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@@ -3394,7 +3394,7 @@
 <p>设元素数量为 <span class="arithmatex">\(n\)</span> ，则最后一个元素入堆的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> 。在依次添加元素时，堆的平均长度为 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此该方法的总体时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
 <h2 id="822">8.2.2. &nbsp; 基于堆化操作实现<a class="headerlink" href="#822" title="Permanent link">&para;</a></h2>
 <p>有趣的是，存在一种更高效的建堆方法，其时间复杂度仅为 <span class="arithmatex">\(O(n)\)</span> 。我们先将列表所有元素原封不动添加到堆中，<strong>然后迭代地对各个节点执行“从顶至底堆化”</strong>。当然，<strong>我们不需要对叶节点执行堆化操作</strong>，因为它们没有子节点。</p>
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chapter_heap/heap/index.html

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 <p class="admonition-title">Tip</p>
 <p>类似于排序算法中的“从小到大排列”和“从大到小排列”，我们可以通过修改 Comparator 来实现“小顶堆”与“大顶堆”之间的转换。</p>
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@@ -3781,7 +3781,7 @@
 <p align="center"> Fig. 堆的表示与存储 </p>
 
 <p>我们可以将索引映射公式封装成函数，方便后续使用。</p>
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 <div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="cm">/* 获取左子节点索引 */</span>
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 </div>
 <h3 id="_2">访问堆顶元素<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
 <p>堆顶元素即为二叉树的根节点，也就是列表的首个元素。</p>
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 </div>
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 <p>设节点总数为 <span class="arithmatex">\(n\)</span> ，则树的高度为 <span class="arithmatex">\(O(\log n)\)</span> 。由此可知，堆化操作的循环轮数最多为 <span class="arithmatex">\(O(\log n)\)</span> ，<strong>元素入堆操作的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span></strong> 。</p>
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@@ -4457,7 +4457,7 @@
 </div>
 </div>
 <p>与元素入堆操作相似，堆顶元素出堆操作的时间复杂度也为 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
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chapter_heap/top_k/index.html

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 </div>
 <p>总共执行了 <span class="arithmatex">\(n\)</span> 轮入堆和出堆，堆的最大长度为 <span class="arithmatex">\(k\)</span> ，因此时间复杂度为 <span class="arithmatex">\(O(n \log k)\)</span> 。该方法的效率很高，当 <span class="arithmatex">\(k\)</span> 较小时，时间复杂度趋向 <span class="arithmatex">\(O(n)\)</span> ；当 <span class="arithmatex">\(k\)</span> 较大时，时间复杂度不会超过 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
 <p>另外，该方法适用于动态数据流的使用场景。在不断加入数据时，我们可以持续维护堆内的元素，从而实现最大 <span class="arithmatex">\(k\)</span> 个元素的动态更新。</p>
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 <div class="highlight"><span class="filename">top_k.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 基于堆查找数组中最大的 k 个元素 */</span>
@@ -3542,7 +3542,23 @@
 </code></pre></div>
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 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">top_k</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">topKHeap</span><span class="p">}</span>
+<div class="highlight"><span class="filename">top_k.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 基于堆查找数组中最大的 k 个元素 */</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">topKHeap</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</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">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="p">();</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 将数组的前 k 个元素入堆</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></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">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// 从第 k+1 个元素开始，保持堆的长度为 k</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</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>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// 若当前元素大于堆顶元素，则将堆顶元素出堆、当前元素入堆</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">Peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">heap</span><span class="p">;</span>
+<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
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