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@ -3443,7 +3443,7 @@
<p>在上两节中，我们了解了哈希表的工作原理和哈希冲突的处理方法。然而无论是开放寻址还是链地址法，<strong>它们只能保证哈希表可以在发生冲突时正常工作，但无法减少哈希冲突的发生</strong>。</p>
<p>如果哈希冲突过于频繁，哈希表的性能则会急剧劣化。对于链地址哈希表，理想情况下键值对平均分布在各个桶中，达到最佳查询效率；最差情况下所有键值对都被存储到同一个桶中，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p><img alt="哈希冲突的最佳与最差情况" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></p>
<p align="center"> Fig. 哈希冲突的最佳与最差情况 </p>
<p align="center"> 图：哈希冲突的最佳与最差情况 </p>
<p><strong>键值对的分布情况由哈希函数决定</strong>。回忆哈希函数的计算步骤，先计算哈希值，再对数组长度取模：</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
@ -3844,13 +3844,45 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">simple_hash.dart</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></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">add_hash</span><span class="p">}</span>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></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">mul_hash</span><span class="p">}</span>
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a>
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></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">xor_hash</span><span class="p">}</span>
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></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">rot_hash</span><span class="p">}</span>
<div class="highlight"><span class="filename">simple_hash.dart</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 加法哈希 */</span>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">addHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></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">key</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-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">hash</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="p">}</span>
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a>
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* 乘法哈希 */</span>
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">mulHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></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">key</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-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="m">31</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="p">}</span>
<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a>
<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="cm">/* 异或哈希 */</span>
<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">xorHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></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">key</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-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">^=</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="p">}</span>
<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a>
<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="cm">/* 旋转哈希 */</span>
<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="kt">int</span><span class="w"> </span><span class="n">rotHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></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">key</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-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">((</span><span class="n">hash</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="m">4</span><span class="p">)</span><span class="w"> </span><span class="o">^</span><span class="w"> </span><span class="p">(</span><span class="n">hash</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="m">28</span><span class="p">)</span><span class="w"> </span><span class="o">^</span><span class="w"> </span><span class="n">key</span><span class="p">.</span><span class="n">codeUnitAt</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="p">;</span>
<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@ -4013,7 +4045,7 @@
<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1"># 布尔量 True 的哈希值为 1</span>
<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a>
<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="n">dec</span> <span class="o">=</span> <span class="mf">3.14159</span>
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">hash_dec</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">dec</span><span class="p">)</span>
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">hash_dec</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">dec</span><span class="p">)</span>
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1"># 小数 3.14159 的哈希值为 326484311674566659</span>
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a>
<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="nb">str</span> <span class="o">=</span> <span class="s2">&quot;Hello 算法&quot;</span>