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<p>复杂的回溯问题通常包含一个或多个约束条件，<strong>约束条件通常可用于“剪枝”</strong>。</p>
<div class="admonition question">
<p class="admonition-title">例题三</p>
</div>
<p>在二叉树中搜索所有值为 <span class="arithmatex">\(7\)</span> 的节点，返回根节点到这些节点的路径，<strong>路径中不能包含值为 <span class="arithmatex">\(3\)</span> 的节点</strong>。</p>
</div>
<p><strong>解题思路</strong>：在例题二的基础上添加剪枝操作，当遇到值为 <span class="arithmatex">\(3\)</span> 的节点时，则终止继续搜索。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:10"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label></div>
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</ul>
<p>为了提高操作效率，<strong>可以将链表转换为「AVL 树」或「红黑树」</strong>，将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
<h2 id="623">6.2.3. &nbsp; 开放寻址<a class="headerlink" href="#623" title="Permanent link">&para;</a></h2>
<p>「开放寻址」方法不引入额外的数据结构，而是通过“多次探测”来解决哈希冲突，<strong>探测方主要包括线性探测、平方探测、多次哈希</strong>。</p>
<p>「开放寻址」方法不引入额外的数据结构，而是通过“多次探测”来解决哈希冲突，<strong>探测方式主要包括线性探测、平方探测、多次哈希</strong>。</p>
<h3 id="_1">线性探测<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<p>「线性探测」采用固定步长的线性查找来解决哈希冲突。</p>
<p><strong>插入元素</strong>：若出现哈希冲突，则从冲突位置向后线性遍历（步长通常为 <span class="arithmatex">\(1\)</span> ），直至找到空位，将元素插入其中。</p>
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<li>感谢 @squidfunk 提供的写作排版建议，以及杰出的开源项目 <a href="https://github.com/squidfunk/mkdocs-material/tree/master">Material-for-MkDocs</a> 。</li>
</ul>
<p>在写作过程中，我阅读了许多关于数据结构与算法的教材和文章。这些作品为本书提供了优秀的范本，确保了本书内容的准确性与品质。在此感谢所有老师和前辈们的杰出贡献！</p>
<p>本书倡导“手脑并用”的学习方法，在此方面深受《动手学深度学习》的启发。在此向各位读者强烈推荐这本优秀著作，包括<a href="https://github.com/d2l-ai/d2l-zh">中文版</a>、<a href="https://github.com/d2l-ai/d2l-en">英文版</a>、<a href="https://space.bilibili.com/1567748478">李沐老师 bilibili 主页</a>。</p>
<p>本书倡导手脑并用的学习方式，在这一点上深受<a href="https://github.com/d2l-ai/d2l-zh">《动手学深度学习》</a>的启发。在此向各位读者强烈推荐这本优秀著作。</p>
<p>衷心感谢我的父母，正是你们一直以来的支持与鼓励，让我有机会做这些富有趣味的事。</p>
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<li class="md-nav__item">
<a href="#021" class="md-nav__link">
0.2.1. &nbsp; 算法学习路线
0.2.1. &nbsp; 行文风格约定
</a>
</li>
<li class="md-nav__item">
<a href="#022" class="md-nav__link">
0.2.2. &nbsp; 行文风格约定
0.2.2. &nbsp; 在动画图解中高效学习
</a>
</li>
<li class="md-nav__item">
<a href="#023" class="md-nav__link">
0.2.3. &nbsp; 在动画图解中高效学习
0.2.3. &nbsp; 在代码实践中加深理解
</a>
</li>
<li class="md-nav__item">
<a href="#024" class="md-nav__link">
0.2.4. &nbsp; 在代码实践中加深理解
0.2.4. &nbsp; 在提问讨论中共同成长
</a>
</li>
<li class="md-nav__item">
<a href="#025" class="md-nav__link">
0.2.5. &nbsp; 在提问讨论中共同成长
0.2.5. &nbsp; 算法学习路线
</a>
</li>
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<li class="md-nav__item">
<a href="#021" class="md-nav__link">
0.2.1. &nbsp; 算法学习路线
0.2.1. &nbsp; 行文风格约定
</a>
</li>
<li class="md-nav__item">
<a href="#022" class="md-nav__link">
0.2.2. &nbsp; 行文风格约定
0.2.2. &nbsp; 在动画图解中高效学习
</a>
</li>
<li class="md-nav__item">
<a href="#023" class="md-nav__link">
0.2.3. &nbsp; 在动画图解中高效学习
0.2.3. &nbsp; 在代码实践中加深理解
</a>
</li>
<li class="md-nav__item">
<a href="#024" class="md-nav__link">
0.2.4. &nbsp; 在代码实践中加深理解
0.2.4. &nbsp; 在提问讨论中共同成长
</a>
</li>
<li class="md-nav__item">
<a href="#025" class="md-nav__link">
0.2.5. &nbsp; 在提问讨论中共同成长
0.2.5. &nbsp; 算法学习路线
</a>
</li>
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<p class="admonition-title">Tip</p>
<p>为了获得最佳的阅读体验，建议您通读本节内容。</p>
</div>
<h2 id="021">0.2.1. &nbsp; 算法学习路线<a class="headerlink" href="#021" title="Permanent link">&para;</a></h2>
<p>从总体上看，我们可以将学习数据结构与算法的过程划分为三个阶段：</p>
<ol>
<li><strong>算法入门</strong>。我们需要熟悉各种数据结构的特点和用法，学习不同算法的原理、流程、用途和效率等方面内容。</li>
<li><strong>刷算法题</strong>。建议从热门题目开刷，如<a href="https://leetcode.cn/problem-list/xb9nqhhg/">剑指 Offer</a>和<a href="https://leetcode.cn/problem-list/2cktkvj/">LeetCode Hot 100</a>，先积累至少 100 道题目，熟悉主流的算法问题。初次刷题时，“知识遗忘”可能是一个挑战，但请放心，这是很正常的。我们可以按照“艾宾浩斯遗忘曲线”来复习题目，通常在进行 3-5 轮的重复后，就能将其牢记在心。</li>
<li><strong>搭建知识体系</strong>。在学习方面，我们可以阅读算法专栏文章、解题框架和算法教材，以不断丰富知识体系。在刷题方面，可以尝试采用进阶刷题策略，如按专题分类、一题多解、一解多题等，相关的刷题心得可以在各个社区找到。</li>
</ol>
<p>作为一本入门教程，本书内容主要涵盖“第一阶段”，旨在帮助你更高效地展开第二和第三阶段的学习。</p>
<p><img alt="算法学习路线" src="../suggestions.assets/learning_route.png" /></p>
<p align="center"> Fig. 算法学习路线 </p>
<h2 id="022">0.2.2. &nbsp; 行文风格约定<a class="headerlink" href="#022" title="Permanent link">&para;</a></h2>
<h2 id="021">0.2.1. &nbsp; 行文风格约定<a class="headerlink" href="#021" title="Permanent link">&para;</a></h2>
<p>标题后标注 <code>*</code> 的是选读章节，内容相对困难。如果你的时间有限，建议可以先跳过。</p>
<p>文章中的重要名词会用 <code>「 」</code> 括号标注，例如 <code>「数组 Array」</code> 。请务必记住这些名词，包括英文翻译，以便后续阅读文献时使用。</p>
<p><strong>加粗的文字</strong> 表示重点内容或总结性语句，这类文字值得特别关注。</p>
@ -2088,13 +2077,13 @@
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</div>
<h2 id="023">0.2.3. &nbsp; 在动画图解中高效学习<a class="headerlink" href="#023" title="Permanent link">&para;</a></h2>
<h2 id="022">0.2.2. &nbsp; 在动画图解中高效学习<a class="headerlink" href="#022" title="Permanent link">&para;</a></h2>
<p>相较于文字，视频和图片具有更高的信息密度和结构化程度，因此更易于理解。在本书中，<strong>重点和难点知识将主要通过动画和图解形式展示</strong>，而文字则作为动画和图片的解释与补充。</p>
<p>在阅读本书时，如果发现某段内容提供了动画或图解，<strong>建议以图为主线</strong>，以文字（通常位于图像上方）为辅，综合两者来理解内容。</p>
<p><img alt="动画图解示例" src="../../index.assets/animation.gif" /></p>
<p align="center"> Fig. 动画图解示例 </p>
<h2 id="024">0.2.4. &nbsp; 在代码实践中加深理解<a class="headerlink" href="#024" title="Permanent link">&para;</a></h2>
<h2 id="023">0.2.3. &nbsp; 在代码实践中加深理解<a class="headerlink" href="#023" title="Permanent link">&para;</a></h2>
<p>本书的配套代码托管在<a href="https://github.com/krahets/hello-algo">GitHub 仓库</a>，<strong>源代码包含详细注释，并附有测试样例，可直接运行</strong>。</p>
<p>如果学习时间有限，建议你至少通读并运行所有代码。如果时间充裕，<strong>建议参照代码自行敲一遍</strong>。与仅阅读代码相比，编写代码的过程往往能带来更多收获。</p>
<p><img alt="运行代码示例" src="../../index.assets/running_code.gif" /></p>
@ -2112,12 +2101,23 @@
<p><img alt="代码块与对应的源代码文件" src="../suggestions.assets/code_md_to_repo.png" /></p>
<p align="center"> Fig. 代码块与对应的源代码文件 </p>
<h2 id="025">0.2.5. &nbsp; 在提问讨论中共同成长<a class="headerlink" href="#025" title="Permanent link">&para;</a></h2>
<h2 id="024">0.2.4. &nbsp; 在提问讨论中共同成长<a class="headerlink" href="#024" title="Permanent link">&para;</a></h2>
<p>阅读本书时，请不要“惯着”那些没学明白的知识点。<strong>欢迎在评论区提出你的问题</strong>，我和其他小伙伴们将竭诚为你解答，一般情况下可在两天内得到回复。</p>
<p>同时，也希望您能在评论区多花些时间。一方面，您可以了解大家遇到的问题，从而查漏补缺，这将有助于激发更深入的思考。另一方面，希望您能慷慨地回答其他小伙伴的问题、分享您的见解，让大家共同学习和进步。</p>
<p><img alt="评论区示例" src="../../index.assets/comment.gif" /></p>
<p align="center"> Fig. 评论区示例 </p>
<h2 id="025">0.2.5. &nbsp; 算法学习路线<a class="headerlink" href="#025" title="Permanent link">&para;</a></h2>
<p>从总体上看，我们可以将学习数据结构与算法的过程划分为三个阶段：</p>
<ol>
<li><strong>算法入门</strong>。我们需要熟悉各种数据结构的特点和用法，学习不同算法的原理、流程、用途和效率等方面内容。</li>
<li><strong>刷算法题</strong>。建议从热门题目开刷，如<a href="https://leetcode.cn/problem-list/xb9nqhhg/">剑指 Offer</a>和<a href="https://leetcode.cn/problem-list/2cktkvj/">LeetCode Hot 100</a>，先积累至少 100 道题目，熟悉主流的算法问题。初次刷题时，“知识遗忘”可能是一个挑战，但请放心，这是很正常的。我们可以按照“艾宾浩斯遗忘曲线”来复习题目，通常在进行 3-5 轮的重复后，就能将其牢记在心。</li>
<li><strong>搭建知识体系</strong>。在学习方面，我们可以阅读算法专栏文章、解题框架和算法教材，以不断丰富知识体系。在刷题方面，可以尝试采用进阶刷题策略，如按专题分类、一题多解、一解多题等，相关的刷题心得可以在各个社区找到。</li>
</ol>
<p>作为一本入门教程，本书内容主要涵盖“第一阶段”，旨在帮助你更高效地展开第二和第三阶段的学习。</p>
<p><img alt="算法学习路线" src="../suggestions.assets/learning_route.png" /></p>
<p align="center"> Fig. 算法学习路线 </p>
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