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@ -1974,7 +1990,18 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">preorder_traversal_i_compact.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">preOrder</span><span class="p">}</span>
<div class="highlight"><span class="filename">preorder_traversal_i_compact.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 前序遍历：例题一 */</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">func</span> <span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?)</span> <span class="p">{</span>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="k">guard</span> <span class="kd">let</span> <span class="nv">root</span> <span class="p">=</span> <span class="n">root</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="k">return</span>
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="p">}</span>
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="k">if</span> <span class="n">root</span><span class="p">.</span><span class="n">val</span> <span class="p">==</span> <span class="mi">7</span> <span class="p">{</span>
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="c1">// 记录解</span>
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a> <span class="p">}</span>
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@ -2086,7 +2113,22 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">preorder_traversal_ii_compact.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">preOrder</span><span class="p">}</span>
<div class="highlight"><span class="filename">preorder_traversal_ii_compact.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 前序遍历：例题二 */</span>
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span> <span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?)</span> <span class="p">{</span>
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a> <span class="k">guard</span> <span class="kd">let</span> <span class="nv">root</span> <span class="p">=</span> <span class="n">root</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a> <span class="k">return</span>
<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a> <span class="p">}</span>
<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a> <span class="c1">// 尝试</span>
<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a> <span class="n">path</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a> <span class="k">if</span> <span class="n">root</span><span class="p">.</span><span class="n">val</span> <span class="p">==</span> <span class="mi">7</span> <span class="p">{</span>
<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a> <span class="c1">// 记录解</span>
<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a> <span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a> <span class="p">}</span>
<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a> <span class="c1">// 回退</span>
<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a> <span class="n">path</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@ -2235,7 +2277,23 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">preorder_traversal_iii_compact.swift</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">preOrder</span><span class="p">}</span>
<div class="highlight"><span class="filename">preorder_traversal_iii_compact.swift</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* 前序遍历：例题三 */</span>
<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kd">func</span> <span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?)</span> <span class="p">{</span>
<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a> <span class="c1">// 剪枝</span>
<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a> <span class="k">guard</span> <span class="kd">let</span> <span class="nv">root</span> <span class="p">=</span> <span class="n">root</span><span class="p">,</span> <span class="n">root</span><span class="p">.</span><span class="n">val</span> <span class="o">!=</span> <span class="mi">3</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a> <span class="k">return</span>
<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a> <span class="p">}</span>
<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a> <span class="c1">// 尝试</span>
<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a> <span class="n">path</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a> <span class="k">if</span> <span class="n">root</span><span class="p">.</span><span class="n">val</span> <span class="p">==</span> <span class="mi">7</span> <span class="p">{</span>
<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a> <span class="c1">// 记录解</span>
<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a> <span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a> <span class="p">}</span>
<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
<a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a> <span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a> <span class="c1">// 回退</span>
<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a> <span class="n">path</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
@ -2563,17 +2621,51 @@
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">preorder_traversal_iii_template.swift</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">isSolution</span><span class="p">}</span>
<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a>
<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">recordSolution</span><span class="p">}</span>
<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>
<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">isValid</span><span class="p">}</span>
<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>
<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">makeChoice</span><span class="p">}</span>
<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>
<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">undoChoice</span><span class="p">}</span>
<div class="highlight"><span class="filename">preorder_traversal_iii_template.swift</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="cm">/* 判断当前状态是否为解 */</span>
<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="kd">func</span> <span class="nf">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">])</span> <span class="p">-&gt;</span> <span class="nb">Bool</span> <span class="p">{</span>
<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a> <span class="o">!</span><span class="n">state</span><span class="p">.</span><span class="bp">isEmpty</span> <span class="o">&amp;&amp;</span> <span class="n">state</span><span class="p">.</span><span class="bp">last</span><span class="p">!.</span><span class="n">val</span> <span class="p">==</span> <span class="mi">7</span>
<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a><span class="p">}</span>
<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>
<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a><span class="cm">/* 记录解 */</span>
<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a><span class="kd">func</span> <span class="nf">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="n">TreeNode</span><span class="p">]])</span> <span class="p">{</span>
<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a> <span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="p">}</span>
<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>
<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a><span class="cm">/* 判断在当前状态下，该选择是否合法 */</span>
<a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a><span class="kd">func</span> <span class="nf">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?)</span> <span class="p">-&gt;</span> <span class="nb">Bool</span> <span class="p">{</span>
<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a> <span class="n">choice</span> <span class="o">!=</span> <span class="kc">nil</span> <span class="o">&amp;&amp;</span> <span class="n">choice</span><span class="p">!.</span><span class="n">val</span> <span class="o">!=</span> <span class="mi">3</span>
<a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a><span class="p">}</span>
<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>
<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a><span class="cm">/* 更新状态 */</span>
<a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a><span class="kd">func</span> <span class="nf">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a> <span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
<a id="__codelineno-39-19" name="__codelineno-39-19" href="#__codelineno-39-19"></a><span class="p">}</span>
<a id="__codelineno-39-20" name="__codelineno-39-20" href="#__codelineno-39-20"></a>
<a id="__codelineno-39-21" name="__codelineno-39-21" href="#__codelineno-39-21"></a><span class="cm">/* 恢复状态 */</span>
<a id="__codelineno-39-22" name="__codelineno-39-22" href="#__codelineno-39-22"></a><span class="kd">func</span> <span class="nf">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-39-23" name="__codelineno-39-23" href="#__codelineno-39-23"></a> <span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
<a id="__codelineno-39-24" name="__codelineno-39-24" href="#__codelineno-39-24"></a><span class="p">}</span>
<a id="__codelineno-39-25" name="__codelineno-39-25" href="#__codelineno-39-25"></a>
<a id="__codelineno-39-26" name="__codelineno-39-26" href="#__codelineno-39-26"></a><span class="cm">/* 回溯算法：例题三 */</span>
<a id="__codelineno-39-27" name="__codelineno-39-27" href="#__codelineno-39-27"></a><span class="kd">func</span> <span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="n">TreeNode</span><span class="p">]])</span> <span class="p">{</span>
<a id="__codelineno-39-28" name="__codelineno-39-28" href="#__codelineno-39-28"></a> <span class="c1">// 检查是否为解</span>
<a id="__codelineno-39-29" name="__codelineno-39-29" href="#__codelineno-39-29"></a> <span class="k">if</span> <span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="n">state</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-39-30" name="__codelineno-39-30" href="#__codelineno-39-30"></a> <span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-39-31" name="__codelineno-39-31" href="#__codelineno-39-31"></a> <span class="k">return</span>
<a id="__codelineno-39-32" name="__codelineno-39-32" href="#__codelineno-39-32"></a> <span class="p">}</span>
<a id="__codelineno-39-33" name="__codelineno-39-33" href="#__codelineno-39-33"></a> <span class="c1">// 遍历所有选择</span>
<a id="__codelineno-39-34" name="__codelineno-39-34" href="#__codelineno-39-34"></a> <span class="k">for</span> <span class="n">choice</span> <span class="k">in</span> <span class="n">choices</span> <span class="p">{</span>
<a id="__codelineno-39-35" name="__codelineno-39-35" href="#__codelineno-39-35"></a> <span class="c1">// 剪枝：检查选择是否合法</span>
<a id="__codelineno-39-36" name="__codelineno-39-36" href="#__codelineno-39-36"></a> <span class="k">if</span> <span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">:</span> <span class="n">choice</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-39-37" name="__codelineno-39-37" href="#__codelineno-39-37"></a> <span class="c1">// 尝试：做出选择，更新状态</span>
<a id="__codelineno-39-38" name="__codelineno-39-38" href="#__codelineno-39-38"></a> <span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">:</span> <span class="n">choice</span><span class="p">)</span>
<a id="__codelineno-39-39" name="__codelineno-39-39" href="#__codelineno-39-39"></a> <span class="c1">// 进行下一轮选择</span>
<a id="__codelineno-39-40" name="__codelineno-39-40" href="#__codelineno-39-40"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">:</span> <span class="p">[</span><span class="n">choice</span><span class="p">.</span><span class="kr">left</span><span class="p">,</span> <span class="n">choice</span><span class="p">.</span><span class="kr">right</span><span class="p">].</span><span class="n">compactMap</span> <span class="p">{</span> <span class="nv">$0</span> <span class="p">},</span> <span class="n">res</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
<a id="__codelineno-39-41" name="__codelineno-39-41" href="#__codelineno-39-41"></a> <span class="c1">// 回退：撤销选择，恢复到之前的状态</span>
<a id="__codelineno-39-42" name="__codelineno-39-42" href="#__codelineno-39-42"></a> <span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">:</span> <span class="n">choice</span><span class="p">)</span>
<a id="__codelineno-39-43" name="__codelineno-39-43" href="#__codelineno-39-43"></a> <span class="p">}</span>
<a id="__codelineno-39-44" name="__codelineno-39-44" href="#__codelineno-39-44"></a> <span class="p">}</span>
<a id="__codelineno-39-45" name="__codelineno-39-45" href="#__codelineno-39-45"></a><span class="p">}</span>
</code></pre></div>
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<h1>N queens problem</h1>
<h1 id="133-n">13.3. &nbsp; N 皇后问题<a class="headerlink" href="#133-n" title="Permanent link">&para;</a></h1>
<div class="admonition question">
<p class="admonition-title">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>
</div>
<p>如下图所示，当 <span class="arithmatex">\(n = 4\)</span> 时，共可以找到两个解。从回溯算法的角度看，<span class="arithmatex">\(n \times n\)</span> 大小的棋盘共有 <span class="arithmatex">\(n^2\)</span> 个格子，给出了所有的选择 <code>choices</code> 。在逐个放置皇后的过程中，棋盘状态在不断地变化，每个时刻的棋盘就是状态 <code>state</code> 。</p>
<p><img alt="4 皇后问题的解" src="../n_queens_problem.assets/solution_4_queens.png" /></p>
<p align="center"> Fig. 4 皇后问题的解 </p>
<p>本题共有三个约束条件：<strong>多个皇后不能在同一行、同一列和同一对角线</strong>。值得注意的是，对角线分为主对角线 <code>\</code> 和副对角线 <code>/</code> 两种。</p>
<p><img alt="n 皇后问题的约束条件" src="../n_queens_problem.assets/n_queens_constraints.png" /></p>
<p align="center"> Fig. n 皇后问题的约束条件 </p>
<p>皇后的数量和棋盘的行数都为 <span class="arithmatex">\(n\)</span> ，因此我们容易得到第一个推论：<strong>棋盘每行都允许且只允许放置一个皇后</strong>。这意味着，我们可以采取逐行放置策略：从第一行开始，在每行放置一个皇后，直至最后一行结束。<strong>此策略起到了剪枝的作用</strong>，它避免了同一行出现多个皇后的所有搜索分支。</p>
<p>下图展示了 <span class="arithmatex">\(4\)</span> 皇后问题的逐行放置过程。受篇幅限制，下图仅展开了第一行的一个搜索分支。在搜索过程中，我们将不满足列约束和对角线约束的方案都剪枝了。</p>
<p><img alt="逐行放置策略" src="../n_queens_problem.assets/n_queens_placing.png" /></p>
<p align="center"> Fig. 逐行放置策略 </p>
<p>为了实现根据列约束剪枝，我们可以利用一个长度为 <span class="arithmatex">\(n\)</span> 的布尔型数组 <code>cols</code> 记录每一列是否有皇后。在每次决定放置前，我们通过 <code>cols</code> 将已有皇后的列剪枝，并在回溯中动态更新 <code>cols</code> 的状态。</p>
<p>那么，如何处理对角线约束呢？设棋盘中某个格子的行列索引为 <code>(row, col)</code> ，观察矩阵的某条主对角线，<strong>我们发现该对角线上所有格子的行索引减列索引相等</strong>，即 <code>row - col</code> 为恒定值。换句话说，若两个格子满足 <code>row1 - col1 == row2 - col2</code> ，则这两个格子一定处在一条主对角线上。</p>
<p>利用该性质，我们可以借助一个数组 <code>diag1</code> 来记录每条主对角线上是否有皇后。注意，<span class="arithmatex">\(n\)</span> 维方阵 <code>row - col</code> 的范围是 <span class="arithmatex">\([-n + 1, n - 1]\)</span> ，因此共有 <span class="arithmatex">\(2n - 1\)</span> 条主对角线。</p>
<p><img alt="处理列约束和对角线约束" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></p>
<p align="center"> Fig. 处理列约束和对角线约束 </p>
<p>同理，<strong>次对角线上的所有格子的 <code>row + col</code> 是恒定值</strong>。我们可以使用同样的方法，借助数组 <code>diag2</code> 来处理次对角线约束。</p>
<p>根据以上分析，我们便可以写出 <span class="arithmatex">\(n\)</span> 皇后的解题代码。</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>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="n">copyState</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">sRow</span><span class="p">));</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></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">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<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-0-20" 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>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></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-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></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;#&quot;</span><span class="p">);</span>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></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">false</span><span class="p">;</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="p">}</span>
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="cm">/* 求解 N 皇后 */</span>
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></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-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></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="mi">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">n</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-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="s">&quot;#&quot;</span><span class="p">);</span>
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
<a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</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="o">]</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-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</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="o">]</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-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a>
<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></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="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-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a>
<a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 回溯算法：N 皇后 */</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">cols</span><span class="p">,</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="c1">// 当放置完所有行时，记录解</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="c1">// 遍历所有列</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></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">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<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-1-16" 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>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="c1">// 放置下一行</span>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></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-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="c1">// 回退：将该格子恢复为空位</span>
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></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;#&quot;</span><span class="p">;</span>
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></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">false</span><span class="p">;</span>
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* 求解 N 皇后 */</span>
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></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-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;#&quot;</span><span class="p">));</span>
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cols</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// 记录列是否有皇后</span>
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="p">(</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="nb">false</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-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="p">(</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="nb">false</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-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a>
<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></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="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-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a>
<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">def</span> <span class="nf">backtrack</span><span class="p">(</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a> <span class="n">row</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]],</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="n">cols</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="n">diags1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="n">diags2</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]],</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="p">):</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;回溯算法：N 皇后&quot;&quot;&quot;</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="c1"># 当放置完所有行时，记录解</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">if</span> <span class="n">row</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">state</span><span class="p">])</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="k">return</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="c1"># 遍历所有列</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<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-2-22" 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>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="c1"># 放置下一行</span>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="c1"># 回退：将该格子恢复为空位</span>
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></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;#&quot;</span>
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></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">False</span>
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="k">def</span> <span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]]:</span>
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;求解 N 皇后&quot;&quot;&quot;</span>
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a> <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-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[[</span><span class="s2">&quot;#&quot;</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a> <span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span> <span class="c1"># 记录列是否有皇后</span>
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a> <span class="n">diags1</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># 记录主对角线是否有皇后</span>
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a> <span class="n">diags2</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># 记录副对角线是否有皇后</span>
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a> <span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a>
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a> <span class="k">return</span> <span class="n">res</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">nQueens</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">nQueens</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">backtrack</span><span class="p">}</span>
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">nQueens</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></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">backtrack</span><span class="p">}</span>
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></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>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">n_queens</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">backtrack</span><span class="p">}</span>
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">n_queens</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>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">backtrack</span><span class="p">}</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">nQueens</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">n_queens.zig</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></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">backtrack</span><span class="p">}</span>
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></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>
</div>
</div>
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<h2 id="1331">13.3.1. &nbsp; 复杂度分析<a class="headerlink" href="#1331" 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>
<h2 id="__comments">评论</h2>
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<p><img alt="两种剪枝条件的作用范围" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
<p align="center"> Fig. 两种剪枝条件的作用范围 </p>
<h2 id="1323">13.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#1323" title="Permanent link">&para;</a></h2>
<p>假设元素两两之间互不相同，则 <span class="arithmatex">\(n\)</span> 个元素共有 <span class="arithmatex">\(n!\)</span> 种排列（阶乘）；在记录结果时，需要复制长度为 <span class="arithmatex">\(n\)</span> 的列表，使用 <span class="arithmatex">\(O(n)\)</span> 时间。因此，<strong>时间复杂度为 <span class="arithmatex">\(O(n!n)\)</span></strong> 。</p>
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<a id="__codelineno-78-28" name="__codelineno-78-28" href="#__codelineno-78-28"></a> <span class="p">}</span>
<a id="__codelineno-78-29" name="__codelineno-78-29" href="#__codelineno-78-29"></a> <span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-78-30" name="__codelineno-78-30" href="#__codelineno-78-30"></a> <span class="c1">// 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点</span>
<a id="__codelineno-78-31" name="__codelineno-78-31" href="#__codelineno-78-31"></a> <span class="kd">let</span> <span class="nv">temp</span> <span class="p">=</span> <span class="n">node</span><span class="p">?.</span><span class="kr">right</span>
<a id="__codelineno-78-31" name="__codelineno-78-31" href="#__codelineno-78-31"></a> <span class="kd">var</span> <span class="nv">temp</span> <span class="p">=</span> <span class="n">node</span><span class="p">?.</span><span class="kr">right</span>
<a id="__codelineno-78-32" name="__codelineno-78-32" href="#__codelineno-78-32"></a> <span class="k">while</span> <span class="n">temp</span><span class="p">?.</span><span class="kr">left</span> <span class="o">!=</span> <span class="kc">nil</span> <span class="p">{</span>
<a id="__codelineno-78-33" name="__codelineno-78-33" href="#__codelineno-78-33"></a> <span class="n">temp</span> <span class="p">=</span> <span class="n">temp</span><span class="p">?.</span><span class="kr">left</span>
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<a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a> <span class="c1">// 子节点数量 = 2</span>
<a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a> <span class="c1">// 获取中序遍历中 cur 的下一个节点</span>
<a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a> <span class="kd">let</span> <span class="nv">tmp</span> <span class="p">=</span> <span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
<a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a> <span class="kd">var</span> <span class="nv">tmp</span> <span class="p">=</span> <span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a> <span class="k">while</span> <span class="n">tmp</span><span class="p">?.</span><span class="kr">left</span> <span class="o">!=</span> <span class="kc">nil</span> <span class="p">{</span>
<a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a> <span class="n">tmp</span> <span class="p">=</span> <span class="n">tmp</span><span class="p">?.</span><span class="kr">left</span>
<a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a> <span class="p">}</span>

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