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<title>12.3.   N 皇后问题 - Hello 算法</title>
<title>12.4.   N 皇后问题 - Hello 算法</title>
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12.3.1. &nbsp; 复杂度分析
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12.4.1. &nbsp; 复杂度分析
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12.3.1. &nbsp; 复杂度分析
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12.4.1. &nbsp; 复杂度分析
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<h1 id="123-n">12.3. &nbsp; N 皇后问题<a class="headerlink" href="#123-n" title="Permanent link">&para;</a></h1>
<h1 id="124-n">12.4. &nbsp; N 皇后问题<a class="headerlink" href="#124-n" title="Permanent link">&para;</a></h1>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>根据国际象棋的规则,皇后可以攻击与之处在同一行或同一列或同一斜线上的棋子。给定 <span class="arithmatex">\(n\)</span> 个皇后和一个 <span class="arithmatex">\(n \times n\)</span> 大小的棋盘,寻找使得所有皇后之间无法相互攻击的摆放方案。</p>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></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-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="kt">int</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">col</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>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="w"> </span><span class="kt">int</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-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></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-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</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="o">[</span><span class="n">diag1</span><span class="o">]</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="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;Q&quot;</span><span class="p">);</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="w"> </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
@ -2182,8 +2198,8 @@
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></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-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="kt">int</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">col</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>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="kt">int</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-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</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="o">||</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="o">||</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="p">{</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></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-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</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>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </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">=</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="o">=</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="nb">true</span><span class="p">;</span>
@ -2231,8 +2247,8 @@
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="c1"># 计算该格子对应的主对角线和副对角线</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="n">diag1</span> <span class="o">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">diag2</span> <span class="o">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="c1"># 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="k">if</span> <span class="ow">not</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="ow">or</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="ow">or</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]):</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <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-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]:</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="c1"># 尝试:将皇后放置在该格子</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">&quot;Q&quot;</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</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="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
@ -2273,8 +2289,8 @@
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></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-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!((</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></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-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">&quot;Q&quot;</span>
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </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><span class="w"> </span><span class="kc">true</span>
@ -2304,8 +2320,8 @@
<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></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-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
<a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span>
<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!((</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></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-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">!(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-3-52" name="__codelineno-3-52" href="#__codelineno-3-52"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">&quot;Q&quot;</span>
<a id="__codelineno-3-53" name="__codelineno-3-53" href="#__codelineno-3-53"></a><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </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><span class="w"> </span><span class="kc">true</span>
@ -2351,8 +2367,8 @@
<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></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-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></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-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">cols</span><span class="p">[</span><span class="nx">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="nx">diags1</span><span class="p">[</span><span class="nx">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="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">&#39;Q&#39;</span><span class="p">;</span>
<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
@ -2400,8 +2416,8 @@
<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></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-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="p">(</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></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-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">cols</span><span class="p">[</span><span class="nx">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="nx">diags1</span><span class="p">[</span><span class="nx">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="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">&#39;Q&#39;</span><span class="p">;</span>
<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w"> </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
@ -2452,8 +2468,8 @@
<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></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-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w"> </span><span class="kt">int</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">col</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="m">1</span><span class="p">;</span>
<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w"> </span><span class="kt">int</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-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w"> </span><span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</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="o">||</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="o">||</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="p">{</span>
<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></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-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w"> </span><span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</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>
<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w"> </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">=</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="o">=</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="k">true</span><span class="p">;</span>
@ -2501,8 +2517,8 @@
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="c1">// 计算该格子对应的主对角线和副对角线</span>
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="kd">let</span> <span class="nv">diag1</span> <span class="p">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a> <span class="kd">let</span> <span class="nv">diag2</span> <span class="p">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <span class="c1">// 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后</span>
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a> <span class="k">if</span> <span class="o">!</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="o">||</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</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="p">{</span>
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <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-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a> <span class="k">if</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="o">&amp;&amp;</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="o">&amp;&amp;</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="p">{</span>
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a> <span class="c1">// 尝试:将皇后放置在该格子</span>
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="s">&quot;Q&quot;</span>
<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="p">=</span> <span class="kc">true</span>
@ -2548,7 +2564,7 @@
</div>
</div>
</div>
<h2 id="1231">12.3.1. &nbsp; 复杂度分析<a class="headerlink" href="#1231" title="Permanent link">&para;</a></h2>
<h2 id="1241">12.4.1. &nbsp; 复杂度分析<a class="headerlink" href="#1241" title="Permanent link">&para;</a></h2>
<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>
<p><code>state</code> 使用 <span class="arithmatex">\(O(n^2)\)</span> 空间,<code>cols</code> , <code>diags1</code> , <code>diags2</code> 皆使用 <span class="arithmatex">\(O(n)\)</span> 空间。最大递归深度为 <span class="arithmatex">\(n\)</span> ,使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。因此,<strong>空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span></strong></p>
@ -2628,7 +2644,7 @@
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@ -2637,7 +2653,7 @@
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@ -18,7 +18,7 @@
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10.1. &nbsp; 二分查找New
10.1. &nbsp; 二分查找
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@ -1488,7 +1488,7 @@
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10.2. &nbsp; 二分查找边界New
10.2. &nbsp; 二分查找边界
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@ -1631,7 +1631,7 @@
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12.3. &nbsp; 子集和问题New
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12.3. &nbsp; 子集和问题New
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
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12.3. &nbsp; 子集和问题New
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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<li class="md-nav__item">
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10.2. &nbsp; 二分查找边界
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<li class="md-nav__item">
<a href="../chapter_sorting/heap_sort/" class="md-nav__link">
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11.7. &nbsp; 堆排序
</a>
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<div class="md-nav__link md-nav__link--index ">
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<a href="../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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10.1. &nbsp; 二分查找
</a>
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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10.2. &nbsp; 二分查找边界
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<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
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12.3. &nbsp; 子集和问题New
</a>
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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10.2. &nbsp; 二分查找边界
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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<li class="md-nav__item">
<a href="../chapter_searching/binary_search_edge/" class="md-nav__link">
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10.2. &nbsp; 二分查找边界
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<li class="md-nav__item">
<a href="../chapter_sorting/heap_sort/" class="md-nav__link">
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<div class="md-nav__link md-nav__link--index ">
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<a href="../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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</a>
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<a href="../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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@ -1500,7 +1500,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<div class="md-nav__link md-nav__link--index ">
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<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
</li>
@ -1545,7 +1545,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<li class="md-nav__item">
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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11.7. &nbsp; 堆排序
</a>
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<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
@ -1911,9 +1913,23 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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</a>
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@ -1500,7 +1500,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<title>10.1.   二分查找New - Hello 算法</title>
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12.3. &nbsp; 子集和问题New
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题
12.4. &nbsp; N 皇后问题
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<title>10.2.   二分查找边界New - Hello 算法</title>
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@ -113,7 +113,7 @@
<div class="md-header__topic" data-md-component="header-topic">
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<li class="md-nav__item">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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@ -1758,7 +1758,7 @@
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12.3. &nbsp; N 皇后问题
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@ -2678,7 +2694,7 @@
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</a>
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<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题
12.4. &nbsp; N 皇后问题
</a>
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<li class="md-nav__item">
<a href="../chapter_searching/binary_search/" class="md-nav__link">
10.1. &nbsp; 二分查找New
10.1. &nbsp; 二分查找
</a>
</li>
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<li class="md-nav__item">
<a href="../chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界New
10.2. &nbsp; 二分查找边界
</a>
</li>
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<li class="md-nav__item">
<a href="../chapter_sorting/selection_sort/" class="md-nav__link">
11.2. &nbsp; 选择排序New
11.2. &nbsp; 选择排序
</a>
</li>
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<li class="md-nav__item">
<a href="../chapter_sorting/heap_sort/" class="md-nav__link">
11.7. &nbsp; 堆排序New
11.7. &nbsp; 堆排序
</a>
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<div class="md-nav__link md-nav__link--index ">
<a href="../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
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<li class="md-nav__item">
<a href="../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
</li>
<li class="md-nav__item">
<a href="../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题
12.4. &nbsp; N 皇后问题
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<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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10.1. &nbsp; 二分查找
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界New
10.2. &nbsp; 二分查找边界
</a>
</li>
@ -1643,7 +1643,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
11.2. &nbsp; 选择排序New
11.2. &nbsp; 选择排序
</a>
</li>
@ -1713,7 +1713,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
11.7. &nbsp; 堆排序New
11.7. &nbsp; 堆排序
</a>
</li>
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<div class="md-nav__link md-nav__link--index ">
<a href="../../chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
@ -1866,9 +1868,23 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题
12.4. &nbsp; N 皇后问题
</a>
</li>

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@ -1467,7 +1467,7 @@
<li class="md-nav__item">
<a href="chapter_searching/binary_search/" class="md-nav__link">
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10.1. &nbsp; 二分查找
</a>
</li>
@ -1481,7 +1481,7 @@
<li class="md-nav__item">
<a href="chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界New
10.2. &nbsp; 二分查找边界
</a>
</li>
@ -1624,7 +1624,7 @@
<li class="md-nav__item">
<a href="chapter_sorting/selection_sort/" class="md-nav__link">
11.2. &nbsp; 选择排序New
11.2. &nbsp; 选择排序
</a>
</li>
@ -1694,7 +1694,7 @@
<li class="md-nav__item">
<a href="chapter_sorting/heap_sort/" class="md-nav__link">
11.7. &nbsp; 堆排序New
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</a>
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<div class="md-nav__link md-nav__link--index ">
<a href="chapter_backtracking/">12. &nbsp; &nbsp; 回溯</a>
@ -1847,9 +1849,23 @@
<li class="md-nav__item">
<a href="chapter_backtracking/subset_sum_problem/" class="md-nav__link">
12.3. &nbsp; 子集和问题New
</a>
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<li class="md-nav__item">
<a href="chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题
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