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2025-07-29 05:13:14 +08:00 · 2023-09-21 20:44:52 +08:00
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--- a/chapter_heap/heap/index.html
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@ -3512,7 +3512,32 @@
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@ -3523,6 +3548,13 @@
  


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 <h1 id="81">8.1 &nbsp; 堆<a class="headerlink" href="#81" title="Permanent link">&para;</a></h1>
 <p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为图 8-1 所示的两种类型。</p>
 <ul>
@ -4600,19 +4632,18 @@
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">        </span><span class="c1">// 判断节点 i, l, r 中值最大的节点，记为 ma</span>
 <a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">        </span><span class="c1">// 若节点 i 最大或索引 l, r 越界，则无须继续堆化，跳出</span>
-<a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
-<a id="__codelineno-49-22" name="__codelineno-49-22" href="#__codelineno-49-22"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
-<a id="__codelineno-49-23" name="__codelineno-49-23" href="#__codelineno-49-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
-<a id="__codelineno-49-24" name="__codelineno-49-24" href="#__codelineno-49-24"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-49-25" name="__codelineno-49-25" href="#__codelineno-49-25"></a><span class="w">        </span><span class="c1">// 若节点 i 最大或索引 l, r 越界，则无须继续堆化，跳出</span>
-<a id="__codelineno-49-26" name="__codelineno-49-26" href="#__codelineno-49-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-49-27" name="__codelineno-49-27" href="#__codelineno-49-27"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-49-28" name="__codelineno-49-28" href="#__codelineno-49-28"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">]);</span>
-<a id="__codelineno-49-29" name="__codelineno-49-29" href="#__codelineno-49-29"></a><span class="w">        </span><span class="c1">// 循环向下堆化</span>
-<a id="__codelineno-49-30" name="__codelineno-49-30" href="#__codelineno-49-30"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
-<a id="__codelineno-49-31" name="__codelineno-49-31" href="#__codelineno-49-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-32" name="__codelineno-49-32" href="#__codelineno-49-32"></a><span class="p">}</span>
+<a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
+<a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
+<a id="__codelineno-49-22" name="__codelineno-49-22" href="#__codelineno-49-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
+<a id="__codelineno-49-23" name="__codelineno-49-23" href="#__codelineno-49-23"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
+<a id="__codelineno-49-24" name="__codelineno-49-24" href="#__codelineno-49-24"></a><span class="w">        </span><span class="c1">// 若节点 i 最大或索引 l, r 越界，则无须继续堆化，跳出</span>
+<a id="__codelineno-49-25" name="__codelineno-49-25" href="#__codelineno-49-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
+<a id="__codelineno-49-26" name="__codelineno-49-26" href="#__codelineno-49-26"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
+<a id="__codelineno-49-27" name="__codelineno-49-27" href="#__codelineno-49-27"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">]);</span>
+<a id="__codelineno-49-28" name="__codelineno-49-28" href="#__codelineno-49-28"></a><span class="w">        </span><span class="c1">// 循环向下堆化</span>
+<a id="__codelineno-49-29" name="__codelineno-49-29" href="#__codelineno-49-29"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
+<a id="__codelineno-49-30" name="__codelineno-49-30" href="#__codelineno-49-30"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-49-31" name="__codelineno-49-31" href="#__codelineno-49-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@ -5004,11 +5035,73 @@
 <li><strong>获取最大的 <span class="arithmatex">\(k\)</span> 个元素</strong>：这是一个经典的算法问题，同时也是一种典型应用，例如选择热度前 10 的新闻作为微博热搜，选取销量前 10 的商品等。</li>
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