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2026-03-29 05:07:03 +08:00
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chapter_backtracking/subset_sum_problem/index.html
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@@ -35,18 +35,23 @@
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.1">
<meta name="generator" content="zensical-0.0.29">
<title>13.3   子集和问题 - Hello 算法</title>
<title>13.3 子集和问题 - Hello 算法</title>
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/main.6f2fbb28.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/palette.dfe2e883.min.css">
@@ -60,29 +65,20 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Noto Sans SC";--md-code-font:"JetBrains Mono"}</style>
<link rel="stylesheet" href="../../stylesheets/extra.css">
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,t)=>(e<<5)-e+t.charCodeAt(0)),0),__md_get=(e,t=localStorage,a=__md_scope)=>JSON.parse(t.getItem(("__palette"===e?"/":a.pathname)+"."+e)),__md_set=(e,t,a=localStorage,_=__md_scope)=>{try{a.setItem(("__palette"===e?"/":_.pathname)+"."+e,JSON.stringify(t))}catch(e){}},document.documentElement.setAttribute("data-platform",navigator.platform)</script>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
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</head>
@@ -97,7 +93,7 @@
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
<label class="md-overlay" for="__drawer"></label>
<label class="md-overlay" for="__drawer" aria-label="导航栏"></label>
<div data-md-component="skip">
@@ -139,12 +135,12 @@
<nav class="md-header__inner md-grid" aria-label="页眉">
<a href="../.." title="Hello 算法" class="md-header__button md-logo" aria-label="Hello 算法" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello 算法">
</a>
<label class="md-header__button md-icon" for="__drawer">
<label class="md-header__button md-icon" for="__drawer" aria-label="导航栏">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-menu" viewBox="0 0 24 24"><path d="M4 5h16M4 12h16M4 19h16"/></svg>
</label>
<div class="md-header__title" data-md-component="header-title">
<div class="md-header__ellipsis">
@@ -169,7 +165,7 @@
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="深色模式" type="radio" name="__palette" id="__palette_0">
<input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="深色模式" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="深色模式" for="__palette_1" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -179,7 +175,7 @@
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<label class="md-header__button md-icon" title="浅色模式" for="__palette_0" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -197,7 +193,7 @@
<div class="md-select">
<button class="md-header__button md-icon" aria-label="选择当前语言">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.5 17.5 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11zM18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2zm-2.62 7 1.62-4.33L19.12 17z"/></svg>
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</button>
<div class="md-select__inner">
<ul class="md-select__list">
@@ -240,64 +236,30 @@
<label class="md-header__button md-icon" for="__search">
<label class="md-header__button md-icon" for="__search" aria-label="查找">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-search" viewBox="0 0 24 24"><path d="m21 21-4.34-4.34"/><circle cx="11" cy="11" r="8"/></svg>
</label>
<div class="md-search" data-md-component="search" role="dialog">
<label class="md-search__overlay" for="__search"></label>
<div class="md-search__inner" role="search">
<form class="md-search__form" name="search">
<input type="text" class="md-search__input" name="query" aria-label="搜索" placeholder="搜索" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
<label class="md-search__icon md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
</label>
<nav class="md-search__options" aria-label="查找">
<a href="javascript:void(0)" class="md-search__icon md-icon" title="分享" aria-label="分享" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7s-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91s2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08"/></svg>
</a>
<button type="reset" class="md-search__icon md-icon" title="清空当前内容" aria-label="清空当前内容" tabindex="-1">
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</button>
</nav>
<div class="md-search__suggest" data-md-component="search-suggest"></div>
</form>
<div class="md-search__output">
<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
<div class="md-search-result" data-md-component="search-result">
<div class="md-search-result__meta">
正在初始化搜索引擎
</div>
<ol class="md-search-result__list" role="presentation"></ol>
</div>
</div>
</div>
</div>
<div class="md-search" data-md-component="search" role="dialog" aria-label="查找">
<button type="button" class="md-search__button">
查找
</button>
</div>
<div class="md-header__source">
<div class="md-header__source">
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
<div class="md-source__icon md-icon">
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</div>
<div class="md-source__repository">
krahets/hello-algo
</div>
</a>
</div>
</div>
</nav>
</header>
@@ -325,7 +287,7 @@
<label class="md-nav__title" for="__drawer">
<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello 算法">
</a>
Hello 算法
@@ -335,7 +297,7 @@
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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krahets/hello-algo
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<nav class="md-nav" aria-label="13.3.1   无重复元素的情况">
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<nav class="md-nav" aria-label="13.3.2   考虑重复元素的情况">
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<nav class="md-nav" aria-label="13.3.1   无重复元素的情况">
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<nav class="md-nav" aria-label="13.3.2   考虑重复元素的情况">
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</a>
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</details>
<p>向以上代码输入数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，输出结果为 <span class="arithmatex">\([3, 3, 3], [4, 5], [5, 4]\)</span> 。<strong>虽然成功找出了所有和为 <span class="arithmatex">\(9\)</span> 的子集，但其中存在重复的子集 <span class="arithmatex">\([4, 5]\)</span> 和 <span class="arithmatex">\([5, 4]\)</span></strong> 。</p>
<p>这是因为搜索过程是区分选择顺序的，然而子集不区分选择顺序。如图 13-10 所示，先选 <span class="arithmatex">\(4\)</span> 后选 <span class="arithmatex">\(5\)</span> 与先选 <span class="arithmatex">\(5\)</span> 后选 <span class="arithmatex">\(4\)</span> 是不同的分支，但对应同一个子集。</p>
<p><a class="glightbox" href="../subset_sum_problem.assets/subset_sum_i_naive.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="子集搜索与越界剪枝" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></a></p>
<p><img alt="子集搜索与越界剪枝" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
<p align="center"> 图 13-10 &nbsp; 子集搜索与越界剪枝 </p>
<p>为了去除重复子集，<strong>一种直接的思路是对结果列表进行去重</strong>。但这个方法效率很低，有两方面原因。</p>
@@ -5097,7 +5061,7 @@
<li>前两轮选择 <span class="arithmatex">\(4\)</span> 和 <span class="arithmatex">\(5\)</span> ，生成子集 <span class="arithmatex">\([4, 5, \dots]\)</span> 。</li>
<li>若第一轮选择 <span class="arithmatex">\(5\)</span> ，<strong>则第二轮应该跳过 <span class="arithmatex">\(3\)</span> 和 <span class="arithmatex">\(4\)</span></strong> ，因为子集 <span class="arithmatex">\([5, 3, \dots]\)</span> 和 <span class="arithmatex">\([5, 4, \dots]\)</span> 与第 <code>1.</code> 步和第 <code>2.</code> 步中描述的子集完全重复。</li>
</ol>
<p><a class="glightbox" href="../subset_sum_problem.assets/subset_sum_i_pruning.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="不同选择顺序导致的重复子集" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></a></p>
<p><img alt="不同选择顺序导致的重复子集" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></p>
<p align="center"> 图 13-11 &nbsp; 不同选择顺序导致的重复子集 </p>
<p>总结来看，给定输入数组 <span class="arithmatex">\([x_1, x_2, \dots, x_n]\)</span> ，设搜索过程中的选择序列为 <span class="arithmatex">\([x_{i_1}, x_{i_2}, \dots, x_{i_m}]\)</span> ，则该选择序列需要满足 <span class="arithmatex">\(i_1 \leq i_2 \leq \dots \leq i_m\)</span> ，<strong>不满足该条件的选择序列都会造成重复，应当剪枝</strong>。</p>
@@ -5607,7 +5571,7 @@
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20target%3A%20int,%20choices%3A%20list%5Bint%5D,%20start%3A%20int,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%EF%BC%9A%E5%AD%90%E9%9B%86%E5%92%8C%20I%22%22%22%0A%20%20%20%20%23%20%E5%AD%90%E9%9B%86%E5%92%8C%E7%AD%89%E4%BA%8E%20target%20%E6%97%B6%EF%BC%8C%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20if%20target%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E6%89%80%E6%9C%89%E9%80%89%E6%8B%A9%0A%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%E4%BA%8C%EF%BC%9A%E4%BB%8E%20start%20%E5%BC%80%E5%A7%8B%E9%81%8D%E5%8E%86%EF%BC%8C%E9%81%BF%E5%85%8D%E7%94%9F%E6%88%90%E9%87%8D%E5%A4%8D%E5%AD%90%E9%9B%86%0A%20%20%20%20for%20i%20in%20range%28start,%20len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%E4%B8%80%EF%BC%9A%E8%8B%A5%E5%AD%90%E9%9B%86%E5%92%8C%E8%B6%85%E8%BF%87%20target%20%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E7%BB%93%E6%9D%9F%E5%BE%AA%E7%8E%AF%0A%20%20%20%20%20%20%20%20%23%20%E8%BF%99%E6%98%AF%E5%9B%A0%E4%B8%BA%E6%95%B0%E7%BB%84%E5%B7%B2%E6%8E%92%E5%BA%8F%EF%BC%8C%E5%90%8E%E8%BE%B9%E5%85%83%E7%B4%A0%E6%9B%B4%E5%A4%A7%EF%BC%8C%E5%AD%90%E9%9B%86%E5%92%8C%E4%B8%80%E5%AE%9A%E8%B6%85%E8%BF%87%20target%0A%20%20%20%20%20%20%20%20if%20target%20-%20choices%5Bi%5D%20%3C%200%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%EF%BC%9A%E5%81%9A%E5%87%BA%E9%80%89%E6%8B%A9%EF%BC%8C%E6%9B%B4%E6%96%B0%20target,%20start%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20%E8%BF%9B%E8%A1%8C%E4%B8%8B%E4%B8%80%E8%BD%AE%E9%80%89%E6%8B%A9%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target%20-%20choices%5Bi%5D,%20choices,%20i,%20res%29%0A%20%20%20%20%20%20%20%20%23%20%E5%9B%9E%E9%80%80%EF%BC%9A%E6%92%A4%E9%94%80%E9%80%89%E6%8B%A9%EF%BC%8C%E6%81%A2%E5%A4%8D%E5%88%B0%E4%B9%8B%E5%89%8D%E7%9A%84%E7%8A%B6%E6%80%81%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_i%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22%E6%B1%82%E8%A7%A3%E5%AD%90%E9%9B%86%E5%92%8C%20I%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20%E7%8A%B6%E6%80%81%EF%BC%88%E5%AD%90%E9%9B%86%EF%BC%89%0A%20%20%20%20nums.sort%28%29%20%20%23%20%E5%AF%B9%20nums%20%E8%BF%9B%E8%A1%8C%E6%8E%92%E5%BA%8F%0A%20%20%20%20start%20%3D%200%20%20%23%20%E9%81%8D%E5%8E%86%E8%B5%B7%E5%A7%8B%E7%82%B9%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20%E7%BB%93%E6%9E%9C%E5%88%97%E8%A1%A8%EF%BC%88%E5%AD%90%E9%9B%86%E5%88%97%E8%A1%A8%EF%BC%89%0A%20%20%20%20backtrack%28state,%20target,%20nums,%20start,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B3,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_i%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22%E8%BE%93%E5%85%A5%E6%95%B0%E7%BB%84%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22%E6%89%80%E6%9C%89%E5%92%8C%E7%AD%89%E4%BA%8E%20%7Btarget%7D%20%E7%9A%84%E5%AD%90%E9%9B%86%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p>图 13-12 所示为将数组 <span class="arithmatex">\([3, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 输入以上代码后的整体回溯过程。</p>
<p><a class="glightbox" href="../subset_sum_problem.assets/subset_sum_i.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="子集和 I 回溯过程" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i.png" /></a></p>
<p><img alt="子集和 I 回溯过程" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
<p align="center"> 图 13-12 &nbsp; 子集和 I 回溯过程 </p>
<h2 id="1332">13.3.2 &nbsp; 考虑重复元素的情况<a class="headerlink" href="#1332" title="Permanent link">&para;</a></h2>
@@ -5617,7 +5581,7 @@
</div>
<p>相比于上题，<strong>本题的输入数组可能包含重复元素</strong>，这引入了新的问题。例如，给定数组 <span class="arithmatex">\([4, \hat{4}, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> ，则现有代码的输出结果为 <span class="arithmatex">\([4, 5], [\hat{4}, 5]\)</span> ，出现了重复子集。</p>
<p><strong>造成这种重复的原因是相等元素在某轮中被多次选择</strong>。在图 13-13 中，第一轮共有三个选择，其中两个都为 <span class="arithmatex">\(4\)</span> ，会产生两个重复的搜索分支，从而输出重复子集；同理，第二轮的两个 <span class="arithmatex">\(4\)</span> 也会产生重复子集。</p>
<p><a class="glightbox" href="../subset_sum_problem.assets/subset_sum_ii_repeat.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="相等元素导致的重复子集" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii_repeat.png" /></a></p>
<p><img alt="相等元素导致的重复子集" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii_repeat.png" /></p>
<p align="center"> 图 13-13 &nbsp; 相等元素导致的重复子集 </p>
<h3 id="1_1">1. &nbsp; 相等元素剪枝<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
@@ -6186,7 +6150,7 @@
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</details>
<p>图 13-14 展示了数组 <span class="arithmatex">\([4, 4, 5]\)</span> 和目标元素 <span class="arithmatex">\(9\)</span> 的回溯过程，共包含四种剪枝操作。请你将图示与代码注释相结合，理解整个搜索过程，以及每种剪枝操作是如何工作的。</p>
<p><a class="glightbox" href="../subset_sum_problem.assets/subset_sum_ii.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="子集和 II 回溯过程" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii.png" /></a></p>
<p><img alt="子集和 II 回溯过程" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii.png" /></p>
<p align="center"> 图 13-14 &nbsp; 子集和 II 回溯过程 </p>
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