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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">// 向顶点列表中添加新顶点的值</span>
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="c1">// 在邻接矩阵中添加一行</span>
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="n">adjMat</span><span class="p">.</span><span class="n">emplace_back</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="n">adjMat</span><span class="p">.</span><span class="n">emplace_back</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w"> </span><span class="c1">// 在邻接矩阵中添加一列</span>
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">row</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>

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<h1 id="62">6.2. &nbsp; 哈希冲突<a class="headerlink" href="#62" title="Permanent link">&para;</a></h1>
<p>在理想情况下，哈希函数为每个输入生成唯一的输出，实现 key 和数组索引的一一对应。但实际上，<strong>哈希函数的输入空间通常远大于输出空间</strong>，因此多个输入产生相同输出的情况是不可避免的。例如，输入空间为全体整数，输出空间为数组容量大小，则必然有多个整数映射至同一数组索引。</p>
<p>这种多个输入对应同一输出索引的现象被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误，严重影响哈希表的可用性。哈希冲突的处理方法主要有两种：</p>
<ul>
<li><strong>扩大哈希表容量</strong>：哈希表容量越大，键值对聚集的概率就越低。极端情况下，当输入空间和输出空间大小相等时，哈希表等同于数组，每个 key 都对应唯一的数组索引。</li>
<li><strong>优化哈希表结构</strong>：常用方法包括链式地址和开放寻址。这类方法的思路是通过改良数据结构，使得哈希表可以在发生哈希冲突时仍然可以正常工作。当然，这些优化往往是以牺牲时间效率为代价的。</li>
</ul>
<h2 id="621">6.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
<p>哈希函数的最后一步通常是对桶数量 <span class="arithmatex">\(n\)</span> 取余，作用是将哈希值映射到桶索引范围，从而将 key 放入对应的桶中。当哈希表容量越大（即 <span class="arithmatex">\(n\)</span> 越大）时，多个 key 被分配到同一个桶中的概率就越低，冲突就越少。因此，<strong>当哈希表内的冲突总体较为严重时，编程语言通常通过扩容哈希表来缓解冲突</strong>。类似于数组扩容，哈希表扩容需将所有键值对从原哈希表迁移至新哈希表，开销较大。</p>
<p>编程语言通常使用「负载因子 Load Factor」来衡量哈希冲突的严重程度，<strong>定义为哈希表中元素数量除以桶数量</strong>，常作为哈希表扩容的触发条件。在 Java 中，当负载因子超过 <span class="arithmatex">\(0.75\)</span> 时，系统会将 HashMap 容量扩展为原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
<h2 id="622">6.2.2. &nbsp; 链式地址<a class="headerlink" href="#622" title="Permanent link">&para;</a></h2>
<p>上节提到，<strong>通常情况下哈希函数的输入空间远大于输出空间</strong>，因此哈希冲突是不可避免的。例如，输入空间为全体整数，输出空间为数组容量大小，则必然有多个整数映射至同一数组索引。</p>
<p>哈希冲突会导致查询结果错误，严重影响哈希表的可用性。为解决该问题，我们可以每当遇到哈希冲突时就进行哈希表扩容，直至冲突消失为止。此方法简单粗暴且有效，但效率太低，因为哈希表扩容需要进行大量的数据搬运与哈希值计算。为了提升效率，我们换一种思路：</p>
<ol>
<li>改良哈希表数据结构，<strong>使得哈希表可以在存在哈希冲突时正常工作</strong>。</li>
<li>仅在必要时，即当哈希冲突比较严重时，执行扩容操作。</li>
</ol>
<p>哈希表的结构改良方法主要包括链式地址和开放寻址。</p>
<h2 id="621">6.2.1. &nbsp; 链式地址<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
<p>在原始哈希表中，每个桶仅能存储一个键值对。「链式地址 Separate Chaining」将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中。</p>
<p><img alt="链式地址哈希表" src="../hash_collision.assets/hash_table_chaining.png" /></p>
<p align="center"> Fig. 链式地址哈希表 </p>
<p>链式地址下，哈希表的操作方法包括：</p>
<ul>
<li><strong>查询元素</strong>：输入 key ，经过哈希函数得到数组索引，即可访问链表头节点，然后遍历链表并对比 key 以查找目标键值对。</li>
<li><strong>查询元素</strong>：输入 <code>key</code> ，经过哈希函数得到数组索引，即可访问链表头节点，然后遍历链表并对比 <code>key</code> 以查找目标键值对。</li>
<li><strong>添加元素</strong>：先通过哈希函数访问链表头节点，然后将节点（即键值对）添加到链表中。</li>
<li><strong>删除元素</strong>：根据哈希函数的结果访问链表头部，接着遍历链表以查找目标节点，并将其删除。</li>
</ul>
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<p class="admonition-title">Tip</p>
<p>为了提高效率，<strong>我们可以将链表转换为「AVL 树」或「红黑树」</strong>，从而将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
</div>
<h2 id="623">6.2.3. &nbsp; 开放寻址<a class="headerlink" href="#623" title="Permanent link">&para;</a></h2>
<h2 id="622">6.2.2. &nbsp; 开放寻址<a class="headerlink" href="#622" title="Permanent link">&para;</a></h2>
<p>「开放寻址 Open Addressing」不引入额外的数据结构，而是通过“多次探测”来处理哈希冲突，探测方式主要包括线性探测、平方探测、多次哈希。</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进行探测，对应的哈希表操作方法为：</p>
<ul>
<li><strong>插入元素</strong>：通过哈希函数计算数组索引，若发现桶内已有元素，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。</li>
<li><strong>查找元素</strong>：若发现哈希冲突，则使用相同步长向后线性遍历，直到找到对应元素，返回 value 即可；或者若遇到空位，说明目标键值对不在哈希表中，返回 <span class="arithmatex">\(\text{None}\)</span> 。</li>
<li><strong>查找元素</strong>：若发现哈希冲突，则使用相同步长向后线性遍历，直到找到对应元素，返回 <code>value</code> 即可；或者若遇到空位，说明目标键值对不在哈希表中，返回 <span class="arithmatex">\(\text{None}\)</span> 。</li>
</ul>
<p><img alt="线性探测" src="../hash_collision.assets/hash_table_linear_probing.png" /></p>
<p align="center"> Fig. 线性探测 </p>
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