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0.3   小结
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第 1 章   初识算法
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1.1   算法无处不在
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1.2   算法是什么
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1.3   小结
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1.   重点回顾
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第 2 章   复杂度分析
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第 2 章   复杂度分析
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2.1   算法效率评估
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2.2   迭代与递归
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2.3   时间复杂度
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2.4   空间复杂度
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2.5   小结
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2.6   练习
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第 3 章   数据结构
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第 3 章   数据结构
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3.1   数据结构分类
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3.2   基本数据类型
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3.3   数字编码 *
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3.4   字符编码 *
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3.5   小结
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3.6   练习
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第 4 章   数组与链表
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第 4 章   数组与链表
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4.1   数组
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4.2   链表
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4.3   列表
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4.4   内存与缓存 *
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4.5   小结
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4.6   练习
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第 5 章   栈与队列
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第 5 章   栈与队列
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5.1   栈
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5.2   队列
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5.3   双向队列
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5.4   小结
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5.5   练习
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第 6 章   哈希表
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第 6 章   哈希表
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6.1   哈希表
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6.2   哈希冲突
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6.3   哈希算法
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6.4   小结
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6.5   练习
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第 7 章   树
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第 7 章   树
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7.1   二叉树
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7.2   二叉树遍历
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7.3   二叉树数组表示
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7.4   二叉搜索树
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7.5   AVL 树 *
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7.6   小结
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7.7   练习
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第 8 章   堆
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第 8 章   堆
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/heap/">
<span class="md-ellipsis">
8.1   堆
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/build_heap/">
<span class="md-ellipsis">
8.2   建堆操作
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</a>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/top_k/">
<span class="md-ellipsis">
8.3   Top-k 问题
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/summary/">
<span class="md-ellipsis">
8.4   小结
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</a>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_heap/exercises/">
<span class="md-ellipsis">
8.5   练习
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第 9 章   图
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第 9 章   图
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph/">
<span class="md-ellipsis">
9.1   图
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph_operations/">
<span class="md-ellipsis">
9.2   图基础操作
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/graph_traversal/">
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9.3   图的遍历
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<a class="md-nav__link" href="../../chapter_graph/summary/">
<span class="md-ellipsis">
9.4   小结
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</a>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_graph/exercises/">
<span class="md-ellipsis">
9.5   练习
</span>
</a>
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<a class="md-nav__link" href="../../chapter_searching/">
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第 10 章   搜索
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第 10 章   搜索
</label>
<ul class="md-nav__list" data-md-scrollfix="">
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search/">
<span class="md-ellipsis">
10.1   二分查找
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search_insertion/">
<span class="md-ellipsis">
10.2   二分查找插入点
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/binary_search_edge/">
<span class="md-ellipsis">
10.3   二分查找边界
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</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/replace_linear_by_hashing/">
<span class="md-ellipsis">
10.4   哈希优化策略
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/searching_algorithm_revisited/">
<span class="md-ellipsis">
10.5   重识搜索算法
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/summary/">
<span class="md-ellipsis">
10.6   小结
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_searching/exercises/">
<span class="md-ellipsis">
10.7   练习
</span>
</a>
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</ul>
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第 11 章   排序
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第 11 章   排序
</label>
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/sorting_algorithm/">
<span class="md-ellipsis">
11.1   排序算法
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/selection_sort/">
<span class="md-ellipsis">
11.2   选择排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/bubble_sort/">
<span class="md-ellipsis">
11.3   冒泡排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/insertion_sort/">
<span class="md-ellipsis">
11.4   插入排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/quick_sort/">
<span class="md-ellipsis">
11.5   快速排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/merge_sort/">
<span class="md-ellipsis">
11.6   归并排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/heap_sort/">
<span class="md-ellipsis">
11.7   堆排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/bucket_sort/">
<span class="md-ellipsis">
11.8   桶排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/counting_sort/">
<span class="md-ellipsis">
11.9   计数排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/radix_sort/">
<span class="md-ellipsis">
11.10   基数排序
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/summary/">
<span class="md-ellipsis">
11.11   小结
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_sorting/exercises/">
<span class="md-ellipsis">
11.12   练习
</span>
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第 12 章   分治
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第 12 章   分治
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<a class="md-nav__link" href="../../chapter_divide_and_conquer/divide_and_conquer/">
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12.1   分治算法
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</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/binary_search_recur/">
<span class="md-ellipsis">
12.2   分治搜索策略
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/build_binary_tree_problem/">
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12.3   构建树问题
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</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/hanota_problem/">
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12.4   汉诺塔问题
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/summary/">
<span class="md-ellipsis">
12.5   小结
</span>
</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_divide_and_conquer/exercises/">
<span class="md-ellipsis">
12.6   练习
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第 13 章   回溯
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第 13 章   回溯
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<a class="md-nav__link" href="../../chapter_backtracking/backtracking_algorithm/">
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13.1   回溯算法
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<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_backtracking/permutations_problem/">
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13.2   全排列问题
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<a class="md-nav__link" href="../../chapter_backtracking/subset_sum_problem/">
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13.3   子集和问题
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13.4   N 皇后问题
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13.5   小结
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<a class="md-nav__link" href="../../chapter_backtracking/exercises/">
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13.6   练习
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第 14 章   动态规划
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第 14 章   动态规划
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<a class="md-nav__link" href="../../chapter_dynamic_programming/intro_to_dynamic_programming/">
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14.1   初探动态规划
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</a>
</li>
<li class="md-nav__item">
<a class="md-nav__link" href="../../chapter_dynamic_programming/dp_problem_features/">
<span class="md-ellipsis">
14.2   DP 问题特性
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<a class="md-nav__link" href="../../chapter_dynamic_programming/dp_solution_pipeline/">
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14.3   DP 解题思路
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</li>
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<a class="md-nav__link" href="../../chapter_dynamic_programming/knapsack_problem/">
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14.4   0-1 背包问题
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<a class="md-nav__link" href="../../chapter_dynamic_programming/unbounded_knapsack_problem/">
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14.5   完全背包问题
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<a class="md-nav__link" href="../../chapter_dynamic_programming/edit_distance_problem/">
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14.6   编辑距离问题
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<a class="md-nav__link" href="../../chapter_dynamic_programming/summary/">
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14.7   小结
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14.8   练习
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<h1 id="13">1.3   小结<a class="headerlink" href="#13" title="Permanent link"></a></h1>
<h3 id="1">1.   重点回顾<a class="headerlink" href="#1" title="Permanent link"></a></h3>
<ul>
<li>算法在日常生活中无处不在,并不是遥不可及的高深知识。实际上,我们已经在不知不觉中学会了许多算法,用以解决生活中的大小问题。</li>
<li>查字典的原理与二分查找算法相一致。二分查找算法体现了分而治之的重要算法思想。</li>
<li>整理扑克的过程与插入排序算法非常类似。插入排序算法适合排序小型数据集。</li>
<li>货币找零的步骤本质上是贪心算法,每一步都采取当前看来最好的选择。</li>
<li>算法是在有限时间内解决特定问题的一组指令或操作步骤,而数据结构是计算机中组织和存储数据的方式。</li>
<li>数据结构与算法紧密相连。数据结构是算法的基石,而算法为数据结构注入生命力。</li>
<li>我们可以将数据结构与算法类比为拼装积木,积木代表数据,积木的形状和连接方式等代表数据结构,拼装积木的步骤则对应算法。</li>
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<h3 id="2-q-a">2.   Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link"></a></h3>
<p><strong>Q</strong>:作为一名程序员,我在日常工作中从未用算法解决过问题,常用算法都被编程语言封装好了,直接用就可以了;这是否意味着我们工作中的问题还没有到达需要算法的程度?</p>
<p>如果把具体的工作技能比作是武功的“招式”的话,那么基础科目应该更像是“内功”。</p>
<p>我认为学算法(以及其他基础科目)的意义不是在于在工作中从零实现它,而是基于学到的知识,在解决问题时能够作出专业的反应和判断,从而提升工作的整体质量。举一个简单例子,每种编程语言都内置了排序函数:</p>
<ul>
<li>如果我们没有学过数据结构与算法,那么给定任何数据,我们可能都塞给这个排序函数去做了。运行顺畅、性能不错,看上去并没有什么问题。</li>
<li>但如果学过算法,我们就会知道内置排序函数的时间复杂度是 <span class="arithmatex">\(O(n \log n)\)</span> ;而如果给定的数据是固定位数的整数(例如学号),那么我们就可以用效率更高的“基数排序”来做,将时间复杂度降为 <span class="arithmatex">\(O(nk)\)</span> ,其中 <span class="arithmatex">\(k\)</span> 为位数。当数据体量很大时,节省出来的运行时间就能创造较大价值(成本降低、体验变好等)。</li>
</ul>
<p>在工程领域中,大量问题是难以达到最优解的,许多问题只是被“差不多”地解决了。问题的难易程度一方面取决于问题本身的性质,另一方面也取决于观测问题的人的知识储备。人的知识越完备、经验越多,分析问题就会越深入,问题就能被解决得更优雅。</p>
<p><strong>Q</strong>:现在几乎所有代码都让 AI Agent 写了,我们还有必要学习算法吗?</p>
<p>引用一下李沐老师的话:“agent 是能力放大器,就是能力 x10,而不是加法器,不是能力 +10。”</p>
<p>既然 AI 是能力放大器,那么作为基数的“能力”也很重要。我觉得两方面能力尤其值得积累:一是表达能力,尤其是和 AI 协作的能力。讲清目标、补足背景、及时纠偏。向 AI 解释问题,也是在帮自己想清楚问题。二是基础知识。细节可以让 AI 补,但基本概念得自己懂,才能问到点上、判断对错,把它的输出变成有用的成果。</p>
<p>AI 很特别的一点是,它不像过去的工具那样有清晰的功能边界。它有点像水,到了不同的人手里,会呈现出不同的形状。人的能力越强、taste 越好,AI 的放大效应就越明显。</p>
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