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krahets/hello-algo
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</a>
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0. 写在前面
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<li class="md-nav__item">
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<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
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0.1. 关于本书
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0.3. 小结
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1. 引言
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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1.1. 算法无处不在
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</a>
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1.2. 算法是什么
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<a href="../../chapter_introduction/summary/" class="md-nav__link">
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1.3. 小结
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 复杂度分析
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</label>
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<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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<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. 数据结构简介
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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3. 数据结构简介
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/data_and_memory/" class="md-nav__link">
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3.1. 数据与内存
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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3.2. 数据结构分类
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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3.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
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4. 数组与链表
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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4.1. 数组
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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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</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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</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">
|
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8. 树
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_9">
|
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<span class="md-nav__icon md-icon"></span>
|
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8. 树
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
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8.1. 二叉树
|
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</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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8.2. 二叉树遍历
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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8.3. 二叉树数组表示
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|
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8.4. 二叉搜索树
|
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</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
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8.5. AVL 树 *
|
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|
</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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8.6. 小结
|
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_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>
|
|
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|
|
<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>
|
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|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
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|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
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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">
|
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|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
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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>
|
|
|
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|
|
<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">
|
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|
<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>
|
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</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">
|
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|
|
11.5. 归并排序
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|
|
11.6. 桶排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
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|
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|
|
11.7. 计数排序
|
|
|
</a>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|
|
11.8. 基数排序
|
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<li class="md-nav__item">
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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11.9. 小结
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</a>
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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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</label>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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12.1. 搜索算法（New）
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</a>
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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>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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12.3. 小结
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</a>
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
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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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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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13. 回溯算法
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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="../backtracking_algorithm/" class="md-nav__link">
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13.1. 回溯算法（New）
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</a>
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</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
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13.2. 全排列问题（New）
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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13.2. 全排列问题（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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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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目录
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#1321" class="md-nav__link">
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13.2.1. 无重复的情况
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</a>
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<li class="md-nav__item">
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<a href="#1322" class="md-nav__link">
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13.2.2. 考虑重复的情况
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</a>
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<li class="md-nav__item">
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<a href="#1323" class="md-nav__link">
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13.2.3. 复杂度分析
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</a>
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<a href="../n_queens_problem/" class="md-nav__link">
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13.3. N 皇后问题（New）
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</a>
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</li>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
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14. 附录
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<span class="md-nav__icon md-icon"></span>
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</label>
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<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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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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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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</li>
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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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</li>
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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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<label class="md-nav__title" for="__nav_16">
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<span class="md-nav__icon md-icon"></span>
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参考文献
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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</div>
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</div>
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</div>
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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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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<label class="md-nav__title" for="__toc">
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目录
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#1321" class="md-nav__link">
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13.2.1. 无重复的情况
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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="#1322" class="md-nav__link">
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13.2.2. 考虑重复的情况
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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="#1323" class="md-nav__link">
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13.2.3. 复杂度分析
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</a>
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</li>
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</ul>
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</nav>
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<div class="md-content" data-md-component="content">
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<article class="md-content__inner md-typeset">
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_backtracking/permutations_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="132">13.2. 全排列问题<a class="headerlink" href="#132" title="Permanent link">¶</a></h1>
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<p>全排列问题是回溯算法的一个典型应用。它的定义是在给定一个集合（如一个数组或字符串）的情况下，找出这个集合中元素的所有可能的排列。</p>
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<p>如下表所示，列举了几个示例数组和其对应的所有排列。</p>
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<div class="center-table">
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<table>
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<thead>
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<tr>
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<th align="left">输入数组</th>
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<th align="left">所有排列</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td align="left">[1]</td>
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<td align="left">[1]</td>
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</tr>
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<tr>
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<td align="left">[1, 2]</td>
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<td align="left">[1, 2], [2, 1]</td>
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</tr>
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<tr>
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<td align="left">[1, 2, 3]</td>
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<td align="left">[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]</td>
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</tr>
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</tbody>
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</table>
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</div>
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<h2 id="1321">13.2.1. 无重复的情况<a class="headerlink" href="#1321" title="Permanent link">¶</a></h2>
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<div class="admonition question">
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<p class="admonition-title">输入一个整数数组，数组中不包含重复元素，返回所有可能的排列。</p>
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</div>
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<p><strong>从回溯算法的角度看，我们可以把生成排列的过程想象成一系列选择的结果</strong>。假设输入数组为 <code>[1, 2, 3]</code> ，如果我们先选择 <code>1</code> 、再选择 <code>3</code> 、最后选择 <code>2</code> ，则获得排列 <code>[1, 3, 2]</code> 。回退表示撤销一个选择，之后继续尝试其他选择。</p>
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<p>从回溯算法代码的角度看，候选集合 <code>choices</code> 是输入数组中的所有元素，状态 <code>state</code> 是直至目前已被选择的元素。注意，每个元素只允许被选择一次，<strong>因此在遍历选择时，应当排除已经选择过的元素</strong>。</p>
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<p>如下图所示，我们可以将搜索过程展开成一个递归树，树中的每个节点代表当前状态 <code>state</code> 。从根节点开始，经过三轮选择后到达叶节点，每个叶节点都对应一个排列。</p>
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<p><img alt="全排列的递归树" src="../permutations_problem.assets/permutations_i.png" /></p>
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<p align="center"> Fig. 全排列的递归树 </p>
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<p>想清楚以上信息之后，我们就可以在框架代码中做“完形填空”了。为了缩短代码行数，我们不单独实现框架代码中的各个函数，而是将他们展开在 <code>backtrack()</code> 函数中。</p>
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<div class="highlight"><span class="filename">permutations_i.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：全排列 I */</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="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</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">selected</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">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </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="c1">// 当状态长度等于元素数量时，记录解</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</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="n">res</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="n">Integer</span><span class="o">></span><span class="p">(</span><span class="n">state</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="k">return</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="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="c1">// 遍历所有选择</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="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">choices</span><span class="p">.</span><span class="na">length</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-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</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="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</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-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</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="n">selected</span><span class="o">[</span><span class="n">i</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-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></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">choice</span><span class="p">);</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="c1">// 进行下一轮选择</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="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</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">// 回退：撤销选择，恢复到之前的状态</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="n">selected</span><span class="o">[</span><span class="n">i</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-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</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-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w"> </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="p">}</span>
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<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="p">}</span>
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<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>
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<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="cm">/* 全排列 I */</span>
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<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="nf">permutationsI</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </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">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</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="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></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="n">backtrack</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="n">Integer</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</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">nums</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">res</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="k">return</span><span class="w"> </span><span class="n">res</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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</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">permutations_i.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：全排列 I */</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="n">vector</span><span class="o"><</span><span class="kt">int</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="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">choices</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">selected</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="kt">int</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="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="c1">// 当状态长度等于元素数量时，记录解</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</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="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-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="k">return</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="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="c1">// 遍历所有选择</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="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">choices</span><span class="p">.</span><span class="n">size</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-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</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">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </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="c1">// 尝试：做出选择，更新状态</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="n">selected</span><span class="p">[</span><span class="n">i</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-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</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">// 进行下一轮选择</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">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</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="c1">// 回退：撤销选择，恢复到之前的状态</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="n">selected</span><span class="p">[</span><span class="n">i</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-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</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="p">}</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="p">}</span>
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<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="p">}</span>
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<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a>
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<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="cm">/* 全排列 I */</span>
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<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsI</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">;</span>
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<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></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">selected</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
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<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></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="kt">int</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-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</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="k">return</span><span class="w"> </span><span class="n">res</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="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">permutations_i.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">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <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">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="p">):</span>
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<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="sd">"""回溯算法：全排列 I"""</span>
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<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="c1"># 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</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">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">state</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="k">return</span>
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<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="c1"># 遍历所有选择</span>
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<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
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<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="c1"># 剪枝：不允许重复选择元素</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="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</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="c1"># 尝试：做出选择，更新状态</span>
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<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
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<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
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<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="c1"># 进行下一轮选择</span>
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<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
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<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="c1"># 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
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<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
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<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a>
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<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="k">def</span> <span class="nf">permutations_i</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</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">int</span><span class="p">]]:</span>
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<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w"> </span><span class="sd">"""全排列 I"""</span>
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<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
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<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="p">[],</span> <span class="n">choices</span><span class="o">=</span><span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">),</span> <span class="n">res</span><span class="o">=</span><span class="n">res</span><span class="p">)</span>
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<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></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">permutations_i.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">permutationsI</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">permutations_i.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsI</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">permutations_i.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsI</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">permutations_i.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">permutationsI</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">permutations_i.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：全排列 I */</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="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</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">selected</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">int</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </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="c1">// 当状态长度等于元素数量时，记录解</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">)</span><span class="w"> </span><span class="p">{</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="n">res</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">int</span><span class="o">></span><span class="p">(</span><span class="n">state</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="k">return</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="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="c1">// 遍历所有选择</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="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">choices</span><span class="p">.</span><span class="n">Length</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-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</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="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><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">// 尝试：做出选择，更新状态</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="n">selected</span><span class="p">[</span><span class="n">i</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-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></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">choice</span><span class="p">);</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="c1">// 进行下一轮选择</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="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</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">// 回退：撤销选择，恢复到之前的状态</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="n">selected</span><span class="p">[</span><span class="n">i</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-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</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-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w"> </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="p">}</span>
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<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="p">}</span>
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<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a>
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<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* 全排列 I */</span>
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<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsI</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </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">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">int</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="kt">int</span><span class="o">>></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="n">backtrack</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">int</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</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">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">],</span><span class="w"> </span><span class="n">res</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="k">return</span><span class="w"> </span><span class="n">res</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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</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">permutations_i.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：全排列 I */</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">state</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">selected</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">res</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="nb">Int</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">// 当状态长度等于元素数量时，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">state</span><span class="p">.</span><span class="bp">count</span> <span class="p">==</span> <span class="n">choices</span><span class="p">.</span><span class="bp">count</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">// 遍历所有选择</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="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span> <span class="k">in</span> <span class="n">choices</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</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">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="k">if</span> <span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">{</span>
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<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a> <span class="c1">// 尝试：做出选择，更新状态</span>
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<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
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<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a> <span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</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">// 进行下一轮选择</span>
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<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a> <span class="n">backtrack</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">choices</span><span class="p">:</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">:</span> <span class="p">&</span><span class="n">selected</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>
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<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a> <span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">=</span> <span class="kc">false</span>
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<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a> <span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
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<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a> <span class="p">}</span>
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<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a> <span class="p">}</span>
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<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="p">}</span>
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<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a>
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<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="cm">/* 全排列 I */</span>
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<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="kd">func</span> <span class="nf">permutationsI</span><span class="p">(</span><span class="n">nums</span><span class="p">:</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">Int</span><span class="p">]]</span> <span class="p">{</span>
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<a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a> <span class="kd">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">]</span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a> <span class="kd">var</span> <span class="nv">selected</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">nums</span><span class="p">.</span><span class="bp">count</span><span class="p">)</span>
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<a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a> <span class="kd">var</span> <span class="nv">res</span><span class="p">:</span> <span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a> <span class="n">backtrack</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">choices</span><span class="p">:</span> <span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="p">:</span> <span class="p">&</span><span class="n">selected</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>
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<a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a> <span class="k">return</span> <span class="n">res</span>
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<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></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">permutations_i.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">permutationsI</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>需要重点关注的是，我们引入了一个布尔型数组 <code>selected</code> ，它的长度与输入数组长度相等，其中 <code>selected[i]</code> 表示 <code>choices[i]</code> 是否已被选择。我们利用 <code>selected</code> 避免某个元素被重复选择，从而实现剪枝。</p>
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<p>如下图所示，假设我们第一轮选择 1 ，第二轮选择 3 ，第三轮选择 2 ，则需要在第二轮剪掉元素 1 的分支，在第三轮剪掉元素 1, 3 的分支。<strong>从本质上理解，此剪枝操作可将搜索空间大小从 <span class="arithmatex">\(O(n^n)\)</span> 降低至 <span class="arithmatex">\(O(n!)\)</span></strong> 。</p>
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<p><img alt="全排列剪枝示例" src="../permutations_problem.assets/permutations_i_pruning.png" /></p>
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<p align="center"> Fig. 全排列剪枝示例 </p>
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<h2 id="1322">13.2.2. 考虑重复的情况<a class="headerlink" href="#1322" title="Permanent link">¶</a></h2>
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<div class="admonition question">
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<p class="admonition-title">输入一个整数数组，<strong>数组中可能包含重复元素</strong>，返回所有不重复的排列。</p>
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</div>
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<p>假设输入数组为 <code>[1, 1, 2]</code> 。为了方便区分两个重复的元素 <code>1</code> ，接下来我们将第二个元素记为 <code>1'</code> 。如下图所示，上述方法生成的排列有一半都是重复的。</p>
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<p><img alt="重复排列" src="../permutations_problem.assets/permutations_ii.png" /></p>
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<p align="center"> Fig. 重复排列 </p>
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<p>那么，如何去除重复的排列呢？最直接地，我们可以借助一个哈希表，直接对排列结果进行去重。然而，这样做不够优雅，因为生成重复排列的搜索分支是没有必要的，应当被提前识别并剪枝，这样可以提升算法效率。</p>
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<p>观察发现，在第一轮中，选择 <code>1</code> 或选择 <code>1'</code> 是等价的，因为在这两个选择之下生成的所有排列都是重复的。因此，我们应该把 <code>1'</code> 剪枝掉。同理，在第一轮选择 <code>2</code> 后，第二轮选择中的 <code>1</code> 和 <code>1'</code> 也会产生重复分支，因此也需要将第二轮的 <code>1'</code> 剪枝。</p>
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<p><img alt="重复排列剪枝" src="../permutations_problem.assets/permutations_ii_pruning.png" /></p>
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<p align="center"> Fig. 重复排列剪枝 </p>
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<p>本质上看，<strong>我们的目标是实现在某一轮选择中，多个相等的元素仅被选择一次</strong>。因此，在上一题的代码的基础上，我们考虑在每一轮选择中开启一个哈希表 <code>duplicated</code> ，用于记录该轮中已经尝试过的元素，并将重复元素剪枝。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">permutations_ii.java</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 回溯算法：全排列 II */</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</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">selected</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">Integer</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="c1">// 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></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="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
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<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
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<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w"> </span><span class="n">Set</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="w"> </span><span class="n">duplicated</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">HashSet</span><span class="o"><</span><span class="n">Integer</span><span class="o">></span><span class="p">();</span>
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<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></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">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">choices</span><span class="p">.</span><span class="na">length</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-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</span>
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<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录选择过的元素值</span>
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<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</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-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></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">choice</span><span class="p">);</span>
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<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
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<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</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-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</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-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="p">}</span>
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<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a>
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<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="cm">/* 全排列 II */</span>
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<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="w"> </span><span class="nf">permutationsII</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></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">Integer</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="n">List</span><span class="o"><</span><span class="n">Integer</span><span class="o">>></span><span class="p">();</span>
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<a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w"> </span><span class="n">backtrack</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="n">Integer</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</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">nums</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></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-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></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">permutations_ii.cpp</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 回溯算法：全排列 II */</span>
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<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</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="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">choices</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">selected</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="kt">int</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="p">{</span>
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<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="c1">// 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></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-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
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<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="n">unordered_set</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">duplicated</span><span class="p">;</span>
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<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></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">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">choices</span><span class="p">.</span><span class="n">size</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-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
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<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w"> </span><span class="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">find</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">end</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</span>
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<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录选择过的元素值</span>
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<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</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-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
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<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
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<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</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-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
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<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="p">}</span>
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<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a>
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<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="cm">/* 全排列 II */</span>
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<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsII</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">;</span>
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<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></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">selected</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
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<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></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="kt">int</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-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></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-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></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">permutations_ii.py</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
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<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <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">int</span><span class="p">]]</span>
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<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="p">):</span>
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<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="sd">"""回溯算法：全排列 II"""</span>
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<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a> <span class="c1"># 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
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<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></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">state</span><span class="p">))</span>
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<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a> <span class="k">return</span>
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<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a> <span class="c1"># 遍历所有选择</span>
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<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a> <span class="n">duplicated</span> <span class="o">=</span> <span class="nb">set</span><span class="p">[</span><span class="nb">int</span><span class="p">]()</span>
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<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
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<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a> <span class="c1"># 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">and</span> <span class="n">choice</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">duplicated</span><span class="p">:</span>
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<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a> <span class="c1"># 尝试：做出选择，更新状态</span>
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<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a> <span class="n">duplicated</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span> <span class="c1"># 记录选择过的元素值</span>
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<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
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<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
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<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a> <span class="c1"># 进行下一轮选择</span>
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<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
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<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a> <span class="c1"># 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
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<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
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<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a>
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<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="k">def</span> <span class="nf">permutations_ii</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</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">int</span><span class="p">]]:</span>
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<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w"> </span><span class="sd">"""全排列 II"""</span>
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<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
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<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="o">=</span><span class="p">[],</span> <span class="n">choices</span><span class="o">=</span><span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">),</span> <span class="n">res</span><span class="o">=</span><span class="n">res</span><span class="p">)</span>
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<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></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">permutations_ii.go</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>
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<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">permutationsII</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">permutations_ii.js</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a>
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<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsII</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">permutations_ii.ts</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
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<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a>
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<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">permutationsII</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">permutations_ii.c</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></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-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a>
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<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></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">permutationsII</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">permutations_ii.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 回溯算法：全排列 II */</span>
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<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</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">selected</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">int</span><span class="o">>></span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w"> </span><span class="c1">// 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></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="k">new</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
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<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
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<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w"> </span><span class="n">ISet</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">duplicated</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">HashSet</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="p">();</span>
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<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></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">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">choices</span><span class="p">.</span><span class="n">Length</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-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
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<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w"> </span><span class="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="n">Contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新状态</span>
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<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录选择过的元素值</span>
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<a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</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-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></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">choice</span><span class="p">);</span>
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<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
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<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w"> </span><span class="n">selected</span><span class="p">[</span><span class="n">i</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-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</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-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="p">}</span>
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<a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a>
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<a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="cm">/* 全排列 II */</span>
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<a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">permutationsII</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></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">int</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="kt">int</span><span class="o">>></span><span class="p">();</span>
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<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w"> </span><span class="n">backtrack</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">int</span><span class="o">></span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</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">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
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<a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></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-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></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">permutations_ii.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 回溯算法：全排列 II */</span>
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<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span> <span class="nf">backtrack</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">Int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">selected</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">res</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span> <span class="p">{</span>
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<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a> <span class="c1">// 当状态长度等于元素数量时，记录解</span>
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<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a> <span class="k">if</span> <span class="n">state</span><span class="p">.</span><span class="bp">count</span> <span class="p">==</span> <span class="n">choices</span><span class="p">.</span><span class="bp">count</span> <span class="p">{</span>
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<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></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-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a> <span class="k">return</span>
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<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a> <span class="p">}</span>
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<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a> <span class="c1">// 遍历所有选择</span>
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<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a> <span class="kd">var</span> <span class="nv">duplicated</span><span class="p">:</span> <span class="n">Set</span><span class="p"><</span><span class="nb">Int</span><span class="p">></span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a> <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span> <span class="k">in</span> <span class="n">choices</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</span> <span class="p">{</span>
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<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a> <span class="c1">// 剪枝：不允许重复选择元素 且 不允许重复选择相等元素</span>
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<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a> <span class="k">if</span> <span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="bp">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span> <span class="p">{</span>
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<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a> <span class="c1">// 尝试：做出选择，更新状态</span>
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<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a> <span class="n">duplicated</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span> <span class="c1">// 记录选择过的元素值</span>
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<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
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<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a> <span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
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<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a> <span class="c1">// 进行下一轮选择</span>
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<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a> <span class="n">backtrack</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">choices</span><span class="p">:</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">:</span> <span class="p">&</span><span class="n">selected</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>
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<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a> <span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
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<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">=</span> <span class="kc">false</span>
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<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a> <span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
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<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a> <span class="p">}</span>
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<a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a> <span class="p">}</span>
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<a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="p">}</span>
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<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a>
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<a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="cm">/* 全排列 II */</span>
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<a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="kd">func</span> <span class="nf">permutationsII</span><span class="p">(</span><span class="n">nums</span><span class="p">:</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">Int</span><span class="p">]]</span> <span class="p">{</span>
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<a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a> <span class="kd">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">]</span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a> <span class="kd">var</span> <span class="nv">selected</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">nums</span><span class="p">.</span><span class="bp">count</span><span class="p">)</span>
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<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a> <span class="kd">var</span> <span class="nv">res</span><span class="p">:</span> <span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span> <span class="p">=</span> <span class="p">[]</span>
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<a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a> <span class="n">backtrack</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">choices</span><span class="p">:</span> <span class="n">nums</span><span class="p">,</span> <span class="n">selected</span><span class="p">:</span> <span class="p">&</span><span class="n">selected</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>
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<a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a> <span class="k">return</span> <span class="n">res</span>
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<a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></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">permutations_ii.zig</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></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-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a>
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<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></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">permutationsII</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>注意，虽然 <code>selected</code> 和 <code>duplicated</code> 都起到剪枝的作用，但他们剪掉的是不同的分支：</p>
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<ul>
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<li><strong>剪枝条件一</strong>：整个搜索过程中只有一个 <code>selected</code> 。它记录的是当前状态中包含哪些元素，作用是避免某个元素在 <code>state</code> 中重复出现。</li>
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<li><strong>剪枝条件二</strong>：每轮选择（即每个开启的 <code>backtrack</code> 函数）都包含一个 <code>duplicated</code> 。它记录的是在遍历中哪些元素已被选择过，作用是保证相等元素只被选择一次，以避免产生重复的搜索分支。</li>
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</ul>
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<p>下图展示了两个剪枝条件的生效范围。注意，树中的每个节点代表一个选择，从根节点到叶节点的路径上的各个节点构成一个排列。</p>
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<p><img alt="两种剪枝条件的作用范围" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
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<p align="center"> Fig. 两种剪枝条件的作用范围 </p>
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<h2 id="1323">13.2.3. 复杂度分析<a class="headerlink" href="#1323" title="Permanent link">¶</a></h2>
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<p>假设元素两两之间互不相同，则 <span class="arithmatex">\(n\)</span> 个元素共有 <span class="arithmatex">\(n!\)</span> 种排列（阶乘）；在记录结果时，需要复制长度为 <span class="arithmatex">\(n\)</span> 的列表，使用 <span class="arithmatex">\(O(n)\)</span> 时间。因此，<strong>时间复杂度为 <span class="arithmatex">\(O(n!n)\)</span></strong> 。</p>
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<p>最大递归深度为 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。<code>selected</code> 使用 <span class="arithmatex">\(O(n)\)</span> 空间。同一时刻最多共有 <span class="arithmatex">\(n\)</span> 个 <code>duplicated</code> ，使用 <span class="arithmatex">\(O(n^2)\)</span> 空间。因此，<strong>全排列 I 的空间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，全排列 II 的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span></strong> 。</p>
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