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第 8 章 &nbsp; 堆
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第 8 章 &nbsp; 堆
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8.1 &nbsp; 堆
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8.2 &nbsp; 建堆操作
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第 9 章 &nbsp; 图
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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9.2 &nbsp; 图基础操作
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9.3 &nbsp; 图的遍历
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第 10 章 &nbsp; 搜索
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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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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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序算法
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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11.10 &nbsp; 基数排序
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11.11 &nbsp; 小结
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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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12.2 &nbsp; 分治搜索策略
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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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12.5 &nbsp; 小结
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第 13 章 &nbsp; 回溯
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13.1 &nbsp; 回溯算法
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13.2 &nbsp; 全排列问题
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13.3 &nbsp; 子集和问题
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13.4 &nbsp; N 皇后问题
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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<h1 id="11">1.1 &nbsp; 算法无处不在<a class="headerlink" href="#11" title="Permanent link">&para;</a></h1>
<p>当我们听到“算法”这个词时，很自然地会想到数学。然而实际上，许多算法并不涉及复杂数学，而是更多地依赖基本逻辑，这些逻辑在我们的日常生活中处处可见。</p>
<p>在正式探讨算法之前，有一个有趣的事实值得分享：<strong>你已经在不知不觉中学会了许多算法，并习惯将它们应用到日常生活中了</strong>。下面我将举几个具体的例子来证实这一点。</p>
<p><strong>例一：查字典</strong>。在字典里，每个汉字都对应一个拼音，而字典是按照拼音字母顺序排列的。假设我们需要查找一个拼音首字母为 <span class="arithmatex">\(r\)</span> 的字，通常会按照图 1-1 所示的方式实现。</p>
<ol>
<li>翻开字典约一半的页数，查看该页的首字母是什么，假设首字母为 <span class="arithmatex">\(m\)</span> 。</li>
<li>由于在拼音字母表中 <span class="arithmatex">\(r\)</span> 位于 <span class="arithmatex">\(m\)</span> 之后，所以排除字典前半部分，查找范围缩小到后半部分。</li>
<li>不断重复步骤 <code>1.</code> 和 步骤 <code>2.</code> ，直至找到拼音首字母为 <span class="arithmatex">\(r\)</span> 的页码为止。</li>
</ol>
<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label></div>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="查字典步骤" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="binary_search_dictionary_step2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="binary_search_dictionary_step3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="binary_search_dictionary_step4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="binary_search_dictionary_step5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></a></p>
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<p align="center"> 图 1-1 &nbsp; 查字典步骤 </p>
<p>查字典这个小学生必备技能，实际上就是著名的“二分查找”算法。从数据结构的角度，我们可以把字典视为一个已排序的“数组”；从算法的角度，我们可以将上述查字典的一系列操作看作“二分查找”。</p>
<p><strong>例二：整理扑克</strong>。我们在打牌时，每局都需要整理手中的扑克牌，使其从小到大排列，实现流程如图 1-2 所示。</p>
<ol>
<li>将扑克牌划分为“有序”和“无序”两部分，并假设初始状态下最左 1 张扑克牌已经有序。</li>
<li>在无序部分抽出一张扑克牌，插入至有序部分的正确位置；完成后最左 2 张扑克已经有序。</li>
<li>不断循环步骤 <code>2.</code> ，每一轮将一张扑克牌从无序部分插入至有序部分，直至所有扑克牌都有序。</li>
</ol>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="扑克排序步骤" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
<p align="center"> 图 1-2 &nbsp; 扑克排序步骤 </p>
<p>上述整理扑克牌的方法本质上是“插入排序”算法，它在处理小型数据集时非常高效。许多编程语言的排序库函数中都有插入排序的身影。</p>
<p><strong>例三：货币找零</strong>。假设我们在超市购买了 <span class="arithmatex">\(69\)</span> 元的商品，给了收银员 <span class="arithmatex">\(100\)</span> 元，则收银员需要找我们 <span class="arithmatex">\(31\)</span> 元。他会很自然地完成如图 1-3 所示的思考。</p>
<ol>
<li>可选项是比 <span class="arithmatex">\(31\)</span> 元面值更小的货币，包括 <span class="arithmatex">\(1\)</span> 元、<span class="arithmatex">\(5\)</span> 元、<span class="arithmatex">\(10\)</span> 元、<span class="arithmatex">\(20\)</span> 元。</li>
<li>从可选项中拿出最大的 <span class="arithmatex">\(20\)</span> 元，剩余 <span class="arithmatex">\(31 - 20 = 11\)</span> 元。</li>
<li>从剩余可选项中拿出最大的 <span class="arithmatex">\(10\)</span> 元，剩余 <span class="arithmatex">\(11 - 10 = 1\)</span> 元。</li>
<li>从剩余可选项中拿出最大的 <span class="arithmatex">\(1\)</span> 元，剩余 <span class="arithmatex">\(1 - 1 = 0\)</span> 元。</li>
<li>完成找零，方案为 <span class="arithmatex">\(20 + 10 + 1 = 31\)</span> 元。</li>
</ol>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/greedy_change.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="货币找零过程" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></a></p>
<p align="center"> 图 1-3 &nbsp; 货币找零过程 </p>
<p>在以上步骤中，我们每一步都采取当前看来最好的选择（尽可能用大面额的货币），最终得到了可行的找零方案。从数据结构与算法的角度看，这种方法本质上是“贪心”算法。</p>
<p>小到烹饪一道菜，大到星际航行，几乎所有问题的解决都离不开算法。计算机的出现使得我们能够通过编程将数据结构存储在内存中，同时编写代码调用 CPU 和 GPU 执行算法。这样一来，我们就能把生活中的问题转移到计算机上，以更高效的方式解决各种复杂问题。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>如果你对数据结构、算法、数组和二分查找等概念仍感到一知半解，请继续往下阅读，本书将引导你迈入数据结构与算法的知识殿堂。</p>
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