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2842 lines
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<meta name="description" content="在动画与代码中掌握数据结构与算法">
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<link rel="canonical" href="https://www.hello-algo.com/chapter_backtracking/subset_sum_problem/">
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<title>12.3. 子集和问题（New） - Hello 算法</title>
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跳转至
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Hello 算法
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<span class="md-ellipsis">
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12.3. 子集和问题（New）
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</span>
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</div>
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<a href="javascript:void(0)" class="md-search__icon md-icon" title="分享" aria-label="分享" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
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<div class="md-search-result__meta">
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正在初始化搜索引擎
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<ol class="md-search-result__list" role="presentation"></ol>
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<div class="md-header__source">
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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</div>
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<div class="md-source__repository">
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krahets/hello-algo
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</div>
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<main class="md-main" data-md-component="main">
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<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
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Hello 算法
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<div class="md-nav__source">
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
|
|||
|
|
3.3. 数字编码 *
|
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|
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</a>
|
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|
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</li>
|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
|
|||
|
|
3.4. 字符编码 *
|
|||
|
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</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
|
|||
|
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3.5. 小结
|
|||
|
|
</a>
|
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</li>
|
|||
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|
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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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<div class="md-nav__link md-nav__link--index ">
|
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|
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<a href="../../chapter_array_and_linkedlist/">4. 数组与链表</a>
|
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|
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|
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<label for="__nav_5">
|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
|
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|
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<label class="md-nav__title" for="__nav_5">
|
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|
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<span class="md-nav__icon md-icon"></span>
|
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|
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4. 数组与链表
|
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</label>
|
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|
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<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
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|
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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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|
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</li>
|
|||
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|
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|
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|
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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">
|
|||
|
|
4.2. 链表
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
|||
|
|
4.3. 列表
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
|||
|
|
4.4. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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</ul>
|
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</nav>
|
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
|
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|
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|
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|
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<div class="md-nav__link md-nav__link--index ">
|
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|
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<a href="../../chapter_stack_and_queue/">5. 栈与队列</a>
|
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|
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|
|
<label for="__nav_6">
|
|||
|
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<span class="md-nav__icon md-icon"></span>
|
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|
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</label>
|
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|
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|
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</div>
|
|||
|
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|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_6">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
5. 栈与队列
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
|||
|
|
5.1. 栈
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
|||
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
|||
|
|
5.2. 队列
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
|||
|
|
5.3. 双向队列
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
|||
|
|
5.4. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
|
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</ul>
|
|||
|
|
</nav>
|
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</li>
|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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|
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|
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|
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|
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|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
|
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|
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|
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|
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|
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<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_hashing/">6. 散列表</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_7">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
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|
|
</label>
|
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|
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|
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|
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</div>
|
|||
|
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|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_7">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
6. 散列表
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
|||
|
|
6.1. 哈希表（New）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
|||
|
|
6.2. 哈希冲突（New）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
|||
|
|
6.3. 哈希算法（New）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
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|
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|
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|
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|
|||
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
|||
|
|
6.4. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
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|
|||
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
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|
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|
|||
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|
|||
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|
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|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
|
|||
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|
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|
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|
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|
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|
|||
|
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<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_tree/">7. 树</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_8">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
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|
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</label>
|
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|
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|
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</div>
|
|||
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|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_8">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
7. 树
|
|||
|
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</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|||
|
|
7.1. 二叉树
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
|||
|
|
7.2. 二叉树遍历
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
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|
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|
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|
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|
|||
|
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|
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|
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|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
|||
|
|
7.3. 二叉树数组表示
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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|
|||
|
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|||
|
|
7.4. 二叉搜索树
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
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|
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|
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|
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|
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|||
|
|
7.5. AVL 树 *
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
|||
|
|
7.6. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
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|
|||
|
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|
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|
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|
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|
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</ul>
|
|||
|
|
</nav>
|
|||
|
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</li>
|
|||
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|
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|
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|
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|
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|
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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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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_heap/">8. 堆</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_9">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_9">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
8. 堆
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
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|
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|
|||
|
|
|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|||
|
|
8.1. 堆
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|||
|
|
8.2. 建堆操作
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
|
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
|||
|
|
8.3. Top-K 问题（New）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|||
|
|
8.4. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_graph/">9. 图</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_10">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<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. 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">
|
|||
|
|
9.1. 图
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
|||
|
|
9.2. 图基础操作
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|||
|
|
9.3. 图的遍历
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|||
|
|
9.4. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|
|||
|
|
|
|||
|
|
|
|||
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|
|
|||
|
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|
|||
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|
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_searching/">10. 搜索</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_11">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<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. 搜索
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|||
|
|
10.1. 二分查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
|||
|
|
10.2. 二分查找边界
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
|||
|
|
10.3. 哈希优化策略
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
|||
|
|
10.4. 重识搜索算法
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|||
|
|
10.5. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
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|
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
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|
|||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_sorting/">11. 排序</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_12">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_12">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
11. 排序
|
|||
|
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</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
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|
|
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|
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|
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|
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|
|||
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
|||
|
|
11.1. 排序算法
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
|||
|
|
11.2. 选择排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|||
|
|
11.3. 冒泡排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
|||
|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|||
|
|
11.4. 插入排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
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|
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|
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|
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|
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|
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|
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|
|||
|
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|
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|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|||
|
|
11.5. 快速排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|||
|
|
11.6. 归并排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
|||
|
|
11.7. 堆排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|||
|
|
11.8. 桶排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|||
|
|
11.9. 计数排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|||
|
|
11.10. 基数排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|||
|
|
11.11. 小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../">12. 回溯</a>
|
|||
|
|
|
|||
|
|
<label for="__nav_13">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
|
|||
|
|
<label class="md-nav__title" for="__nav_13">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
12. 回溯
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../backtracking_algorithm/" class="md-nav__link">
|
|||
|
|
12.1. 回溯算法
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../permutations_problem/" class="md-nav__link">
|
|||
|
|
12.2. 全排列问题
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--active">
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|||
|
|
12.3. 子集和问题（New）
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|||
|
|
12.3. 子集和问题（New）
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__title" for="__toc">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
目录
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#1231" class="md-nav__link">
|
|||
|
|
12.3.1. 参考全排列解法
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#1232" class="md-nav__link">
|
|||
|
|
12.3.2. 重复子集剪枝
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#1233" class="md-nav__link">
|
|||
|
|
12.3.3. 考虑相等元素
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../n_queens_problem/" class="md-nav__link">
|
|||
|
|
12.4. N 皇后问题
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
|
|||
|
|
13. 附录
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_14">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
13. 附录
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_appendix/installation/" class="md-nav__link">
|
|||
|
|
13.1. 编程环境安装
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
|
|||
|
|
13.2. 一起参与创作
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_reference/">参考文献</a>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
|
|||
|
|
<label class="md-nav__title" for="__nav_15">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
参考文献
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
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|
|||
|
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12.3.1. 参考全排列解法
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12.3.2. 重复子集剪枝
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12.3.3. 考虑相等元素
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
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<h1 id="123">12.3. 子集和问题<a class="headerlink" href="#123" title="Permanent link">¶</a></h1>
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<div class="admonition question">
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<p class="admonition-title">Question</p>
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<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。给定数组无重复元素，每个元素可以被选取多次。请以列表形式返回这些组合，列表中不应包含重复组合。</p>
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</div>
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<p>例如，输入集合 <span class="arithmatex">\(\{3, 4, 5\}\)</span> 和目标整数 <span class="arithmatex">\(9\)</span> ，由于集合中的数字可以被重复选取，因此解为 <span class="arithmatex">\(\{3, 3, 3\}, \{4, 5\}\)</span> 。请注意，子集是不区分元素顺序的，例如 <span class="arithmatex">\(\{4, 5\}\)</span> 和 <span class="arithmatex">\(\{5, 4\}\)</span> 是同一个子集。</p>
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<h2 id="1231">12.3.1. 参考全排列解法<a class="headerlink" href="#1231" title="Permanent link">¶</a></h2>
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<p>从回溯算法的角度看，我们可以把子集的生成过程想象成一系列选择的结果，并在选择过程中实时更新“元素和”，当元素和等于 <code>target</code> 时，就将子集记录至结果列表。</p>
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<p>与上节全排列问题不同的是，本题允许重复选取同一元素，因此无需借助 <code>selected</code> 布尔列表来记录元素是否已被选择。我们可以对全排列代码进行小幅修改，初步得到解题代码。</p>
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<div class="highlight"><span class="filename">subset_sum_i_naive.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：子集和 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="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</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="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">// 子集和等于 target 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">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</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="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>
|
|||
|
|
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="c1">// 剪枝：若子集和超过 target ，则跳过该选择</span>
|
|||
|
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</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="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="k">continue</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="p">}</span>
|
|||
|
|
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新元素和 total</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">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
|
|||
|
|
<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">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</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><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<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">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-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </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="p">}</span>
|
|||
|
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<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>
|
|||
|
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<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="cm">/* 求解子集和 I（包含重复子集） */</span>
|
|||
|
|
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></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">subsetSumINaive</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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="w"> </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="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</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="c1">// 子集和</span>
|
|||
|
|
<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="p">();</span><span class="w"> </span><span class="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<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="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</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">res</span><span class="p">);</span>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
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<div class="highlight"><span class="filename">subset_sum_i_naive.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：子集和 I */</span>
|
|||
|
|
<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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="p">,</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="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>
|
|||
|
|
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// 子集和等于 target 时，记录解</span>
|
|||
|
|
<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">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<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>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<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>
|
|||
|
|
<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">size_t</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>
|
|||
|
|
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="c1">// 剪枝：若子集和超过 target ，则跳过该选择</span>
|
|||
|
|
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</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><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新元素和 total</span>
|
|||
|
|
<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">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
|
|||
|
|
<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>
|
|||
|
|
<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">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</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><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
|
|||
|
|
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a>
|
|||
|
|
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="cm">/* 求解子集和 I（包含重复子集） */</span>
|
|||
|
|
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></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">subsetSumINaive</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">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></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><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</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="c1">// 子集和</span>
|
|||
|
|
<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="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><span class="w"> </span><span class="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></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">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</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">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.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>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
|
|||
|
|
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">total</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
|
|||
|
|
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <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>
|
|||
|
|
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span>
|
|||
|
|
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="sd">"""回溯算法：子集和 I"""</span>
|
|||
|
|
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="c1"># 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="k">if</span> <span class="n">total</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></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>
|
|||
|
|
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">return</span>
|
|||
|
|
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="c1"># 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
|
|||
|
|
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="c1"># 剪枝：若子集和超过 target ，则跳过该选择</span>
|
|||
|
|
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="k">if</span> <span class="n">total</span> <span class="o">+</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">></span> <span class="n">target</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="k">continue</span>
|
|||
|
|
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="c1"># 尝试：做出选择，更新元素和 total</span>
|
|||
|
|
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
|||
|
|
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="c1"># 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">total</span> <span class="o">+</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
|
|||
|
|
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a>
|
|||
|
|
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="k">def</span> <span class="nf">subset_sum_i_naive</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="n">target</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>
|
|||
|
|
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w"> </span><span class="sd">"""求解子集和 I（包含重复子集）"""</span>
|
|||
|
|
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">total</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># 子集和</span>
|
|||
|
|
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">total</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a> <span class="k">return</span> <span class="n">res</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.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">backtrackINaive</span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
|
|||
|
|
<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">subsetSumINaive</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.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>
|
|||
|
|
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
|
|||
|
|
<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">subsetSumINaive</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.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>
|
|||
|
|
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
|
|||
|
|
<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">subsetSumINaive</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.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>
|
|||
|
|
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
|
|||
|
|
<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">subsetSumINaive</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i_naive.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_i_naive</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-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_i_naive</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">subsetSumINaive</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">subset_sum_i_naive.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">subsetSumINaive</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">subset_sum_i_naive.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">subsetSumINaive</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">subset_sum_i_naive.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></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-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></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">subsetSumINaive</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>向以上代码输入数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，输出结果为 <span class="arithmatex">\([3, 3, 3], [4, 5], [5, 4]\)</span> 。<strong>虽然成功找出了所有和为 <span class="arithmatex">\(9\)</span> 的子集，但其中存在重复的子集 <span class="arithmatex">\([4, 5]\)</span> 和 <span class="arithmatex">\([5, 4]\)</span></strong> 。这是因为搜索过程是区分选择顺序的，如下图所示，先选 <span class="arithmatex">\(4\)</span> 后选 <span class="arithmatex">\(5\)</span> 与先选 <span class="arithmatex">\(5\)</span> 后选 <span class="arithmatex">\(4\)</span> 是两种不同的情况。</p>
|
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|
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<p><img alt="子集搜索与越界剪枝" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
|
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|
|
<p align="center"> Fig. 子集搜索与越界剪枝 </p>
|
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|
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<h2 id="1232">12.3.2. 重复子集剪枝<a class="headerlink" href="#1232" title="Permanent link">¶</a></h2>
|
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|
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<p>为了去除重复子集，<strong>一种直接的思路是对结果列表进行去重</strong>。但这个方法效率很低，因为：</p>
|
|||
|
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<ul>
|
|||
|
|
<li>当数组元素较多，尤其是当 <code>target</code> 较大时，搜索过程会产生大量的重复子集。</li>
|
|||
|
|
<li>比较子集（数组）的异同是很耗时的，需要先排序数组，再比较数组中每个元素的异同。</li>
|
|||
|
|
</ul>
|
|||
|
|
<p>为了达到最佳效率，<strong>我们希望在搜索过程中通过剪枝进行去重</strong>。观察下图，重复子集是在以不同顺序选择数组元素时产生的，具体来看：</p>
|
|||
|
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<ol>
|
|||
|
|
<li>第一轮和第二轮分别选择 <span class="arithmatex">\(3\)</span> , <span class="arithmatex">\(4\)</span> ，会生成包含这两个元素的所有子集，记为 <span class="arithmatex">\([3, 4, \cdots]\)</span> 。</li>
|
|||
|
|
<li>若第一轮选择 <span class="arithmatex">\(4\)</span> ，<strong>则第二轮应该跳过 <span class="arithmatex">\(3\)</span></strong> ，因为该选择产生的子集 <span class="arithmatex">\([4, 3, \cdots]\)</span> 和 <code>1.</code> 中提到的子集完全重复。</li>
|
|||
|
|
<li>同理，若第一轮选择 <span class="arithmatex">\(5\)</span> ，<strong>则第二轮应该跳过 <span class="arithmatex">\(3\)</span> 和 <span class="arithmatex">\(4\)</span></strong> ，因为子集 <span class="arithmatex">\([5, 3, \cdots]\)</span> 和子集 <span class="arithmatex">\([5, 4, \cdots]\)</span> 和之前的子集重复。</li>
|
|||
|
|
</ol>
|
|||
|
|
<p><img alt="不同选择顺序导致的重复子集" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></p>
|
|||
|
|
<p align="center"> Fig. 不同选择顺序导致的重复子集 </p>
|
|||
|
|
|
|||
|
|
<p>总结来看，给定输入数组 <span class="arithmatex">\([x_1, x_2, \cdots, x_n]\)</span> ，设搜索过程中的选择序列为 <span class="arithmatex">\([x_{i_1}, x_{i_2}, \cdots , x_{i_m}]\)</span> ，则该选择序列需要满足 <span class="arithmatex">\(i_1 \leq i_2 \leq \cdots \leq i_m\)</span> 。<strong>不满足该条件的选择序列都是重复子集</strong>。</p>
|
|||
|
|
<p>为实现该剪枝，我们初始化变量 <code>start</code> ，用于指示遍历起点。<strong>当做出选择 <span class="arithmatex">\(x_{i}\)</span> 后，设定下一轮从索引 <span class="arithmatex">\(i\)</span> 开始遍历</strong>，从而完成子集去重。</p>
|
|||
|
|
<p>除此之外，我们还对代码进行了两项优化。首先，我们在开启搜索前将数组 <code>nums</code> 排序，在搜索过程中，<strong>当子集和超过 <code>target</code> 时直接结束循环</strong>，因为后边的元素更大，其子集和都一定会超过 <code>target</code> 。其次，<strong>我们通过在 <code>target</code> 上执行减法来统计元素和</strong>，当 <code>target</code> 等于 <span class="arithmatex">\(0\)</span> 时记录解，省去了元素和变量 <code>total</code> 。</p>
|
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<div class="tabbed-set tabbed-alternate" data-tabs="2:11"><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" /><input id="__tabbed_2_11" 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><label for="__tabbed_2_11">Dart</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">subset_sum_i.java</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：子集和 I */</span>
|
|||
|
|
<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">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="w"> </span><span class="n">target</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">int</span><span class="w"> </span><span class="n">start</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>
|
|||
|
|
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="c1">// 子集和等于 target 时，记录解</span>
|
|||
|
|
<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">target</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="p">{</span>
|
|||
|
|
<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="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="c1">// 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<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="n">start</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>
|
|||
|
|
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w"> </span><span class="c1">// 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w"> </span><span class="c1">// 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<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="n">target</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="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="p">{</span>
|
|||
|
|
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<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="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
|
|||
|
|
<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>
|
|||
|
|
<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">target</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><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<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>
|
|||
|
|
<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></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>
|
|||
|
|
<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a>
|
|||
|
|
<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="cm">/* 求解子集和 I */</span>
|
|||
|
|
<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></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">subsetSumI</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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w"> </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="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</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="c1">// 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></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="p">();</span><span class="w"> </span><span class="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></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">target</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">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i.cpp</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="cm">/* 回溯算法：子集和 I */</span>
|
|||
|
|
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</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="kt">int</span><span class="w"> </span><span class="n">start</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>
|
|||
|
|
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w"> </span><span class="c1">// 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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="p">{</span>
|
|||
|
|
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></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>
|
|||
|
|
<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w"> </span><span class="c1">// 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></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="n">start</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>
|
|||
|
|
<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w"> </span><span class="c1">// 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w"> </span><span class="c1">// 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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><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="p">{</span>
|
|||
|
|
<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></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">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
|
|||
|
|
<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></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">target</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><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
|
|||
|
|
<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a>
|
|||
|
|
<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="cm">/* 求解子集和 I */</span>
|
|||
|
|
<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></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">subsetSumI</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">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></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><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w"> </span><span class="c1">// 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</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="c1">// 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></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><span class="w"> </span><span class="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></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">target</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">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
|
|||
|
|
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></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">target</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">start</span><span class="p">:</span> <span class="nb">int</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>
|
|||
|
|
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w"> </span><span class="sd">"""回溯算法：子集和 I"""</span>
|
|||
|
|
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a> <span class="c1"># 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a> <span class="k">if</span> <span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></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>
|
|||
|
|
<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a> <span class="k">return</span>
|
|||
|
|
<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a> <span class="c1"># 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a> <span class="c1"># 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
|
|||
|
|
<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a> <span class="c1"># 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a> <span class="c1"># 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a> <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a> <span class="k">break</span>
|
|||
|
|
<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a> <span class="c1"># 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
|||
|
|
<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a> <span class="c1"># 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a> <span class="c1"># 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
|
|||
|
|
<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>
|
|||
|
|
<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a><span class="k">def</span> <span class="nf">subset_sum_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="n">target</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>
|
|||
|
|
<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w"> </span><span class="sd">"""求解子集和 I"""</span>
|
|||
|
|
<a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a> <span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span> <span class="c1"># 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a> <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a> <span class="k">return</span> <span class="n">res</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">subset_sum_i.go</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="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">backtrackI</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="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">subsetSumI</span><span class="p">}</span>
|
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</code></pre></div>
|
|||
|
|
</div>
|
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|
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">subset_sum_i.js</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">subsetSumI</span><span class="p">}</span>
|
|||
|
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</code></pre></div>
|
|||
|
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</div>
|
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|
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<div class="tabbed-block">
|
|||
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<div class="highlight"><span class="filename">subset_sum_i.ts</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="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-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="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">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</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">subset_sum_i.c</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></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-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a>
|
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<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></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">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">subset_sum_i.cs</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_i</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-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a>
|
|||
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<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_i</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">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">subset_sum_i.swift</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="kd">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="n">backtrack</span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a>
|
|||
|
|
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i.zig</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></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>
|
|||
|
|
<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a>
|
|||
|
|
<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></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">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_i.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></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>
|
|||
|
|
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a>
|
|||
|
|
<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></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">subsetSumI</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>如下图所示，为将数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 输入到以上代码后的整体回溯过程。</p>
|
|||
|
|
<p><img alt="子集和 I 回溯过程" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
|
|||
|
|
<p align="center"> Fig. 子集和 I 回溯过程 </p>
|
|||
|
|
|
|||
|
|
<h2 id="1233">12.3.3. 考虑相等元素<a class="headerlink" href="#1233" title="Permanent link">¶</a></h2>
|
|||
|
|
<div class="admonition question">
|
|||
|
|
<p class="admonition-title">Question</p>
|
|||
|
|
<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。<strong>给定数组可能包含重复元素，每个元素只可被选择一次</strong>。请以列表形式返回这些组合，列表中不应包含重复组合。</p>
|
|||
|
|
</div>
|
|||
|
|
<p>相比于上题，<strong>本题的输入数组可能包含重复元素</strong>，这引入了新的问题。例如，给定数组 <span class="arithmatex">\([4, \hat{4}, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，则现有代码的输出结果为 <span class="arithmatex">\([4, 5], [\hat{4}, 5]\)</span> ，也出现了重复子集。<strong>造成这种重复的原因是相等元素在某轮中被多次选择</strong>。如下图所示，第一轮共有三个选择，其中两个都为 <span class="arithmatex">\(4\)</span> ，会产生两个重复的搜索分支，从而输出重复子集；同理，第二轮的两个 <span class="arithmatex">\(4\)</span> 也会产生重复子集。</p>
|
|||
|
|
<p><img alt="相等元素导致的重复子集" src="../subset_sum_problem.assets/subset_sum_ii_repeat.png" /></p>
|
|||
|
|
<p align="center"> Fig. 相等元素导致的重复子集 </p>
|
|||
|
|
|
|||
|
|
<p>为解决此问题，<strong>我们需要限制相等元素在每一轮中只被选择一次</strong>。实现方式比较巧妙：由于数组是已排序的，因此相等元素都是相邻的。利用该特性，在某轮选择中，若当前元素与其左边元素相等，则说明它已经被选择过，因此直接跳过当前元素。</p>
|
|||
|
|
<p>与此同时，<strong>本题规定数组元素只能被选择一次</strong>。幸运的是，我们也可以利用变量 <code>start</code> 来满足该约束：当做出选择 <span class="arithmatex">\(x_{i}\)</span> 后，设定下一轮从索引 <span class="arithmatex">\(i + 1\)</span> 开始向后遍历。这样即能去除重复子集，也能避免重复选择相等元素。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="3:11"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.java</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 回溯算法：子集和 II */</span>
|
|||
|
|
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></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="w"> </span><span class="n">target</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">int</span><span class="w"> </span><span class="n">start</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>
|
|||
|
|
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w"> </span><span class="c1">// 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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="p">{</span>
|
|||
|
|
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></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="p">(</span><span class="n">state</span><span class="p">));</span>
|
|||
|
|
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w"> </span><span class="c1">// 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w"> </span><span class="c1">// 剪枝三：从 start 开始遍历，避免重复选择同一元素</span>
|
|||
|
|
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></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="n">start</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>
|
|||
|
|
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w"> </span><span class="c1">// 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w"> </span><span class="c1">// 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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="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="p">{</span>
|
|||
|
|
<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w"> </span><span class="c1">// 剪枝四：如果该元素与左边元素相等，说明该搜索分支重复，直接跳过</span>
|
|||
|
|
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">start</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="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="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></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">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></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">target</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><span class="w"> </span><span class="n">choices</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="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></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>
|
|||
|
|
<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a>
|
|||
|
|
<a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="cm">/* 求解子集和 II */</span>
|
|||
|
|
<a id="__codelineno-22-31" name="__codelineno-22-31" href="#__codelineno-22-31"></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">subsetSumII</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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w"> </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="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</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="c1">// 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">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="p">();</span><span class="w"> </span><span class="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></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">target</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">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.cpp</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 回溯算法：子集和 II */</span>
|
|||
|
|
<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></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="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</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="kt">int</span><span class="w"> </span><span class="n">start</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>
|
|||
|
|
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w"> </span><span class="c1">// 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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="p">{</span>
|
|||
|
|
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></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>
|
|||
|
|
<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w"> </span><span class="c1">// 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w"> </span><span class="c1">// 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w"> </span><span class="c1">// 剪枝三：从 start 开始遍历，避免重复选择同一元素</span>
|
|||
|
|
<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></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="n">start</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>
|
|||
|
|
<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w"> </span><span class="c1">// 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w"> </span><span class="c1">// 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</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><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="p">{</span>
|
|||
|
|
<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w"> </span><span class="c1">// 剪枝四：如果该元素与左边元素相等，说明该搜索分支重复，直接跳过</span>
|
|||
|
|
<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">start</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><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="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w"> </span><span class="c1">// 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></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">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
|
|||
|
|
<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w"> </span><span class="c1">// 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></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">target</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><span class="w"> </span><span class="n">choices</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="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w"> </span><span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
|
|||
|
|
<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w"> </span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a>
|
|||
|
|
<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="cm">/* 求解子集和 II */</span>
|
|||
|
|
<a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></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">subsetSumII</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">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|||
|
|
<a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></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><span class="w"> </span><span class="c1">// 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w"> </span><span class="c1">// 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-23-34" name="__codelineno-23-34" href="#__codelineno-23-34"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</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="c1">// 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-23-35" name="__codelineno-23-35" href="#__codelineno-23-35"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="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="c1">// 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></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">target</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">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
|
|||
|
|
<a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
|||
|
|
<a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">subset_sum_ii.py</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
|
|||
|
|
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></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">target</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">start</span><span class="p">:</span> <span class="nb">int</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-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="p">):</span>
|
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<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w"> </span><span class="sd">"""回溯算法：子集和 II"""</span>
|
|||
|
|
<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a> <span class="c1"># 子集和等于 target 时，记录解</span>
|
|||
|
|
<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a> <span class="k">if</span> <span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></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>
|
|||
|
|
<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a> <span class="k">return</span>
|
|||
|
|
<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a> <span class="c1"># 遍历所有选择</span>
|
|||
|
|
<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a> <span class="c1"># 剪枝二：从 start 开始遍历，避免生成重复子集</span>
|
|||
|
|
<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a> <span class="c1"># 剪枝三：从 start 开始遍历，避免重复选择同一元素</span>
|
|||
|
|
<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
|
|||
|
|
<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a> <span class="c1"># 剪枝一：若子集和超过 target ，则直接结束循环</span>
|
|||
|
|
<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a> <span class="c1"># 这是因为数组已排序，后边元素更大，子集和一定超过 target</span>
|
|||
|
|
<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a> <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a> <span class="k">break</span>
|
|||
|
|
<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a> <span class="c1"># 剪枝四：如果该元素与左边元素相等，说明该搜索分支重复，直接跳过</span>
|
|||
|
|
<a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a> <span class="k">if</span> <span class="n">i</span> <span class="o">></span> <span class="n">start</span> <span class="ow">and</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
|
|||
|
|
<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a> <span class="k">continue</span>
|
|||
|
|
<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a> <span class="c1"># 尝试：做出选择，更新 target, start</span>
|
|||
|
|
<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a> <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
|||
|
|
<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a> <span class="c1"># 进行下一轮选择</span>
|
|||
|
|
<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a> <span class="c1"># 回退：撤销选择，恢复到之前的状态</span>
|
|||
|
|
<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a> <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
|
|||
|
|
<a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a>
|
|||
|
|
<a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="k">def</span> <span class="nf">subset_sum_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="n">target</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>
|
|||
|
|
<a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w"> </span><span class="sd">"""求解子集和 II"""</span>
|
|||
|
|
<a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 状态（子集）</span>
|
|||
|
|
<a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a> <span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span> <span class="c1"># 对 nums 进行排序</span>
|
|||
|
|
<a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a> <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># 遍历起始点</span>
|
|||
|
|
<a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># 结果列表（子集列表）</span>
|
|||
|
|
<a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a> <span class="k">return</span> <span class="n">res</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.go</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></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">backtrackII</span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a>
|
|||
|
|
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.js</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></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>
|
|||
|
|
<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a>
|
|||
|
|
<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.ts</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></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>
|
|||
|
|
<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a>
|
|||
|
|
<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.c</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></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>
|
|||
|
|
<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a>
|
|||
|
|
<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_ii</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>
|
|||
|
|
<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a>
|
|||
|
|
<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">subset_sum_ii</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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.swift</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
|
|||
|
|
<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a>
|
|||
|
|
<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.zig</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></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>
|
|||
|
|
<a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a>
|
|||
|
|
<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">subset_sum_ii.dart</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></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>
|
|||
|
|
<a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a>
|
|||
|
|
<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></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">subsetSumII</span><span class="p">}</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>下图展示了数组 <span class="arithmatex">\([4, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 的回溯过程，共包含四种剪枝操作。建议你将图示与代码注释相结合，理解整个搜索过程，以及每种剪枝操作是如何工作的。</p>
|
|||
|
|
<p><img alt="子集和 II 回溯过程" src="../subset_sum_problem.assets/subset_sum_ii.png" /></p>
|
|||
|
|
<p align="center"> Fig. 子集和 II 回溯过程 </p>
|
|||
|
|
|
|||
|
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|
|||
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|
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|
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|
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|
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<h2 id="__comments">评论</h2>
|
|||
|
|
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|
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data-repo="krahets/hello-algo"
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data-repo-id="R_kgDOIXtSqw"
|
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data-category="Announcements"
|
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data-category-id="DIC_kwDOIXtSq84CSZk_"
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data-strict="1"
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data-reactions-enabled="1"
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data-emit-metadata="0"
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data-input-position="top"
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data-theme="preferred_color_scheme"
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data-lang="zh-CN"
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async
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|
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