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统计学
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抽样
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推断
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概率论
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计数
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</a>
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概率概念
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分布
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</span>
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</a>
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贝叶斯
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</span>
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</a>
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信息论
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</span>
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机器学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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机器学习
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经典机器学习
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梯度机器学习
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</span>
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</a>
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深度学习
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</span>
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</a>
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强化学习
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</span>
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</a>
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分布式深度学习
|
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</span>
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|
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|
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|
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</a>
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计算语言学
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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语言学基础
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</a>
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嵌入与序列模型
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</span>
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</a>
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Transformer 与语言模型
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</span>
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|
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</a>
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<span class="md-ellipsis">
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高级文本生成
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
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计算机视觉
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计算机视觉
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图像基础
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</span>
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</a>
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目标检测与分割
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多模态学习
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多模态学习
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多模态表征
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</span>
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</a>
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统一多模态架构
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</span>
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自主系统
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感知
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视觉-语言-动作模型
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自动驾驶
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</a>
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太空与极端机器人
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</span>
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图神经网络
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ML 系统设计
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应用 AI
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药物发现
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智能体系统
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医疗健康
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</span>
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前沿 AI
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命题逻辑
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谓词逻辑与量词
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集合
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函数
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图论基础
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递推关系
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可计算性
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复杂度理论
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<a href="#colab" class="md-nav__link">
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编程任务（使用CoLab或笔记本）
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<article class="md-content__inner md-typeset">
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|
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|
||
<h1 id="_1">离散数学<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>离散数学是关于可数、分离结构的数学，是计算构建的基础。本文涵盖命题逻辑与谓词逻辑、证明技巧、集合、关系、函数、图论基础以及递推关系。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在前面的章节中，我们研究了连续数学：微积分（第3章）、概率分布（第5章）以及实值参数的优化（第6章）。但计算机本质上是<strong>离散</strong>机器。它们存储比特（0或1），处理整数，遵循分支逻辑，并操作有限数据结构。<strong>离散数学</strong>提供了推理这些结构的形式化语言。</p>
|
||
</li>
|
||
<li>
|
||
<p>本章所有内容都建立在离散数学之上：处理器逻辑门是布尔代数，调度算法需要正确性证明，内存管理使用集合运算，算法分析需要递推关系。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_2">命题逻辑<a class="headerlink" href="#_2" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>命题逻辑</strong>是真假语句的代数。一个<strong>命题</strong>是一个要么为真（T）要么为假（F）的陈述，绝不会两者兼有。"天在下雨"是一个命题。"现在几点了？"则不是（它是一个问句，不是具有真值的陈述）。</p>
|
||
</li>
|
||
<li>
|
||
<p>命题可以通过<strong>逻辑连接词</strong>进行组合：</p>
|
||
<ul>
|
||
<li><strong>与</strong>（合取，<span class="arithmatex">\(p \wedge q\)</span>）：仅当<span class="arithmatex">\(p\)</span>和<span class="arithmatex">\(q\)</span>都为真时为真。</li>
|
||
<li><strong>或</strong>（析取，<span class="arithmatex">\(p \vee q\)</span>）：当<span class="arithmatex">\(p\)</span>或<span class="arithmatex">\(q\)</span>至少一个为真时为真。</li>
|
||
<li><strong>非</strong>（否定，<span class="arithmatex">\(\neg p\)</span>）：翻转真值。</li>
|
||
<li><strong>蕴含</strong>（蕴涵，<span class="arithmatex">\(p \to q\)</span>）：仅当<span class="arithmatex">\(p\)</span>为真且<span class="arithmatex">\(q\)</span>为假时为假。"如果下雨，地就是湿的"只有在下了雨而地却是干的时候才被违反。</li>
|
||
<li><strong>当且仅当</strong>（双条件，<span class="arithmatex">\(p \leftrightarrow q\)</span>）：当两者真值相同时为真。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>真值表</strong>穷举列出所有可能的输入组合及相应的输出。对于<span class="arithmatex">\(n\)</span>个命题，该表有<span class="arithmatex">\(2^n\)</span>行。这就是我们验证逻辑等价性的方式：</p>
|
||
</li>
|
||
</ul>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th><span class="arithmatex">\(p\)</span></th>
|
||
<th><span class="arithmatex">\(q\)</span></th>
|
||
<th><span class="arithmatex">\(p \wedge q\)</span></th>
|
||
<th><span class="arithmatex">\(p \vee q\)</span></th>
|
||
<th><span class="arithmatex">\(p \to q\)</span></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>T</td>
|
||
<td>T</td>
|
||
<td>T</td>
|
||
<td>T</td>
|
||
<td>T</td>
|
||
</tr>
|
||
<tr>
|
||
<td>T</td>
|
||
<td>F</td>
|
||
<td>F</td>
|
||
<td>T</td>
|
||
<td>F</td>
|
||
</tr>
|
||
<tr>
|
||
<td>F</td>
|
||
<td>T</td>
|
||
<td>F</td>
|
||
<td>T</td>
|
||
<td>T</td>
|
||
</tr>
|
||
<tr>
|
||
<td>F</td>
|
||
<td>F</td>
|
||
<td>F</td>
|
||
<td>F</td>
|
||
<td>T</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<ul>
|
||
<li>
|
||
<p>蕴含行中<span class="arithmatex">\(p\)</span>为假的情况值得关注：<span class="arithmatex">\(F \to q\)</span>无论<span class="arithmatex">\(q\)</span>为何值都为真。这就是<strong>空真</strong>。"如果猪会飞，那我就是英国国王"在逻辑上为真，因为前提为假。这看起来违反直觉，但对数学推理至关重要。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>逻辑等价式</strong>是对所有真值都成立的恒等式：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>德摩根定律</strong>：<span class="arithmatex">\(\neg(p \wedge q) \equiv \neg p \vee \neg q\)</span> 和 <span class="arithmatex">\(\neg(p \vee q) \equiv \neg p \wedge \neg q\)</span>。要否定一个AND，分别否定每个部分并切换为OR（反之亦然）。这些直接出现在编程中：<code>!(a && b)</code> 等价于 <code>(!a || !b)</code>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>逆否命题</strong>：<span class="arithmatex">\(p \to q \equiv \neg q \to \neg p\)</span>。"如果下雨，地就是湿的"等价于"如果地不是湿的，那么就没下雨。"这是一个强大的证明技巧。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>双重否定</strong>：<span class="arithmatex">\(\neg(\neg p) \equiv p\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>分配律</strong>：<span class="arithmatex">\(p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>一个总是为真（对所有真值指派）的公式是<strong>重言式</strong>。总是为假的公式是<strong>矛盾式</strong>。有时真有时假的公式是<strong>偶然式</strong>。例如，<span class="arithmatex">\(p \vee \neg p\)</span>是重言式，<span class="arithmatex">\(p \wedge \neg p\)</span>是矛盾式。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_3">谓词逻辑与量词<a class="headerlink" href="#_3" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>命题逻辑无法表达关于集合中<em>所有</em>或<em>某些</em>元素的陈述。"每个大于2的素数都是奇数"需要<strong>谓词逻辑</strong>，它用变量、谓词和量词扩展了命题逻辑。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>谓词</strong>是依赖于变量的陈述：<span class="arithmatex">\(P(x)\)</span> = "<span class="arithmatex">\(x\)</span>是偶数。"当给定<span class="arithmatex">\(x\)</span>一个具体值时，它成为一个命题：<span class="arithmatex">\(P(4)\)</span>为真，<span class="arithmatex">\(P(7)\)</span>为假。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>量词</strong>表达范围：</p>
|
||
<ul>
|
||
<li><strong>全称量词</strong>（<span class="arithmatex">\(\forall\)</span>）："对于所有。" <span class="arithmatex">\(\forall x \, P(x)\)</span> 表示"<span class="arithmatex">\(P(x)\)</span>对论域中的每一个<span class="arithmatex">\(x\)</span>成立。"</li>
|
||
<li><strong>存在量词</strong>（<span class="arithmatex">\(\exists\)</span>）："存在。" <span class="arithmatex">\(\exists x \, P(x)\)</span> 表示"至少存在一个<span class="arithmatex">\(x\)</span>使得<span class="arithmatex">\(P(x)\)</span>为真。"</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>否定量词会翻转它们：<span class="arithmatex">\(\neg(\forall x \, P(x)) \equiv \exists x \, \neg P(x)\)</span>。"不是所有人都通过了"意味着"有人没通过。"而 <span class="arithmatex">\(\neg(\exists x \, P(x)) \equiv \forall x \, \neg P(x)\)</span>。"没有完美的算法"意味着"每个算法都有缺陷。"</p>
|
||
</li>
|
||
<li>
|
||
<p>嵌套量词表达复杂关系。<span class="arithmatex">\(\forall x \, \exists y \, (y > x)\)</span> 表示"对于每个数，都有一个更大的数"（对整数成立）。顺序很重要：<span class="arithmatex">\(\exists y \, \forall x \, (y > x)\)</span> 表示"存在一个比所有其他数都大的数"（对整数不成立）。</p>
|
||
</li>
|
||
<li>
|
||
<p>谓词逻辑是形式化规约的语言。当我们说一个算法是"正确"的，意味着 <span class="arithmatex">\(\forall \text{输入} \, x, \, \text{输出}(x) = \text{期望输出}(x)\)</span>。当我们说它"终止"，意味着 <span class="arithmatex">\(\forall x \, \exists t \, \text{终止}(x, t)\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_4">证明技巧<a class="headerlink" href="#_4" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>证明</strong>是确立一个陈述真理性、毫无疑义的逻辑论证。与经验证据（仅展示在某些测试案例下有效）不同，证明保证在所有情况下成立。这是计算机科学中正确性的标准。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>直接证明</strong>：假设前提，通过逻辑步骤推导出结论。要证明"如果<span class="arithmatex">\(n\)</span>是偶数，那么<span class="arithmatex">\(n^2\)</span>是偶数"：假设<span class="arithmatex">\(n = 2k\)</span>对于某个整数<span class="arithmatex">\(k\)</span>，则<span class="arithmatex">\(n^2 = 4k^2 = 2(2k^2)\)</span>，这是偶数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>反证法</strong>：假设该陈述为假，推导出矛盾。要证明<span class="arithmatex">\(\sqrt{2}\)</span>是无理数：假设<span class="arithmatex">\(\sqrt{2} = a/b\)</span>（已约简）。那么<span class="arithmatex">\(2 = a^2/b^2\)</span>，所以<span class="arithmatex">\(a^2 = 2b^2\)</span>，意味着<span class="arithmatex">\(a^2\)</span>是偶数，所以<span class="arithmatex">\(a\)</span>是偶数，设<span class="arithmatex">\(a = 2c\)</span>。那么<span class="arithmatex">\(4c^2 = 2b^2\)</span>，所以<span class="arithmatex">\(b^2 = 2c^2\)</span>，意味着<span class="arithmatex">\(b\)</span>也是偶数。但我们已经假设<span class="arithmatex">\(a/b\)</span>是约简形式——矛盾。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>归纳证明</strong>：通过证明以下两点来证明一个陈述对所有自然数成立：（1）<strong>基础情形</strong>成立（通常<span class="arithmatex">\(n = 0\)</span>或<span class="arithmatex">\(n = 1\)</span>），和（2）<strong>归纳步骤</strong>：如果陈述对<span class="arithmatex">\(n = k\)</span>成立（归纳假设），那么它对<span class="arithmatex">\(n = k + 1\)</span>也成立。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，证明 <span class="arithmatex">\(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\)</span>：</p>
|
||
<ul>
|
||
<li>基础情形：<span class="arithmatex">\(n = 1\)</span>：<span class="arithmatex">\(1 = \frac{1 \cdot 2}{2} = 1\)</span>。成立。</li>
|
||
<li>归纳步骤：假设 <span class="arithmatex">\(\sum_{i=1}^{k} i = \frac{k(k+1)}{2}\)</span>。那么 <span class="arithmatex">\(\sum_{i=1}^{k+1} i = \frac{k(k+1)}{2} + (k+1) = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}\)</span>。这正是<span class="arithmatex">\(n = k+1\)</span>时的公式。证明完成。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>归纳法是证明递归算法和数据结构性质的主力工具。每个递归算法都暗含一个归纳正确性证明：基础情形是终止条件，归纳步骤是递归调用。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>强归纳法</strong>假设该陈述对所有不大于<span class="arithmatex">\(k\)</span>的值都成立（不仅仅是<span class="arithmatex">\(k\)</span>），然后证明它对<span class="arithmatex">\(k + 1\)</span>成立。当递归依赖于多个之前的值时，这很有用。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>鸽巢原理</strong>：如果把<span class="arithmatex">\(n+1\)</span>个物体放入<span class="arithmatex">\(n\)</span>个盒子中，至少有一个盒子包含两个物体。简单但出奇地强大。它证明了在任何13个人中，至少有两个人出生月份相同。在网络中，它证明了当项目数超过桶数时，哈希冲突是不可避免的。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_5">集合<a class="headerlink" href="#_5" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>集合</strong>是不同元素的无序收集。集合是数学中最原始的数据结构，支撑着从类型系统到数据库查询的一切。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>集合运算</strong>（联系第5章，我们在那里用这些进行概率计算）：</p>
|
||
<ul>
|
||
<li><strong>并集</strong> <span class="arithmatex">\(A \cup B\)</span>：在<span class="arithmatex">\(A\)</span>或<span class="arithmatex">\(B\)</span>或两者中的元素。</li>
|
||
<li><strong>交集</strong> <span class="arithmatex">\(A \cap B\)</span>：同时在<span class="arithmatex">\(A\)</span>和<span class="arithmatex">\(B\)</span>中的元素。</li>
|
||
<li><strong>补集</strong> <span class="arithmatex">\(\bar{A}\)</span>：不在<span class="arithmatex">\(A\)</span>中的元素（相对于一个全集）。</li>
|
||
<li><strong>差集</strong> <span class="arithmatex">\(A \setminus B\)</span>：在<span class="arithmatex">\(A\)</span>中但不在<span class="arithmatex">\(B\)</span>中的元素。</li>
|
||
<li><strong>笛卡尔积</strong> <span class="arithmatex">\(A \times B\)</span>：所有有序对<span class="arithmatex">\((a, b)\)</span>，其中<span class="arithmatex">\(a \in A, b \in B\)</span>。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>幂集</strong> <span class="arithmatex">\(\mathcal{P}(A)\)</span> 是<span class="arithmatex">\(A\)</span>的所有子集构成的集合。如果 <span class="arithmatex">\(|A| = n\)</span>，那么 <span class="arithmatex">\(|\mathcal{P}(A)| = 2^n\)</span>。对于 <span class="arithmatex">\(A = \{1, 2\}\)</span>：<span class="arithmatex">\(\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>基数</strong>衡量集合大小。有限集具有整数基数。无限集有不同的大小：自然数<span class="arithmatex">\(\mathbb{N}\)</span>和有理数<span class="arithmatex">\(\mathbb{Q}\)</span>是<strong>可数无穷</strong>（可以列举），而实数<span class="arithmatex">\(\mathbb{R}\)</span>是<strong>不可数无穷</strong>（无法列举，由康托尔的对角线论证证明）。这种区别在可计算性理论中很重要：存在不可数多个函数，但只有可数多个程序，因此大多数函数是不可计算的。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_6">关系<a class="headerlink" href="#_6" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>集合<span class="arithmatex">\(A\)</span>上的<strong>关系</strong><span class="arithmatex">\(R\)</span>是<span class="arithmatex">\(A \times A\)</span>的一个子集：指定哪些元素相关联的有序对集合。例如，整数上的<span class="arithmatex">\(\leq\)</span>是集合 <span class="arithmatex">\(\{(a, b) : a \leq b\}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>关系的重要性质：</p>
|
||
<ul>
|
||
<li><strong>自反性</strong>：每个元素与自身相关。对所有<span class="arithmatex">\(a\)</span>有<span class="arithmatex">\(a R a\)</span>。例：<span class="arithmatex">\(\leq\)</span>（每个数<span class="arithmatex">\(\leq\)</span>自身）。</li>
|
||
<li><strong>对称性</strong>：如果<span class="arithmatex">\(a R b\)</span>则<span class="arithmatex">\(b R a\)</span>。例："是……的兄弟姐妹。"</li>
|
||
<li><strong>反对称性</strong>：如果<span class="arithmatex">\(a R b\)</span>且<span class="arithmatex">\(b R a\)</span>则<span class="arithmatex">\(a = b\)</span>。例：<span class="arithmatex">\(\leq\)</span>。</li>
|
||
<li><strong>传递性</strong>：如果<span class="arithmatex">\(a R b\)</span>且<span class="arithmatex">\(b R c\)</span>则<span class="arithmatex">\(a R c\)</span>。例：<span class="arithmatex">\(<\)</span>、<span class="arithmatex">\(\leq\)</span>、"是……的祖先。"</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>等价关系</strong>是自反、对称且传递的。它将集合划分为<strong>等价类</strong>，其中同一类中的所有元素彼此相关，但与不同类中的元素无关。模运算是一个等价关系：<span class="arithmatex">\(a \equiv b \pmod{n}\)</span> 将整数划分为<span class="arithmatex">\(n\)</span>个类。编程语言中的类型等价是一个等价关系。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>偏序</strong>是自反、反对称且传递的。它定义了一个"小于等于"结构，可能会使某些元素不可比较。文件系统目录构成一个偏序（父-子），但同级目录是不可比较的。<strong>全序</strong>是每一对元素都可比较的偏序（如整数上的<span class="arithmatex">\(\leq\)</span>）。</p>
|
||
</li>
|
||
<li>
|
||
<p>偏序在并发中至关重要：事件上的"先于发生"关系是一个偏序。不由先于发生关系排序的事件是并发的，可能以任意相对顺序执行。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_7">函数<a class="headerlink" href="#_7" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>函数</strong> <span class="arithmatex">\(f: A \to B\)</span> 将<span class="arithmatex">\(A\)</span>（定义域）中的每个元素映射到<span class="arithmatex">\(B\)</span>（陪域）中的恰好一个元素。函数是确定性计算的数学模型：给定一个输入，恰好有一个输出。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>单射</strong>（一对一）：不同的输入总是产生不同的输出。<span class="arithmatex">\(f(a) = f(b) \implies a = b\)</span>。无损压缩是单射的：不同的输入必须压缩成不同的输出（否则无法唯一解压）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>满射</strong>（到上）：<span class="arithmatex">\(B\)</span>中的每个元素都被<span class="arithmatex">\(A\)</span>中的某个元素命中。值域等于陪域。将字符串映射到256位哈希的哈希函数，如果字符串数少于可能的哈希数，则不是满射。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>双射</strong>：既是单射又是满射。<span class="arithmatex">\(A\)</span>和<span class="arithmatex">\(B\)</span>之间的一一对应。双射具有逆函数。加密必须是双射的：每个明文映射到唯一的密文，而解密函数就是逆函数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>复合</strong> <span class="arithmatex">\((g \circ f)(x) = g(f(x))\)</span>：先应用<span class="arithmatex">\(f\)</span>，再应用<span class="arithmatex">\(g\)</span>。函数复合是可结合的（第2章：就像矩阵乘法是可结合的一样）。软件中的管道就是函数复合：数据流经一系列变换。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_8">图论基础<a class="headerlink" href="#_8" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>我们在第12章（图神经网络）中广泛介绍了图，包括邻接矩阵、图类型、拉普拉斯矩阵和谱理论。这里我们专注于与CS相关的<strong>算法</strong>和<strong>结构</strong>性质。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>树</strong>是没有环的连通图。等价地，它有<span class="arithmatex">\(n\)</span>个节点和<span class="arithmatex">\(n-1\)</span>条边。树是文件系统、XML/HTML文档、决策过程和递归分解的结构。<strong>有根树</strong>有一个指定的根节点；每个其他节点恰好有一个父节点。</p>
|
||
</li>
|
||
<li>
|
||
<p>图<span class="arithmatex">\(G\)</span>的<strong>生成树</strong>是包含<span class="arithmatex">\(G\)</span>所有节点并使用其边子集的一棵树。<strong>最小生成树（MST）</strong>最小化总边权。Kruskal算法（对边排序，贪心地添加不形成环的最轻边）和Prim算法（从起始节点开始扩展树，总是添加连接到新节点的最轻边）都能在<span class="arithmatex">\(O(|E| \log |V|)\)</span>内找到MST。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>平面性</strong>：如果一个图可以画在平面上而边不相交，则是平面图。根据<strong>欧拉公式</strong>，对于连通平面图：<span class="arithmatex">\(|V| - |E| + |F| = 2\)</span>，其中<span class="arithmatex">\(|F|\)</span>是面的数量（区域，包括外部面）。这意味着平面图的<span class="arithmatex">\(|E| \leq 3|V| - 6\)</span>，因此平面图是稀疏的。电路板布线和地图着色利用了平面性。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>图着色</strong>为节点分配颜色，使得没有两个相邻节点共享相同的颜色。所需的最小颜色数是<strong>色数</strong> <span class="arithmatex">\(\chi(G)\)</span>。<strong>四色定理</strong>指出任何平面图的 <span class="arithmatex">\(\chi(G) \leq 4\)</span>。在CS中，图着色模拟寄存器分配（将变量分配到CPU寄存器，使得同时活跃的变量获得不同的寄存器）和调度（将任务分配到时间槽，使得冲突的任务不重叠）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>欧拉路径</strong>恰好访问每条边一次。当且仅当图中恰好有0个或2个奇数度节点时，欧拉路径存在。<strong>哈密顿路径</strong>恰好访问每个节点一次。确定哈密顿路径是否存在是NP完全的——这是CS中的经典难题之一。这种对比（欧拉：多项式，哈密顿：NP完全）说明了听起来相似的问题可能具有截然不同的计算复杂度。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_9">递推关系<a class="headerlink" href="#_9" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>递推关系</strong>定义一个序列，其中每一项依赖于前面的项。它们自然地从递归算法中产生。</p>
|
||
</li>
|
||
<li>
|
||
<p>最简单的例子：<span class="arithmatex">\(T(n) = T(n-1) + 1\)</span>，其中 <span class="arithmatex">\(T(0) = 0\)</span>。展开：<span class="arithmatex">\(T(n) = T(n-1) + 1 = T(n-2) + 2 = \cdots = n\)</span>。这是<span class="arithmatex">\(O(n)\)</span>，即简单循环的时间复杂度。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>归并排序</strong>给出 <span class="arithmatex">\(T(n) = 2T(n/2) + O(n)\)</span>：将数组分成两半（两个大小为<span class="arithmatex">\(n/2\)</span>的子问题），递归排序每一半，然后合并（<span class="arithmatex">\(O(n)\)</span>工作）。解为 <span class="arithmatex">\(T(n) = O(n \log n)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>主定理</strong>求解形式为 <span class="arithmatex">\(T(n) = aT(n/b) + O(n^d)\)</span> 的递推式：</p>
|
||
<ul>
|
||
<li>如果 <span class="arithmatex">\(d > \log_b a\)</span>：<span class="arithmatex">\(T(n) = O(n^d)\)</span>（每层的工作占主导）</li>
|
||
<li>如果 <span class="arithmatex">\(d = \log_b a\)</span>：<span class="arithmatex">\(T(n) = O(n^d \log n)\)</span>（工作在各层间平衡）</li>
|
||
<li>如果 <span class="arithmatex">\(d < \log_b a\)</span>：<span class="arithmatex">\(T(n) = O(n^{\log_b a})\)</span>（子问题的数量占主导）</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>对于归并排序：<span class="arithmatex">\(a = 2, b = 2, d = 1\)</span>。由于 <span class="arithmatex">\(d = \log_2 2 = 1\)</span>，我们处于平衡情况：<span class="arithmatex">\(T(n) = O(n \log n)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>斐波那契递推</strong> <span class="arithmatex">\(F(n) = F(n-1) + F(n-2)\)</span>，其中 <span class="arithmatex">\(F(0) = 0, F(1) = 1\)</span>，封闭形式解为 <span class="arithmatex">\(F(n) = \frac{\phi^n - \psi^n}{\sqrt{5}}\)</span>，其中 <span class="arithmatex">\(\phi = \frac{1+\sqrt{5}}{2}\)</span>（黄金比例）且 <span class="arithmatex">\(\psi = \frac{1-\sqrt{5}}{2}\)</span>。这表明斐波那契数列以 <span class="arithmatex">\(O(\phi^n)\)</span> 指数增长，这就是为什么朴素递归斐波那契指数级慢。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>组合数学</strong>（排列、组合、二项式定理和容斥原理）在第5章（概率）中介绍。这些计数技术对算法分析至关重要（有多少种可能的输入？需要多少次比较？），但我们在此不再重复。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_10">可计算性<a class="headerlink" href="#_10" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>并非所有事情都能被计算。这是整个数学中最深刻的结论之一，它设定了计算机能力的基本极限。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>图灵机</strong>是计算的抽象模型：一条无限长的单元格磁带（每个单元格包含一个符号），一个读写头，以及一组带转移规则的有限状态。尽管简单，图灵机可以计算任何实际计算机能计算的任何东西。这就是<strong>邱奇-图灵论题</strong>：任何有效可计算的函数都可以由图灵机计算。</p>
|
||
</li>
|
||
<li>
|
||
<p>每种编程语言（Python、C、Haskell）都是<strong>图灵完备</strong>的：它可以模拟图灵机，从而计算任何可计算的东西。语言之间的区别在于便利性、速度和安全性，而不在于它们根本上能计算什么。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>停机问题</strong>询问：给定一个程序和一个输入，该程序最终会停止，还是永远运行？图灵（1936）证明不存在能普遍解决这个问题的算法。证明采用反证法：假设存在一个停机检测器 <span class="arithmatex">\(H(P, x)\)</span>。构造一个程序 <span class="arithmatex">\(D\)</span>，它运行 <span class="arithmatex">\(H(D, D)\)</span> 并做与 <span class="arithmatex">\(H\)</span> 所说的相反的事。如果 <span class="arithmatex">\(H\)</span> 说 <span class="arithmatex">\(D\)</span> 停机，<span class="arithmatex">\(D\)</span> 就永远循环。如果 <span class="arithmatex">\(H\)</span> 说 <span class="arithmatex">\(D\)</span> 循环，<span class="arithmatex">\(D\)</span> 就停机。矛盾。</p>
|
||
</li>
|
||
<li>
|
||
<p>这不是当前技术的局限；这是一个数学上的不可能性。无论多少计算、多少聪明才智、或多少人工智能，都无法普遍解决停机问题。它是哥德尔不完备定理在计算机科学中的类比。</p>
|
||
</li>
|
||
<li>
|
||
<p>实际后果：你无法编写一个完美的死锁检测器、一个完美的病毒扫描器或一个完美的优化编译器。每一个都需要通用地解决停机问题（或一个等价的不判定问题）。实际工具使用启发式方法和近似方法，在常见情况下有效，但不能保证对所有输入都正确。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果一个问题存在一个总是能给出正确是/否答案并终止的算法，则它是<strong>可判定的</strong>。如果不存在这样的算法，则是<strong>不可判定的</strong>。停机问题是不可判定的。素数测试是可判定的。大多数编程语言中的类型检查是可判定的（通过设计）。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_11">复杂度理论<a class="headerlink" href="#_11" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>即使在可计算的问题中，有些也远比其他的难。<strong>复杂度理论</strong>根据解决问题所需的资源（时间、空间）随输入增长而分类问题。</li>
|
||
</ul>
|
||
<p><img alt="P、NP和NP完全：P包含在NP中，NP完全位于边界处，P是否等于NP是核心开放问题" src="../../images/p_np_complexity.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>P</strong>（多项式时间）：能在 <span class="arithmatex">\(O(n^k)\)</span> 时间内解决的问题，<span class="arithmatex">\(k\)</span>为某个常数。排序（<span class="arithmatex">\(O(n \log n)\)</span>）、最短路径（<span class="arithmatex">\(O(|V|^2)\)</span>）、矩阵乘法（<span class="arithmatex">\(O(n^3)\)</span>）。这些被认为是"高效"或"可处理的。"</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>NP</strong>（非确定性多项式时间）：一个拟议的解答能在多项式时间内<strong>验证</strong>的问题，即使<strong>找到</strong>解答可能需要指数时间。例如，给定一个声称的哈密顿路径，你可以通过检查每条边在 <span class="arithmatex">\(O(n)\)</span> 时间内验证它。但找到一条可能需尝试指数多个可能性。</p>
|
||
</li>
|
||
<li>
|
||
<p>P中的每个问题也在NP中（如果你能快速解决它，你当然能快速验证一个解答）。核心问题是 <span class="arithmatex">\(P = NP\)</span> 是否成立：每个能快速验证解答的问题是否也能快速求解？这是计算机科学中最重要的开放问题，获得克莱数学研究所100万美元的千禧年大奖。</p>
|
||
</li>
|
||
<li>
|
||
<p>大多数专家相信 <span class="arithmatex">\(P \neq NP\)</span>，意味着有些问题本质上比验证更难解决。如果 <span class="arithmatex">\(P = NP\)</span>，密码学将崩溃（破解加密属于NP），而优化、调度和药物设计将变得异常简单。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>NP完全</strong>问题是NP中最难的问题。一个问题如果是NP完全的，则：（1）它在NP中，且（2）所有其他NP问题可以在多项式时间内<strong>归约</strong>到它。如果你能高效解决任何一个NP完全问题，你就能解决所有NP完全问题（从而 <span class="arithmatex">\(P = NP\)</span>）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>归约</strong>将一个问题转换为另一个问题。如果问题A归约到问题B，那么B至少和A一样难。Cook（1971）证明了<strong>SAT</strong>（布尔可满足性：给定一个逻辑公式，是否存在使公式为真的变量赋值？）是NP完全的。Karp（1972）通过将SAT归约到每个问题，证明了其他21个经典问题是NP完全的。</p>
|
||
</li>
|
||
<li>
|
||
<p>著名的NP完全问题：</p>
|
||
<ul>
|
||
<li><strong>旅行商问题（TSP）</strong>：找到访问所有城市恰好一次的最短路线。</li>
|
||
<li><strong>图着色</strong>：用<span class="arithmatex">\(k\)</span>种颜色为节点着色，使得没有相邻节点共享同一颜色（<span class="arithmatex">\(k \geq 3\)</span>）。</li>
|
||
<li><strong>子集和问题</strong>：给定一组整数，是否存在一个子集其和等于目标值？</li>
|
||
<li><strong>布尔可满足性（SAT）</strong>：是否存在使逻辑公式为真的真值赋值？</li>
|
||
<li><strong>哈密顿路径</strong>（上文图论中提到的）。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>当你在实践中遇到NP完全问题时，你不会对大规模输入精确求解。相反，你使用：<strong>近似算法</strong>（找到保证在最优解一定倍数范围内的解）、<strong>启发式方法</strong>（贪心、局部搜索、模拟退火）或<strong>特例求解器</strong>（许多NP完全问题对受限输入很容易）。例如，现代SAT求解器尽管在最坏情况下是指数复杂度，但通过利用实际实例中的结构，通常能解决拥有数百万变量的实例。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>NP困难</strong>问题至少和NP完全问题一样难，但可能不在NP中（它们的解甚至可能不能在多项式时间内验证）。NP完全问题的优化版本通常是NP困难的："找到最短TSP路线"是NP困难的，而"是否存在一条长度小于<span class="arithmatex">\(k\)</span>的TSP路线？"是NP完全的。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab">编程任务（使用CoLab或笔记本）<a class="headerlink" href="#colab" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>构建一个真值表生成器。给定一个逻辑表达式，枚举所有输入组合并计算结果。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">itertools</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">truth_table</span><span class="p">(</span><span class="n">n_vars</span><span class="p">,</span> <span class="n">expr_fn</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="sd">"""为一个n_vars个变量的布尔函数生成真值表。"""</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">headers</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">"p</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_vars</span><span class="p">)]</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="nb">print</span><span class="p">(</span><span class="s2">" | "</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">headers</span> <span class="o">+</span> <span class="p">[</span><span class="s2">"result"</span><span class="p">]))</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="nb">print</span><span class="p">(</span><span class="s2">"-"</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">headers</span><span class="p">)</span> <span class="o">*</span> <span class="mi">4</span> <span class="o">+</span> <span class="mi">10</span><span class="p">))</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">for</span> <span class="n">vals</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">([</span><span class="kc">False</span><span class="p">,</span> <span class="kc">True</span><span class="p">],</span> <span class="n">repeat</span><span class="o">=</span><span class="n">n_vars</span><span class="p">):</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="n">result</span> <span class="o">=</span> <span class="n">expr_fn</span><span class="p">(</span><span class="o">*</span><span class="n">vals</span><span class="p">)</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="n">row</span> <span class="o">=</span> <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">v</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">result</span><span class="p">)[</span><span class="mi">0</span><span class="p">]]</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="nb">print</span><span class="p">(</span><span class="s2">" | "</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">r</span><span class="si">:</span><span class="s2">>2</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">row</span><span class="p">))</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="c1"># 德摩根定律：NOT(p AND q) == (NOT p) OR (NOT q)</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="nb">print</span><span class="p">(</span><span class="s2">"德摩根定律验证："</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">truth_table</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">:</span> <span class="p">(</span><span class="ow">not</span> <span class="p">(</span><span class="n">p</span> <span class="ow">and</span> <span class="n">q</span><span class="p">))</span> <span class="o">==</span> <span class="p">((</span><span class="ow">not</span> <span class="n">p</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="n">q</span><span class="p">)))</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>通过归纳法证明求和公式——对多个值进行数值验证，然后实现封闭形式解。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1"># 验证求和公式：sum(1..n) = n(n+1)/2</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">10000</span><span class="p">]:</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="n">brute</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a> <span class="n">formula</span> <span class="o">=</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"n=</span><span class="si">{</span><span class="n">n</span><span class="si">:</span><span class="s2">5d</span><span class="si">}</span><span class="s2"> sum=</span><span class="si">{</span><span class="n">brute</span><span class="si">:</span><span class="s2">>10d</span><span class="si">}</span><span class="s2"> formula=</span><span class="si">{</span><span class="n">formula</span><span class="si">:</span><span class="s2">>10d</span><span class="si">}</span><span class="s2"> match=</span><span class="si">{</span><span class="n">brute</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">formula</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用主定理求解归并排序递推关系，并通过计数操作进行经验验证。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">merge_sort_ops</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="sd">"""统计归并排序中的比较次数（递推：T(n) = 2T(n/2) + n）。"""</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="k">if</span> <span class="n">n</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="k">return</span> <span class="mi">0</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="n">half</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="k">return</span> <span class="n">merge_sort_ops</span><span class="p">(</span><span class="n">half</span><span class="p">)</span> <span class="o">+</span> <span class="n">merge_sort_ops</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">half</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">64</span><span class="p">,</span> <span class="mi">512</span><span class="p">,</span> <span class="mi">4096</span><span class="p">,</span> <span class="mi">32768</span><span class="p">]:</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="n">ops</span> <span class="o">=</span> <span class="n">merge_sort_ops</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="n">predicted</span> <span class="o">=</span> <span class="n">n</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</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">ratio</span> <span class="o">=</span> <span class="n">ops</span> <span class="o">/</span> <span class="n">predicted</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"n=</span><span class="si">{</span><span class="n">n</span><span class="si">:</span><span class="s2">5d</span><span class="si">}</span><span class="s2"> ops=</span><span class="si">{</span><span class="n">ops</span><span class="si">:</span><span class="s2">>10d</span><span class="si">}</span><span class="s2"> n log n=</span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">predicted</span><span class="p">)</span><span class="si">:</span><span class="s2">>10d</span><span class="si">}</span><span class="s2"> ratio=</span><span class="si">{</span><span class="n">ratio</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
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