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krahets
2023-07-26 10:57:52 +08:00
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@ -3418,7 +3418,7 @@
<h3 id="_3">代码实现<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
<p>请注意,<span class="arithmatex">\(n\)</span> 维方阵中 <span class="arithmatex">\(row - col\)</span> 的范围是 <span class="arithmatex">\([-n + 1, n - 1]\)</span> <span class="arithmatex">\(row + col\)</span> 的范围是 <span class="arithmatex">\([0, 2n - 2]\)</span> ,所以主对角线和次对角线的数量都为 <span class="arithmatex">\(2n - 1\)</span> ,即数组 <code>diag1</code><code>diag2</code> 的长度都为 <span class="arithmatex">\(2n - 1\)</span></p>
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<div class="highlight"><span class="filename">n_queens.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 回溯算法N 皇后 */</span>
@ -3852,6 +3852,60 @@
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></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">nQueens</span><span class="p">}</span>
</code></pre></div>
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<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.rs</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 回溯算法N 皇后 */</span>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">fn</span> <span class="nf">backtrack</span><span class="p">(</span><span class="n">row</span>: <span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">n</span>: <span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">state</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;&gt;</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="n">cols</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">])</span><span class="w"> </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="c1">// 当放置完所有行时,记录解</span>
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</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="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">copy_state</span>: <span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span>::<span class="n">new</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="k">for</span><span class="w"> </span><span class="n">s_row</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><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="n">copy_state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">s_row</span><span class="p">);</span>
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">copy_state</span><span class="p">);</span>
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><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="c1">// 遍历所有列</span>
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </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="c1">// 计算该格子对应的主对角线和副对角线</span>
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</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="kd">let</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</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="c1">// 剪枝:不允许该格子所在列、主对角线、副对角线存在皇后</span>
<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">get_mut</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="n">unwrap</span><span class="p">()[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Q&quot;</span><span class="p">.</span><span class="n">into</span><span class="p">();</span>
<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w"> </span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</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="c1">// 放置下一行</span>
<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</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="c1">// 回退:将该格子恢复为空位</span>
<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">get_mut</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="n">unwrap</span><span class="p">()[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;#&quot;</span><span class="p">.</span><span class="n">into</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><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</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="p">}</span>
<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="p">}</span>
<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a>
<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="cm">/* 求解 N 皇后 */</span>
<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="k">fn</span> <span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span>: <span class="kt">usize</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </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="c1">// 初始化 n*n 大小的棋盘,其中 &#39;Q&#39; 代表皇后,&#39;#&#39; 代表空位</span>
<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">state</span>: <span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span>::<span class="n">new</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="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </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="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">row</span>: <span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span>::<span class="n">new</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">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="s">&quot;#&quot;</span><span class="p">.</span><span class="n">into</span><span class="p">());</span>
<a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
<a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
<a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录主对角线是否有皇后</span>
<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// 记录副对角线是否有皇后</span>
<a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span>: <span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span>::<span class="n">new</span><span class="p">();</span>
<a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a>
<a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
<a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a>
<a id="__codelineno-11-50" name="__codelineno-11-50" href="#__codelineno-11-50"></a><span class="w"> </span><span class="n">res</span>
<a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<p>逐行放置 <span class="arithmatex">\(n\)</span> 次,考虑列约束,则从第一行到最后一行分别有 <span class="arithmatex">\(n, n-1, \cdots, 2, 1\)</span> 个选择,<strong>因此时间复杂度为 <span class="arithmatex">\(O(n!)\)</span></strong> 。实际上,根据对角线约束的剪枝也能够大幅地缩小搜索空间,因而搜索效率往往优于以上时间复杂度。</p>