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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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14.1 &nbsp; 初探动态规划
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<li>小圆盘必须时刻位于大圆盘之上。</li>
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</div>
<p><img alt="汉诺塔问题示例" src="../hanota_problem.assets/hanota_example.png" /></p>
<p><a class="glightbox" href="../hanota_problem.assets/hanota_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="汉诺塔问题示例" src="../hanota_problem.assets/hanota_example.png" /></a></p>
<p align="center"> 图 12-10 &nbsp; 汉诺塔问题示例 </p>
<p><strong>我们将规模为 <span class="arithmatex">\(i\)</span> 的汉诺塔问题记做 <span class="arithmatex">\(f(i)\)</span></strong> 。例如 <span class="arithmatex">\(f(3)\)</span> 代表将 <span class="arithmatex">\(3\)</span> 个圆盘从 <code>A</code> 移动至 <code>C</code> 的汉诺塔问题。</p>
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<p><img alt="规模为 1 问题的解" src="../hanota_problem.assets/hanota_f1_step1.png" /></p>
<p><a class="glightbox" href="../hanota_problem.assets/hanota_f1_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="规模为 1 问题的解" src="../hanota_problem.assets/hanota_f1_step1.png" /></a></p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:4"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label></div>
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<p><img alt="规模为 2 问题的解" src="../hanota_problem.assets/hanota_f2_step1.png" /></p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:4"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">&lt;1&gt;</label><label for="__tabbed_3_2">&lt;2&gt;</label><label for="__tabbed_3_3">&lt;3&gt;</label><label for="__tabbed_3_4">&lt;4&gt;</label></div>
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<li>将 <span class="arithmatex">\(n-1\)</span> 个圆盘借助 <code>A</code> 从 <code>B</code> 移至 <code>C</code> 。</li>
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<p>对于这两个子问题 <span class="arithmatex">\(f(n-1)\)</span> ，<strong>可以通过相同的方式进行递归划分</strong>，直至达到最小子问题 <span class="arithmatex">\(f(1)\)</span> 。而 <span class="arithmatex">\(f(1)\)</span> 的解是已知的，只需一次移动操作即可。</p>
<p><img alt="汉诺塔问题的分治策略" src="../hanota_problem.assets/hanota_divide_and_conquer.png" /></p>
<p><a class="glightbox" href="../hanota_problem.assets/hanota_divide_and_conquer.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="汉诺塔问题的分治策略" src="../hanota_problem.assets/hanota_divide_and_conquer.png" /></a></p>
<p align="center"> 图 12-14 &nbsp; 汉诺塔问题的分治策略 </p>
<h3 id="3">3. &nbsp; 代码实现<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
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<p>如图 12-15 所示，汉诺塔问题形成一个高度为 <span class="arithmatex">\(n\)</span> 的递归树，每个节点代表一个子问题、对应一个开启的 <code>dfs()</code> 函数，<strong>因此时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> ，空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> 。</p>
<p><img alt="汉诺塔问题的递归树" src="../hanota_problem.assets/hanota_recursive_tree.png" /></p>
<p><a class="glightbox" href="../hanota_problem.assets/hanota_recursive_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="汉诺塔问题的递归树" src="../hanota_problem.assets/hanota_recursive_tree.png" /></a></p>
<p align="center"> 图 12-15 &nbsp; 汉诺塔问题的递归树 </p>
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