列</span>
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<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="c1"># 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="n">diag1</span> <span class="o">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
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<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">diag2</span> <span class="o">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
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<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="c1"># 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="ow">or</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="ow">or</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]):</span>
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<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"Q"</span>
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<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
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<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="c1"># 放置下一行</span>
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<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
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<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="c1"># 回退：将该格子恢复为空位</span>
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<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"#"</span>
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<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
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<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
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<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="k">def</span> <span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]]:</span>
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<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w"> </span><span class="sd">"""求解 N 皇后"""</span>
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<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a> <span class="c1"># 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[[</span><span class="s2">"#"</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
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<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a> <span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span> <span class="c1"># 记录列是否有皇后</span>
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<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a> <span class="n">diags1</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># 记录主对角线是否有皇后</span>
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<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a> <span class="n">diags2</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># 记录副对角线是否有皇后</span>
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<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
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<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a> <span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
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<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a>
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<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a> <span class="k">return</span> <span class="n">res</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">bool</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w"> </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">))</span>
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<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">_</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">newState</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w"> </span><span class="nx">newState</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">((</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="mi">0</span><span class="p">]))</span>
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<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w"> </span><span class="nb">copy</span><span class="p">(</span><span class="nx">newState</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">i</span><span class="p">])</span>
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<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a>
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<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
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<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
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<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span>
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<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!((</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"Q"</span>
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<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span>
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<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">)</span>
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<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"#"</span>
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<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span>
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<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="p">}</span>
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<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a>
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<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">bool</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="w"> </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">))</span>
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<a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">_</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">newState</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a><span class="w"> </span><span class="nx">newState</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">((</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="mi">0</span><span class="p">]))</span>
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<a id="__codelineno-3-39" name="__codelineno-3-39" href="#__codelineno-3-39"></a><span class="w"> </span><span class="nb">copy</span><span class="p">(</span><span class="nx">newState</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">i</span><span class="p">])</span>
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<a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a>
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<a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="w"> </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
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<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
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<a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span>
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<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!((</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-3-52" name="__codelineno-3-52" href="#__codelineno-3-52"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"Q"</span>
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<a id="__codelineno-3-53" name="__codelineno-3-53" href="#__codelineno-3-53"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span>
|
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<a id="__codelineno-3-54" name="__codelineno-3-54" href="#__codelineno-3-54"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-3-55" name="__codelineno-3-55" href="#__codelineno-3-55"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">)</span>
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<a id="__codelineno-3-56" name="__codelineno-3-56" href="#__codelineno-3-56"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-3-57" name="__codelineno-3-57" href="#__codelineno-3-57"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"#"</span>
|
|
|
<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span>
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<a id="__codelineno-3-59" name="__codelineno-3-59" href="#__codelineno-3-59"></a><span class="w"> </span><span class="p">}</span>
|
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<a id="__codelineno-3-60" name="__codelineno-3-60" href="#__codelineno-3-60"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-61" name="__codelineno-3-61" href="#__codelineno-3-61"></a><span class="p">}</span>
|
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<a id="__codelineno-3-62" name="__codelineno-3-62" href="#__codelineno-3-62"></a>
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<a id="__codelineno-3-63" name="__codelineno-3-63" href="#__codelineno-3-63"></a><span class="kd">func</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[][][]</span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-64" name="__codelineno-3-64" href="#__codelineno-3-64"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-3-65" name="__codelineno-3-65" href="#__codelineno-3-65"></a><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
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<a id="__codelineno-3-66" name="__codelineno-3-66" href="#__codelineno-3-66"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-67" name="__codelineno-3-67" href="#__codelineno-3-67"></a><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
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<a id="__codelineno-3-68" name="__codelineno-3-68" href="#__codelineno-3-68"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-69" name="__codelineno-3-69" href="#__codelineno-3-69"></a><span class="w"> </span><span class="nx">row</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"#"</span>
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<a id="__codelineno-3-70" name="__codelineno-3-70" href="#__codelineno-3-70"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-71" name="__codelineno-3-71" href="#__codelineno-3-71"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">row</span>
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<a id="__codelineno-3-72" name="__codelineno-3-72" href="#__codelineno-3-72"></a><span class="w"> </span><span class="p">}</span>
|
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|
<a id="__codelineno-3-73" name="__codelineno-3-73" href="#__codelineno-3-73"></a><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-3-74" name="__codelineno-3-74" href="#__codelineno-3-74"></a><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
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<a id="__codelineno-3-75" name="__codelineno-3-75" href="#__codelineno-3-75"></a><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-3-76" name="__codelineno-3-76" href="#__codelineno-3-76"></a><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-3-77" name="__codelineno-3-77" href="#__codelineno-3-77"></a><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
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|
|
<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="nx">diags2</span><span class="p">)</span>
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|
<a id="__codelineno-3-79" name="__codelineno-3-79" href="#__codelineno-3-79"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
|
|
|
<a id="__codelineno-3-80" name="__codelineno-3-80" href="#__codelineno-3-80"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">row</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">row</span><span class="p">.</span><span class="nx">slice</span><span class="p">()));</span>
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<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
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<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
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<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'Q'</span><span class="p">;</span>
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<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
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<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
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<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'#'</span><span class="p">;</span>
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<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
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<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="p">}</span>
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<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a>
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<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="cm">/* 求解 N 皇后 */</span>
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<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="s1">'#'</span><span class="p">));</span>
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<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
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<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a>
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<a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
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<a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
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<a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w"> </span><span class="nx">row</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
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<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w"> </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
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<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w"> </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][],</span>
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<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][][],</span>
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<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="nx">cols</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[],</span>
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<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="nx">diags1</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[],</span>
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<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="nx">diags2</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[]</span>
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<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">row</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nx">row</span><span class="p">.</span><span class="nx">slice</span><span class="p">()));</span>
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<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
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<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
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<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'Q'</span><span class="p">;</span>
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<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
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<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
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<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'#'</span><span class="p">;</span>
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<a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
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<a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="p">}</span>
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<a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a>
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<a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="cm">/* 求解 N 皇后 */</span>
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<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][][]</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="s1">'#'</span><span class="p">));</span>
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<a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-5-41" name="__codelineno-5-41" href="#__codelineno-5-41"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][][]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
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<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a>
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<a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
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<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
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<a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
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<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
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<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">nQueens</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>></span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>></span><span class="p">();</span>
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<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w"> </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">></span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w"> </span><span class="n">copyState</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">></span><span class="p">(</span><span class="n">sRow</span><span class="p">));</span>
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<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
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<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
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<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w"> </span><span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
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<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
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<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w"> </span><span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w"> </span><span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">;</span>
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<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span>
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<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
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<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">;</span>
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<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span>
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<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="p">}</span>
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<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a>
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<a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="cm">/* 求解 N 皇后 */</span>
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<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>>></span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w"> </span><span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>></span><span class="p">();</span>
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<a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">></span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">></span><span class="p">();</span>
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<a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="s">"#"</span><span class="p">);</span>
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<a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
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<a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-7-45" name="__codelineno-7-45" href="#__codelineno-7-45"></a><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-7-46" name="__codelineno-7-46" href="#__codelineno-7-46"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">string</span><span class="o">>>></span><span class="p">();</span>
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<a id="__codelineno-7-47" name="__codelineno-7-47" href="#__codelineno-7-47"></a>
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<a id="__codelineno-7-48" name="__codelineno-7-48" href="#__codelineno-7-48"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
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<a id="__codelineno-7-49" name="__codelineno-7-49" href="#__codelineno-7-49"></a>
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<a id="__codelineno-7-50" name="__codelineno-7-50" href="#__codelineno-7-50"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-7-51" name="__codelineno-7-51" href="#__codelineno-7-51"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">func</span> <span class="nf">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span> <span class="nb">Int</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">Int</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="nb">String</span><span class="p">]],</span> <span class="n">res</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[[</span><span class="nb">String</span><span class="p">]]],</span> <span class="n">cols</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Bool</span><span class="p">],</span> <span class="n">diags1</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Bool</span><span class="p">],</span> <span class="n">diags2</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Bool</span><span class="p">])</span> <span class="p">{</span>
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="c1">// 当放置完所有行时，记录解</span>
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<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="k">if</span> <span class="n">row</span> <span class="p">==</span> <span class="n">n</span> <span class="p">{</span>
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<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
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<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="k">return</span>
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<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="p">}</span>
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<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="c1">// 遍历所有列</span>
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<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a> <span class="k">for</span> <span class="n">col</span> <span class="k">in</span> <span class="mi">0</span> <span class="p">..</span><span class="o"><</span> <span class="n">n</span> <span class="p">{</span>
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<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="c1">// 计算该格子对应的主对角线和副对角线</span>
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<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="kd">let</span> <span class="nv">diag1</span> <span class="p">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
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<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a> <span class="kd">let</span> <span class="nv">diag2</span> <span class="p">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
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<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <span class="c1">// 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
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<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a> <span class="k">if</span> <span class="o">!</span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">||</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">||</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span> <span class="p">{</span>
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<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a> <span class="c1">// 尝试：将皇后放置在该格子</span>
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<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="s">"Q"</span>
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<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
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<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
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<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
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<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a> <span class="c1">// 放置下一行</span>
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<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span> <span class="n">row</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="p">&</span><span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="p">&</span><span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">:</span> <span class="p">&</span><span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">:</span> <span class="p">&</span><span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">:</span> <span class="p">&</span><span class="n">diags2</span><span class="p">)</span>
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<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a> <span class="c1">// 回退：将该格子恢复为空位</span>
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<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="s">"#"</span>
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<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="kc">false</span>
|
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<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="p">=</span> <span class="kc">false</span>
|
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<a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="p">=</span> <span class="kc">false</span>
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|
<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a> <span class="p">}</span>
|
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<a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a> <span class="p">}</span>
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<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="p">}</span>
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<a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a>
|
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<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="cm">/* 求解 N 皇后 */</span>
|
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<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="kd">func</span> <span class="nf">nQueens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-></span> <span class="p">[[[</span><span class="nb">String</span><span class="p">]]]</span> <span class="p">{</span>
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<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a> <span class="c1">// 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位</span>
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<a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a> <span class="kd">var</span> <span class="nv">state</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="s">"#"</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="n">n</span><span class="p">),</span> <span class="bp">count</span><span class="p">:</span> <span class="n">n</span><span class="p">)</span>
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<a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a> <span class="kd">var</span> <span class="nv">cols</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="kc">false</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="n">n</span><span class="p">)</span> <span class="c1">// 记录列是否有皇后</span>
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<a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a> <span class="kd">var</span> <span class="nv">diags1</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="kc">false</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1">// 记录主对角线是否有皇后</span>
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<a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a> <span class="kd">var</span> <span class="nv">diags2</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="kc">false</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1">// 记录副对角线是否有皇后</span>
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<a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a> <span class="kd">var</span> <span class="nv">res</span><span class="p">:</span> <span class="p">[[[</span><span class="nb">String</span><span class="p">]]]</span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-8-39" name="__codelineno-8-39" href="#__codelineno-8-39"></a>
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<a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="p">&</span><span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="p">&</span><span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">:</span> <span class="p">&</span><span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">:</span> <span class="p">&</span><span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">:</span> <span class="p">&</span><span class="n">diags2</span><span class="p">)</span>
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<a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a>
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<a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a> <span class="k">return</span> <span class="n">res</span>
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<a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">n_queens.zig</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
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<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
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<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">nQueens</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<h2 id="1331">13.3.1. 复杂度分析<a class="headerlink" href="#1331" title="Permanent link">¶</a></h2>
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<p>逐行放置 <span class="arithmatex">\(n\)</span> 次，考虑列约束，则从第一行到最后一行分别有 <span class="arithmatex">\(n, n-1, \cdots, 2, 1\)</span> 个选择，<strong>因此时间复杂度为 <span class="arithmatex">\(O(n!)\)</span></strong> 。实际上，根据对角线约束的剪枝也能够大幅地缩小搜索空间，因而搜索效率往往优于以上时间复杂度。</p>
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<p><code>state</code> 使用 <span class="arithmatex">\(O(n^2)\)</span> 空间，<code>cols</code> , <code>diags1</code> , <code>diags2</code> 皆使用 <span class="arithmatex">\(O(n)\)</span> 空间。最大递归深度为 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。因此，<strong>空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span></strong> 。</p>
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