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<div class="md-source__repository">
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krahets/hello-algo
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</a>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
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<span class="md-ellipsis">
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0. 前言
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0. 前言
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<span class="md-ellipsis">
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0.1. 关于本书
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</span>
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</a>
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</li>
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<a href="../../chapter_preface/suggestions/" class="md-nav__link">
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<span class="md-ellipsis">
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0.2. 如何使用本书
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</span>
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</a>
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</li>
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<a href="../../chapter_preface/summary/" class="md-nav__link">
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0.3. 小结
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</span>
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</a>
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</li>
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</nav>
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<a href="../../chapter_introduction/" class="md-nav__link ">
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<span class="md-ellipsis">
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1. 初识算法
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</span>
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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1. 初识算法
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<span class="md-ellipsis">
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1.1. 算法无处不在
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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1.2. 算法是什么
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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1.3. 小结
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</span>
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</a>
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</li>
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</nav>
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</li>
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<a href="../../chapter_computational_complexity/" class="md-nav__link ">
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<span class="md-ellipsis">
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2. 复杂度
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</span>
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</a>
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<label class="md-nav__link " for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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2. 复杂度
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
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<span class="md-ellipsis">
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2.1. 算法效率评估
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.2. 时间复杂度
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.3. 空间复杂度
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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2.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<a href="../../chapter_data_structure/" class="md-nav__link ">
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<span class="md-ellipsis">
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3. 数据结构
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</span>
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</a>
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<label class="md-nav__link " for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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3. 数据结构
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1. 数据结构分类
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
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<span class="md-ellipsis">
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3.2. 基本数据类型
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.3. 数字编码 *
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.4. 字符编码 *
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</span>
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</a>
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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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<span class="md-ellipsis">
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3.5. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
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<span class="md-ellipsis">
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4. 数组与链表
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</span>
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</a>
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<label class="md-nav__link " for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1. 数组
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2. 链表
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3. 列表
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
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<span class="md-ellipsis">
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5. 栈与队列
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</span>
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</a>
|
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<label class="md-nav__link " for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
|
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5. 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1. 栈
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2. 队列
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3. 双向队列
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_hashing/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
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<span class="md-ellipsis">
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6. 散列表
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</span>
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</a>
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<label class="md-nav__link " for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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6. 散列表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1. 哈希表
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2. 哈希冲突
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3. 哈希算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_tree/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
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<span class="md-ellipsis">
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7. 树
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</span>
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</a>
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<label class="md-nav__link " for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
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7. 树
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1. 二叉树
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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7.2. 二叉树遍历
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.3. 二叉树数组表示
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.4. 二叉搜索树
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.5. AVL 树 *
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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7.6. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_heap/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
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<span class="md-ellipsis">
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8. 堆
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</span>
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</a>
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<label class="md-nav__link " for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
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8. 堆
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_heap/heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.1. 堆
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.2. 建堆操作
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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<span class="md-ellipsis">
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8.3. Top-K 问题
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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8.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_graph/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
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<span class="md-ellipsis">
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9. 图
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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9. 图
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1. 图
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2. 图基础操作
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3. 图的遍历
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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9.4. 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_searching/" class="md-nav__link ">
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|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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10. 搜索
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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10. 搜索
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
|
|
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10.1. 二分查找
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2. 二分查找插入点
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3. 二分查找边界
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4. 哈希优化策略
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
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|
|
<span class="md-ellipsis">
|
|
|
10.5. 重识搜索算法
|
|
|
</span>
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</a>
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
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|
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<span class="md-ellipsis">
|
|
|
10.6. 小结
|
|
|
</span>
|
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|
</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
|
|
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|
|
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|
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|
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|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
|
<a href="../../chapter_sorting/" class="md-nav__link ">
|
|
|
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
|
|
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<span class="md-ellipsis">
|
|
|
11. 排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-nav__link " 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
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
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|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
|
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|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.1. 排序算法
|
|
|
</span>
|
|
|
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|
|
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
|
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|
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|
|
<span class="md-ellipsis">
|
|
|
11.2. 选择排序
|
|
|
</span>
|
|
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|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|
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|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.3. 冒泡排序
|
|
|
</span>
|
|
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|
|
</a>
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.4. 插入排序
|
|
|
</span>
|
|
|
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|
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|
|
|
</a>
|
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</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.5. 快速排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
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</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.6. 归并排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
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</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.7. 堆排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
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</li>
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.8. 桶排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
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</li>
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.9. 计数排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
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</li>
|
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|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.10. 基数排序
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
|
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</li>
|
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|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
11.11. 小结
|
|
|
</span>
|
|
|
|
|
|
|
|
|
</a>
|
|
|
</li>
|
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|
|
|
|
</ul>
|
|
|
</nav>
|
|
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|
</li>
|
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|
<li class="md-nav__item md-nav__item--nested">
|
|
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|
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|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__container">
|
|
|
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
|
|
|
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
12. 分治
|
|
|
</span>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<span class="md-status md-status--new" title="最近添加">
|
|
|
</span>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-nav__link " 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="false">
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
12. 分治
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
12.1. 分治算法
|
|
|
</span>
|
|
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|
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|
|
<span class="md-status md-status--new" title="最近添加">
|
|
|
</span>
|
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</a>
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</li>
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
|
|
|
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
12.2. 分治搜索策略
|
|
|
</span>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<span class="md-status md-status--new" title="最近添加">
|
|
|
</span>
|
|
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<span class="md-ellipsis">
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12.3. 构建树问题
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<span class="md-ellipsis">
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12.4. 汉诺塔问题
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12.5. 小结
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13. 回溯
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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13. 回溯
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<ul class="md-nav__list" data-md-scrollfix>
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13.1. 回溯算法
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</span>
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13.2. 全排列问题
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</span>
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3. 子集和问题
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<span class="md-ellipsis">
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13.4. N 皇后问题
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13.5. 小结
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<span class="md-ellipsis">
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14. 动态规划
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<label class="md-nav__title" for="__nav_15">
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<span class="md-nav__icon md-icon"></span>
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14. 动态规划
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<ul class="md-nav__list" data-md-scrollfix>
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14.1. 初探动态规划
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14.2. DP 问题特性
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14.3. DP 解题思路
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14.4. 0-1 背包问题
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<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.5. 完全背包问题
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</span>
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<span class="md-ellipsis">
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14.6. 编辑距离问题
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</span>
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14.7. 小结
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</span>
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<span class="md-status md-status--new" title="最近添加">
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15.1. 贪心算法
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15.2. 分数背包问题
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15.2. 分数背包问题
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正确性证明
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15.3. 最大容量问题
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15.4. 最大切分乘积问题
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15.5. 小结
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16. 附录
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16.1. 编程环境安装
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16.2. 一起参与创作
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_greedy/fractional_knapsack_problem.md" title="编辑此页" class="md-content__button md-icon">
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<h1 id="152">15.2. 分数背包问题<a class="headerlink" href="#152" title="Permanent link">¶</a></h1>
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<p>分数背包是 0-1 背包的一个变种问题。</p>
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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 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span> 、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。每个物品只能选择一次，<strong>但可以选择物品的一部分，价值根据选择的重量比例计算</strong>，问在不超过背包容量下背包中物品的最大价值。</p>
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</div>
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<p><img alt="分数背包问题的示例数据" src="../fractional_knapsack_problem.assets/fractional_knapsack_example.png" /></p>
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<p align="center"> 图：分数背包问题的示例数据 </p>
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<p>本题和 0-1 背包整体上非常相似，状态包含当前物品 <span class="arithmatex">\(i\)</span> 和容量 <span class="arithmatex">\(c\)</span> ，目标是求不超过背包容量下的最大价值。</p>
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<p>不同点在于，本题允许只选择物品的一部分，<strong>这意味着可以对物品任意地进行切分，并按照重量比例来计算物品价值</strong>，因此有：</p>
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<ol>
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<li>对于物品 <span class="arithmatex">\(i\)</span> ，它在单位重量下的价值为 <span class="arithmatex">\(val[i-1] / wgt[i-1]\)</span> ，简称为单位价值。</li>
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<li>假设放入一部分物品 <span class="arithmatex">\(i\)</span> ，重量为 <span class="arithmatex">\(w\)</span> ，则背包增加的价值为 <span class="arithmatex">\(w \times val[i-1] / wgt[i-1]\)</span> 。</li>
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</ol>
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<p><img alt="物品在单位重量下的价值" src="../fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png" /></p>
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<p align="center"> 图：物品在单位重量下的价值 </p>
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<h3 id="_1">贪心策略确定<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
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<p>最大化背包内物品总价值，<strong>本质上是要最大化单位重量下的物品价值</strong>。由此便可推出本题的贪心策略：</p>
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<ol>
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<li>将物品按照单位价值从高到低进行排序。</li>
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<li>遍历所有物品，<strong>每轮贪心地选择单位价值最高的物品</strong>。</li>
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<li>若剩余背包容量不足，则使用当前物品的一部分填满背包即可。</li>
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</ol>
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<p><img alt="分数背包的贪心策略" src="../fractional_knapsack_problem.assets/fractional_knapsack_greedy_strategy.png" /></p>
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<p align="center"> 图：分数背包的贪心策略 </p>
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<h3 id="_2">代码实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
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<p>我们建立了一个物品类 <code>Item</code> ，以便将物品按照单位价值进行排序。循环进行贪心选择，当背包已满时跳出并返回解。</p>
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<div class="highlight"><span class="filename">fractional_knapsack.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kd">class</span> <span class="nc">Item</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="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品重量</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="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="nf">Item</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">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="k">this</span><span class="p">.</span><span class="na">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">v</span><span class="p">;</span>
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<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="p">}</span>
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="kt">double</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</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">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</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">Item</span><span class="o">[]</span><span class="w"> </span><span class="n">items</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">Item</span><span class="o">[</span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="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">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="w"> </span><span class="n">items</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="k">new</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</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">val</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
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<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="w"> </span><span class="p">}</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="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">items</span><span class="p">,</span><span class="w"> </span><span class="n">Comparator</span><span class="p">.</span><span class="na">comparingDouble</span><span class="p">(</span><span class="n">item</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="o">-</span><span class="p">((</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">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="w"> </span><span class="c1">// 循环贪心选择</span>
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<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
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<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="p">;</span>
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<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="p">;</span>
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<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
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<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
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<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
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<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
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<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a><span class="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">fractional_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">Item</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="k">public</span><span class="o">:</span>
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<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品重量</span>
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
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<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">w</span><span class="p">(</span><span class="n">w</span><span class="p">),</span><span class="w"> </span><span class="n">v</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="p">};</span>
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<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
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<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="kt">double</span><span class="w"> </span><span class="nf">fractionalKnapsack</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">wgt</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">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">Item</span><span class="o">></span><span class="w"> </span><span class="n">items</span><span class="p">;</span>
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<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="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">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="n">items</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]));</span>
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<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">items</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">items</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="p">[](</span><span class="n">Item</span><span class="w"> </span><span class="o">&</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">Item</span><span class="w"> </span><span class="o">&</span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="p">});</span>
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<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="c1">// 循环贪心选择</span>
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<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
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<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&</span><span class="n">item</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
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<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
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<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
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<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
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<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
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<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
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<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">fractional_knapsack.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">class</span> <span class="nc">Item</span><span class="p">:</span>
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">"""物品"""</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
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<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">w</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">v</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
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<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">w</span> <span class="o">=</span> <span class="n">w</span> <span class="c1"># 物品重量</span>
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<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="c1"># 物品价值</span>
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<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
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<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="k">def</span> <span class="nf">fractional_knapsack</span><span class="p">(</span><span class="n">wgt</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">val</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">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
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<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w"> </span><span class="sd">"""分数背包：贪心"""</span>
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<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="c1"># 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">Item</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">)]</span>
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<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="c1"># 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="n">items</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">item</span><span class="p">:</span> <span class="n">item</span><span class="o">.</span><span class="n">v</span> <span class="o">/</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="c1"># 循环贪心选择</span>
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<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
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<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">items</span><span class="p">:</span>
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<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span> <span class="o"><=</span> <span class="n">cap</span><span class="p">:</span>
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<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="c1"># 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">res</span> <span class="o">+=</span> <span class="n">item</span><span class="o">.</span><span class="n">v</span>
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<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="n">cap</span> <span class="o">-=</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span>
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<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 若剩余容量不足，则将当前物品的一部分装进背包</span>
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<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">res</span> <span class="o">+=</span> <span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">v</span> <span class="o">/</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">*</span> <span class="n">cap</span>
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<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="c1"># 已无剩余容量，因此跳出循环</span>
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<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="k">break</span>
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<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a> <span class="k">return</span> <span class="n">res</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">fractional_knapsack.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">Item</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="nx">w</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// 物品重量</span>
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<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="p">}</span>
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<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
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<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">fractionalKnapsack</span><span class="p">(</span><span class="nx">wgt</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">float64</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="nx">items</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Item</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">))</span>
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<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">Item</span><span class="p">{</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="p">]}</span>
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<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w"> </span><span class="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w"> </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Slice</span><span class="p">(</span><span class="nx">items</span><span class="p">,</span><span class="w"> </span><span class="kd">func</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">v</span><span class="p">)</span><span class="o">/</span><span class="nb">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="p">></span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">j</span><span class="p">].</span><span class="nx">v</span><span class="p">)</span><span class="o">/</span><span class="nb">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">j</span><span class="p">].</span><span class="nx">w</span><span class="p">)</span>
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<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="p">})</span>
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<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w"> </span><span class="c1">// 循环贪心选择</span>
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<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mf">0.0</span>
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<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">item</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">items</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">)</span>
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<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span>
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<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
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<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">cap</span><span class="p">)</span>
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<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
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<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w"> </span><span class="k">break</span>
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<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
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<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">fractional_knapsack.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="nx">Item</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{}</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">fractionalKnapsack</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">fractional_knapsack.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="nx">Item</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{}</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">fractionalKnapsack</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">fractional_knapsack.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="n">Item</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{}</span>
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<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
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<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">fractionalKnapsack</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">fractional_knapsack.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">Item</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品重量</span>
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<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a>
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<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="nf">Item</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">w</span><span class="p">;</span>
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<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">v</span><span class="p">;</span>
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<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="p">}</span>
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<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a>
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<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="kt">double</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w"> </span><span class="n">Item</span><span class="p">[]</span><span class="w"> </span><span class="n">items</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">Item</span><span class="p">[</span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">];</span>
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<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w"> </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
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<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w"> </span><span class="p">}</span>
|
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|
<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w"> </span><span class="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
|
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<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w"> </span><span class="n">Array</span><span class="p">.</span><span class="n">Sort</span><span class="p">(</span><span class="n">items</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="p">(</span><span class="n">y</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">y</span><span class="p">.</span><span class="n">w</span><span class="p">).</span><span class="n">CompareTo</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="n">w</span><span class="p">));</span>
|
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|
<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w"> </span><span class="c1">// 循环贪心选择</span>
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|
<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
|
|
|
<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w"> </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
|
|
|
<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
|
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<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
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<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
|
|
|
<a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
|
|
|
<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
|
|
|
<a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
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<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w"> </span><span class="p">}</span>
|
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<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></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-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><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">fractional_knapsack.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="n">Item</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</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">fractionalKnapsack</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">fractional_knapsack.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="n">Item</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</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">fractionalKnapsack</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">fractional_knapsack.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">Item</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品重量</span>
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<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a>
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<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">v</span><span class="p">);</span>
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<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="p">}</span>
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<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a>
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<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="kt">double</span><span class="w"> </span><span class="n">fractionalKnapsack</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">Item</span><span class="o">></span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </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">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]));</span>
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<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w"> </span><span class="n">items</span><span class="p">.</span><span class="n">sort</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="o">=></span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="p">).</span><span class="n">compareTo</span><span class="p">(</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="p">));</span>
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<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="c1">// 循环贪心选择</span>
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<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
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<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
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<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
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<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
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<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
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<a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
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<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
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<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="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">fractional_knapsack.rs</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 物品 */</span>
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<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">struct</span> <span class="nc">Item</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="n">w</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// 物品重量</span>
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<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="n">v</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// 物品价值</span>
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<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="p">}</span>
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<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
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<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="k">impl</span><span class="w"> </span><span class="n">Item</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="k">fn</span> <span class="nf">new</span><span class="p">(</span><span class="n">w</span>: <span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">v</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-> <span class="nc">Self</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="bp">Self</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="p">}</span>
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<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a>
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<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="cm">/* 分数背包：贪心 */</span>
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<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="k">fn</span> <span class="nf">fractional_knapsack</span><span class="p">(</span><span class="n">wgt</span>: <span class="kp">&</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span>: <span class="kp">&</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cap</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-> <span class="kt">f64</span> <span class="p">{</span>
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<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="c1">// 创建物品列表，包含两个属性：重量、价值</span>
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<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span>
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<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="p">.</span><span class="n">iter</span><span class="p">()</span>
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<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w"> </span><span class="p">.</span><span class="n">zip</span><span class="p">(</span><span class="n">val</span><span class="p">.</span><span class="n">iter</span><span class="p">())</span>
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<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w"> </span><span class="p">.</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="p">(</span><span class="o">&</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="o">&</span><span class="n">v</span><span class="p">)</span><span class="o">|</span><span class="w"> </span><span class="n">Item</span>::<span class="n">new</span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">))</span>
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<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w"> </span><span class="p">.</span><span class="n">collect</span>::<span class="o"><</span><span class="nb">Vec</span><span class="o"><</span><span class="n">Item</span><span class="o">>></span><span class="p">();</span>
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<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w"> </span><span class="c1">// 按照单位价值 item.v / item.w 从高到低进行排序</span>
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<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w"> </span><span class="n">items</span><span class="p">.</span><span class="n">sort_by</span><span class="p">(</span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">)</span>
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<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w"> </span><span class="p">.</span><span class="n">partial_cmp</span><span class="p">(</span><span class="o">&</span><span class="p">(</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">))</span>
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<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w"> </span><span class="p">.</span><span class="n">unwrap</span><span class="p">()</span>
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<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w"> </span><span class="p">});</span>
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<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w"> </span><span class="c1">// 循环贪心选择</span>
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<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0</span><span class="p">;</span>
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<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&</span><span class="n">items</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w"> </span><span class="c1">// 若剩余容量充足，则将当前物品整个装进背包</span>
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<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">;</span>
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<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
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<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w"> </span><span class="c1">// 若剩余容量不足，则将当前物品的一部分装进背包</span>
|
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<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">;</span>
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<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w"> </span><span class="c1">// 已无剩余容量，因此跳出循环</span>
|
|
|
<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|
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<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w"> </span><span class="n">res</span>
|
|
|
<a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a><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>最差情况下，需要遍历整个物品列表，<strong>因此时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ，其中 <span class="arithmatex">\(n\)</span> 为物品数量。</p>
|
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<p>由于初始化了一个 <code>Item</code> 对象列表，<strong>因此空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
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<h3 id="_3">正确性证明<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
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<p>采用反证法。假设物品 <span class="arithmatex">\(x\)</span> 是单位价值最高的物品，使用某算法求得最大价值为 <code>res</code> ，但该解中不包含物品 <span class="arithmatex">\(x\)</span> 。</p>
|
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<p>现在从背包中拿出单位重量的任意物品，并替换为单位重量的物品 <span class="arithmatex">\(x\)</span> 。由于物品 <span class="arithmatex">\(x\)</span> 的单位价值最高，因此替换后的总价值一定大于 <code>res</code> 。<strong>这与 <code>res</code> 是最优解矛盾，说明最优解中必须包含物品 <span class="arithmatex">\(x\)</span></strong> 。</p>
|
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<p>对于该解中的其他物品，我们也可以构建出上述矛盾。总而言之，<strong>单位价值更大的物品总是更优选择</strong>，这说明贪心策略是有效的。</p>
|
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<p>如下图所示，如果将物品重量和物品单位价值分别看作一个 2D 图表的横轴和纵轴，则分数背包问题可被转化为“求在有限横轴区间下的最大围成面积”。</p>
|
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<p>通过这个类比，我们可以从几何角度理解贪心策略的有效性。</p>
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<p><img alt="分数背包问题的几何表示" src="../fractional_knapsack_problem.assets/fractional_knapsack_area_chart.png" /></p>
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<p align="center"> 图：分数背包问题的几何表示 </p>
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