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chapter_heap/top_k/index.html

@@ -3435,17 +3435,17 @@
 <p>我们可以进行 <span class="arithmatex">\(k\)</span> 轮遍历，分别在每轮中提取第 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(\cdots\)</span> , <span class="arithmatex">\(k\)</span> 大的元素，时间复杂度为 <span class="arithmatex">\(O(nk)\)</span> 。</p>
 <p>该方法只适用于 <span class="arithmatex">\(k \ll n\)</span> 的情况，因为当 <span class="arithmatex">\(k\)</span> 与 <span class="arithmatex">\(n\)</span> 比较接近时，其时间复杂度趋向于 <span class="arithmatex">\(O(n^2)\)</span> ，非常耗时。</p>
 <p><img alt="遍历寻找最大的 k 个元素" src="../top_k.assets/top_k_traversal.png" /></p>
-<p align="center"> Fig. 遍历寻找最大的 k 个元素 </p>
+<p align="center"> 图：遍历寻找最大的 k 个元素 </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>当 <span class="arithmatex">\(k = n\)</span> 时，我们可以得到从大到小的序列，等价于「选择排序」算法。 </p>
+<p>当 <span class="arithmatex">\(k = n\)</span> 时，我们可以得到从大到小的序列，等价于「选择排序」算法。</p>
 </div>
 <h2 id="832">8.3.2. &nbsp; 方法二：排序<a class="headerlink" href="#832" title="Permanent link">&para;</a></h2>
 <p>我们可以对数组 <code>nums</code> 进行排序，并返回最右边的 <span class="arithmatex">\(k\)</span> 个元素，时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
 <p>显然，该方法“超额”完成任务了，因为我们只需要找出最大的 <span class="arithmatex">\(k\)</span> 个元素即可，而不需要排序其他元素。</p>
 <p><img alt="排序寻找最大的 k 个元素" src="../top_k.assets/top_k_sorting.png" /></p>
-<p align="center"> Fig. 排序寻找最大的 k 个元素 </p>
+<p align="center"> 图：排序寻找最大的 k 个元素 </p>
 
 <h2 id="833">8.3.3. &nbsp; 方法三：堆<a class="headerlink" href="#833" title="Permanent link">&para;</a></h2>
 <p>我们可以基于堆更加高效地解决 Top-K 问题，流程如下：</p>
@@ -3486,6 +3486,8 @@
 </div>
 </div>
 </div>
+<p align="center"> 图：基于堆寻找最大的 k 个元素 </p>
+
 <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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@@ -3621,7 +3623,20 @@
 </code></pre></div>
 </div>
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-<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></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">top_k_heap</span><span class="p">}</span>
+<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 基于堆查找数组中最大的 k 个元素 */</span>
+<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">MinHeap</span><span class="w"> </span><span class="n">topKHeap</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</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-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">  </span><span class="c1">// 将数组的前 k 个元素入堆</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">  </span><span class="n">MinHeap</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">sublist</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">));</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></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-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></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-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></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-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></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-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">      </span><span class="n">heap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">      </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">  </span><span class="p">}</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></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-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
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