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<!DOCTYPE html>
<html lang="zh-CN">
<head>
<meta charset="UTF-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>16 图论基础：万物互联的模型</title>
<script src="https://unpkg.com/@tailwindcss/browser@4"></script>
<link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.0/css/all.min.css" rel="stylesheet"/>
<style type="text/tailwindcss">
@theme {
--color-bg-dark: #030712;
--color-text-main: #f1f5f9;
--color-accent: #8b5cf6; /* Violet for Graph */
--color-secondary: #ec4899; /* Pink */
--color-tertiary: #06b6d4; /* Cyan */
--color-warn: #f59e0b;
--color-code-bg: #1e293b;
}
@layer base {
body { @apply bg-black text-text-main font-sans overflow-hidden flex justify-center items-center h-screen; }
}
@layer utilities {
.slide-container { @apply relative w-[1920px] h-[1080px] bg-bg-dark overflow-hidden shadow-2xl; }
.slide { @apply absolute inset-0 hidden flex-col p-14 opacity-0 transition-opacity duration-400 ease-in-out; }
.slide.active { @apply flex opacity-100; }
.bg-decoration { @apply absolute inset-0 z-0 pointer-events-none; background-image: radial-gradient(circle at 15% 15%, rgba(139,92,246,0.05) 0%, transparent 30%), radial-gradient(circle at 85% 85%, rgba(6,182,212,0.05) 0%, transparent 30%); }
.grid-overlay { @apply absolute inset-0 z-0; background-size: 60px 60px; background-image: linear-gradient(to right, rgba(255,255,255,0.02) 1px, transparent 1px), linear-gradient(to bottom, rgba(255,255,255,0.02) 1px, transparent 1px); }
h1 { @apply text-[100px] font-extrabold leading-tight mb-8; background: linear-gradient(135deg, #fff 0%, var(--color-accent) 100%); -webkit-background-clip: text; background-clip: text; -webkit-text-fill-color: transparent; }
h2 { @apply text-[64px] mb-8 text-accent inline-block pb-4 border-b-4 border-accent/30; }
h3 { @apply text-[42px] mb-6 text-secondary; }
p { @apply text-[32px] mb-4 text-slate-300; }
.slide-content li { @apply text-[32px] mb-4 text-slate-300; }
strong { @apply text-accent font-bold; }
.card { @apply bg-white/5 border border-white/10 rounded-3xl p-6 mb-6 backdrop-blur-sm; }
.code-block { @apply font-mono bg-code-bg p-6 rounded-2xl border border-slate-700 text-[22px] leading-relaxed text-slate-200 overflow-x-auto whitespace-pre; }
.code-kw { @apply text-[#c678dd]; }
.code-type { @apply text-[#e5c07b]; }
.code-cm { @apply text-[#64748b] italic; }
.code-num { @apply text-[#d19a66]; }
.btn-primary { @apply px-6 py-3 bg-accent/20 border border-accent/50 rounded-xl text-accent font-bold text-[24px] cursor-pointer transition-all hover:bg-accent/30 hover:scale-105 active:scale-95; }
.btn-secondary { @apply px-6 py-3 bg-slate-700/50 border border-slate-600 rounded-xl text-slate-300 font-bold text-[24px] cursor-pointer transition-all hover:bg-slate-600/50; }
}
</style>
</head>
<body>
<div id="app" class="slide-container">
<div class="bg-decoration"></div>
<div class="grid-overlay"></div>
<!-- P01: 封面 -->
<section class="slide active justify-center items-center text-center" data-title="封面">
<div class="flex flex-col items-center z-10 relative">
<!-- Animated Graph Decoration -->
<svg width="600" height="400" viewBox="0 0 600 400" class="mb-8 overflow-visible">
<defs>
<filter id="glow"><feGaussianBlur stdDeviation="4" result="coloredBlur"/><feMerge><feMergeNode in="coloredBlur"/><feMergeNode in="SourceGraphic"/></feMerge></filter>
<linearGradient id="grad1" x1="0%" y1="0%" x2="100%" y2="100%"><stop offset="0%" style="stop-color:#8b5cf6;stop-opacity:1" /><stop offset="100%" style="stop-color:#06b6d4;stop-opacity:1" /></linearGradient>
</defs>
<!-- Edges -->
<g stroke="#475569" stroke-width="2" opacity="0.5">
<path d="M300 50 L100 200" /><path d="M300 50 L500 200" /><path d="M300 50 L300 350" />
<path d="M100 200 L500 200" /><path d="M100 200 L300 350" /><path d="M500 200 L300 350" />
</g>
<!-- Nodes -->
<g fill="url(#grad1)" filter="url(#glow)">
<circle cx="300" cy="50" r="20"><animate attributeName="r" values="20;25;20" dur="3s" repeatCount="indefinite"/></circle>
<circle cx="100" cy="200" r="20"><animate attributeName="r" values="20;25;20" dur="3s" begin="1s" repeatCount="indefinite"/></circle>
<circle cx="500" cy="200" r="20"><animate attributeName="r" values="20;25;20" dur="3s" begin="2s" repeatCount="indefinite"/></circle>
<circle cx="300" cy="350" r="20"><animate attributeName="r" values="20;25;20" dur="3s" begin="1.5s" repeatCount="indefinite"/></circle>
</g>
</svg>
<div class="bg-purple-500/10 px-6 py-2 rounded-full border border-purple-500/30 mb-6">
<span class="text-purple-400 text-2xl font-mono tracking-widest">GRAPH THEORY</span>
</div>
<h1>数据结构与算法</h1>
<p class="text-[56px] text-secondary">图 (Graph)：万物互联</p>
</div>
</section>
<!-- P02: 实际价值 -->
<section class="slide" data-title="实际价值">
<h2>为什么要学图？</h2>
<div class="grid grid-cols-3 gap-8 h-full">
<!-- Row 1 -->
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-users"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">社交网络</h3>
<p class="text-slate-400 text-[28px]">利用<strong>连通性</strong>做推荐，用<strong>BFS</strong>算六度分隔。</p>
</div>
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-map-marked-alt"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">地图导航</h3>
<p class="text-slate-400 text-[28px]">将路口建模为顶点，用<strong>Dijkstra</strong>算最短路径。</p>
</div>
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-project-diagram"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">任务调度</h3>
<p class="text-slate-400 text-[28px]">解析依赖关系，用<strong>拓扑排序</strong>决定执行顺序。</p>
</div>
<!-- Row 2 -->
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-globe"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">互联网 (Web)</h3>
<p class="text-slate-400 text-[28px]">网页是顶点，链接是边，用<strong>PageRank</strong>算权重。</p>
</div>
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-atom"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">知识图谱</h3>
<p class="text-slate-400 text-[28px]">实体间建立语义关联，支持<strong>复杂推理</strong>。</p>
</div>
<div class="card flex flex-col items-center text-center border-slate-700 bg-slate-800/30 hover:bg-slate-700/30 transition-colors p-8">
<div class="text-[64px] text-accent mb-6"><i class="fas fa-network-wired"></i></div>
<h3 class="text-white text-[36px] font-bold mb-4">网络拓扑</h3>
<p class="text-slate-400 text-[28px]">用<strong>最小生成树</strong>优化布线成本或路由广播。</p>
</div>
</div>
</section>
<!-- P03: 什么是图 -->
<section class="slide" data-title="定义与对比">
<h2>什么是图 (Graph)？</h2>
<div class="grid grid-cols-2 gap-12 h-full">
<div class="flex flex-col justify-center">
<div class="card p-8 border-slate-700 bg-slate-800/30">
<h3 class="text-accent text-[40px] mb-6"><i class="fas fa-quote-left mr-4"></i>定义</h3>
<p class="text-[36px] text-white leading-relaxed">
图是由<strong class="text-white">顶点 (Vertex)</strong> 的有穷非空集合和顶点之间的<strong class="text-white">边 (Edge)</strong> 集合组成的数据结构。
</p>
<div class="mt-8 p-6 bg-black/40 rounded-xl border border-slate-600 font-mono text-[32px] text-center text-accent">
G = (V, E)
</div>
</div>
<div class="mt-8">
<h4 class="text-[38px] text-slate-300 mb-4"><i class="fas fa-lightbulb text-accent mr-3"></i>核心本质</h4>
<p class="text-[32px] text-slate-400">表示 <strong class="text-white">"多对多" (Many-to-Many)</strong> 的关系。</p>
</div>
</div>
<div class="flex flex-col justify-center space-y-8">
<!-- Linear -->
<div class="flex items-center p-8 bg-slate-800/30 rounded-2xl border border-slate-700">
<div class="w-1/3 text-center border-r border-slate-600 pr-6">
<div class="text-[36px] text-slate-400">线性表</div>
<div class="text-[28px] text-slate-500">1 对 1</div>
</div>
<div class="w-2/3 pl-10">
<svg width="300" height="80" viewBox="0 0 300 80">
<circle cx="30" cy="40" r="15" fill="#94a3b8"/>
<line x1="45" y1="40" x2="95" y2="40" stroke="#475569" stroke-width="3"/>
<circle cx="110" cy="40" r="15" fill="#94a3b8"/>
<line x1="125" y1="40" x2="175" y2="40" stroke="#475569" stroke-width="3"/>
<circle cx="190" cy="40" r="15" fill="#94a3b8"/>
</svg>
</div>
</div>
<!-- Tree -->
<div class="flex items-center p-8 bg-slate-800/30 rounded-2xl border border-slate-700">
<div class="w-1/3 text-center border-r border-slate-600 pr-6">
<div class="text-[36px] text-slate-400">树 (Tree)</div>
<div class="text-[28px] text-slate-500">1 对 多</div>
</div>
<div class="w-2/3 pl-10">
<svg width="300" height="140" viewBox="0 0 300 140">
<circle cx="150" cy="25" r="15" fill="#cbd5e1"/>
<line x1="150" y1="40" x2="90" y2="110" stroke="#475569" stroke-width="3"/>
<line x1="150" y1="40" x2="210" y2="110" stroke="#475569" stroke-width="3"/>
<circle cx="90" cy="110" r="15" fill="#cbd5e1"/>
<circle cx="210" cy="110" r="15" fill="#cbd5e1"/>
</svg>
</div>
</div>
<!-- Graph -->
<div class="flex items-center p-8 bg-accent/10 rounded-2xl border border-accent/50">
<div class="w-1/3 text-center border-r border-accent/30 pr-6">
<div class="text-[36px] text-accent font-bold">图 (Graph)</div>
<div class="text-[28px] text-slate-300">多 对 多</div>
</div>
<div class="w-2/3 pl-10">
<svg width="300" height="160" viewBox="0 0 300 160">
<path d="M50 30 L250 30 L150 130 Z" stroke="#8b5cf6" stroke-width="3" fill="none"/>
<circle cx="50" cy="30" r="18" fill="#8b5cf6"/>
<circle cx="250" cy="30" r="18" fill="#8b5cf6"/>
<circle cx="150" cy="130" r="18" fill="#8b5cf6"/>
<path d="M50 30 L150 130" stroke="#8b5cf6" stroke-width="3"/>
</svg>
</div>
</div>
</div>
</div>
</section>
<!-- P04: 课程路线图 -->
<section class="slide" data-title="课程路线图">
<h2>图论课程路线图</h2>
<div class="flex items-center justify-center h-full">
<div class="grid grid-cols-3 gap-10 w-full max-w-[1700px]">
<!-- L1 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-project-diagram"></i></div>
<p class="text-slate-200 text-[42px] font-bold">图论基础</p>
<p class="text-slate-400 text-[30px] mt-3">顶点、边、度、权重、方向</p>
</div>
<!-- L2 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-database"></i></div>
<p class="text-slate-200 text-[42px] font-bold">图的存储</p>
<p class="text-slate-400 text-[30px] mt-3">邻接矩阵 & 邻接表</p>
</div>
<!-- L3 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-search-location"></i></div>
<p class="text-slate-200 text-[42px] font-bold">图的遍历</p>
<p class="text-slate-400 text-[30px] mt-3">BFS & DFS</p>
</div>
<!-- L4 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-tree"></i></div>
<p class="text-slate-200 text-[42px] font-bold">最小生成树</p>
<p class="text-slate-400 text-[30px] mt-3">Prim & Kruskal</p>
</div>
<!-- L5 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-route"></i></div>
<p class="text-slate-200 text-[42px] font-bold">最短路径</p>
<p class="text-slate-400 text-[30px] mt-3">Dijkstra & Floyd</p>
</div>
<!-- L6 -->
<div class="card bg-slate-800/50 border-slate-700 h-[380px] flex flex-col justify-center items-center">
<div class="text-[90px] text-accent mb-6"><i class="fas fa-flag-checkered"></i></div>
<p class="text-slate-200 text-[42px] font-bold">关键路径</p>
<p class="text-slate-400 text-[30px] mt-3">拓扑排序 & AOE网</p>
</div>
</div>
</div>
</section>
<!-- P05: 核心概念1 - 顶点与边 -->
<section class="slide" data-title="概念：方向">
<h2>无向图 vs 有向图</h2>
<div class="grid grid-cols-2 gap-16 h-full items-center px-12">
<!-- Undirected -->
<div class="card border-slate-700 bg-slate-800/30 relative overflow-hidden group p-10">
<div class="absolute top-0 right-0 bg-slate-700 px-6 py-3 rounded-bl-2xl text-slate-200 font-bold text-[28px]">无向图 (Undirected)</div>
<div class="flex justify-center my-10">
<svg width="450" height="300" viewBox="0 0 450 300">
<!-- Edges -->
<line x1="225" y1="50" x2="320" y2="120" stroke="#a855f7" stroke-width="3" />
<line x1="320" y1="120" x2="290" y2="230" stroke="#a855f7" stroke-width="3" />
<line x1="290" y1="230" x2="160" y2="230" stroke="#a855f7" stroke-width="3" />
<line x1="160" y1="230" x2="130" y2="120" stroke="#a855f7" stroke-width="3" />
<line x1="130" y1="120" x2="225" y2="50" stroke="#a855f7" stroke-width="3" />
<line x1="225" y1="50" x2="290" y2="230" stroke="#a855f7" stroke-width="3" />
<line x1="130" y1="120" x2="290" y2="230" stroke="#a855f7" stroke-width="3" />
<!-- Nodes -->
<circle cx="225" cy="50" r="25" fill="#581c87" stroke="#a855f7" stroke-width="3"/>
<text x="225" y="58" text-anchor="middle" fill="#fff" font-size="20">A</text>
<circle cx="320" cy="120" r="25" fill="#581c87" stroke="#a855f7" stroke-width="3"/>
<text x="320" y="128" text-anchor="middle" fill="#fff" font-size="20">B</text>
<circle cx="290" cy="230" r="25" fill="#581c87" stroke="#a855f7" stroke-width="3"/>
<text x="290" y="238" text-anchor="middle" fill="#fff" font-size="20">C</text>
<circle cx="160" cy="230" r="25" fill="#581c87" stroke="#a855f7" stroke-width="3"/>
<text x="160" y="238" text-anchor="middle" fill="#fff" font-size="20">D</text>
<circle cx="130" cy="120" r="25" fill="#581c87" stroke="#a855f7" stroke-width="3"/>
<text x="130" y="128" text-anchor="middle" fill="#fff" font-size="20">E</text>
</svg>
</div>
<ul class="space-y-6 text-[36px]">
<li class="flex items-start"><i class="fas fa-check text-accent mt-3 mr-4"></i><span>边没有方向，是对称的。</span></li>
<li class="flex items-start"><i class="fas fa-code text-slate-500 mt-3 mr-4"></i><span class="font-mono text-slate-300">(A, B) 等同于 (B, A)</span></li>
<li class="flex items-start"><i class="fas fa-example text-slate-500 mt-3 mr-4"></i><span>例子：微信好友，握手。</span></li>
</ul>
</div>
<!-- Directed -->
<div class="card border-slate-700 bg-slate-800/30 relative overflow-hidden p-10">
<div class="absolute top-0 right-0 bg-slate-700 px-6 py-3 rounded-bl-2xl text-slate-200 font-bold text-[28px]">有向图 (Directed)</div>
<div class="flex justify-center my-10">
<svg width="450" height="300" viewBox="0 0 450 300">
<defs>
<marker id="arrow-pink" markerWidth="10" markerHeight="10" refX="18" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#8b5cf6" />
</marker>
</defs>
<!-- Edges -->
<line x1="225" y1="50" x2="320" y2="120" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="320" y1="120" x2="290" y2="230" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="290" y1="230" x2="160" y2="230" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="160" y1="230" x2="130" y2="120" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="130" y1="120" x2="225" y2="50" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="225" y1="50" x2="290" y2="230" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<line x1="160" y1="230" x2="225" y2="50" stroke="#8b5cf6" stroke-width="3" marker-end="url(#arrow-pink)"/>
<!-- Nodes -->
<circle cx="225" cy="50" r="25" fill="#4c1d95" stroke="#8b5cf6" stroke-width="3"/>
<text x="225" y="58" text-anchor="middle" fill="#fff" font-size="20">A</text>
<circle cx="320" cy="120" r="25" fill="#4c1d95" stroke="#8b5cf6" stroke-width="3"/>
<text x="320" y="128" text-anchor="middle" fill="#fff" font-size="20">B</text>
<circle cx="290" cy="230" r="25" fill="#4c1d95" stroke="#8b5cf6" stroke-width="3"/>
<text x="290" y="238" text-anchor="middle" fill="#fff" font-size="20">C</text>
<circle cx="160" cy="230" r="25" fill="#4c1d95" stroke="#8b5cf6" stroke-width="3"/>
<text x="160" y="238" text-anchor="middle" fill="#fff" font-size="20">D</text>
<circle cx="130" cy="120" r="25" fill="#4c1d95" stroke="#8b5cf6" stroke-width="3"/>
<text x="130" y="128" text-anchor="middle" fill="#fff" font-size="20">E</text>
</svg>
</div>
<ul class="space-y-6 text-[36px]">
<li class="flex items-start"><i class="fas fa-check text-accent mt-3 mr-4"></i><span>边有方向，不对称。</span></li>
<li class="flex items-start"><i class="fas fa-code text-slate-500 mt-3 mr-4"></i><span class="font-mono text-slate-300"><A, B> 不等于 <B, A></span></li>
<li class="flex items-start"><i class="fas fa-example text-slate-500 mt-3 mr-4"></i><span>例子：微博关注，网页链接。</span></li>
</ul>
</div>
</div>
</section>
<!-- P06: 核心概念2 - 度 -->
<section class="slide" data-title="概念：度">
<h2>顶点的度 (Degree)</h2>
<div class="grid grid-cols-2 gap-16 h-full">
<!-- Undirected Degree -->
<div class="card bg-slate-800/30 border-slate-700 p-8 flex flex-col">
<h3 class="text-white text-[40px] mb-6 border-b border-slate-700 pb-4">无向图 (Undirected)</h3>
<p class="text-[32px] mb-8">度 = 与该顶点相连的边的数量。</p>
<div class="flex-1 flex items-center justify-center bg-black/30 rounded-2xl relative">
<svg width="300" height="310" viewBox="0 0 300 310">
<line x1="150" y1="150" x2="50" y2="50" stroke="#60a5fa" stroke-width="4" />
<line x1="150" y1="150" x2="250" y2="50" stroke="#60a5fa" stroke-width="4" />
<line x1="150" y1="150" x2="150" y2="280" stroke="#60a5fa" stroke-width="4" />
<!-- Other Nodes (Opacity 0.5) -->
<circle cx="50" cy="50" r="20" fill="#0f172a" stroke="#60a5fa" stroke-width="4""/>
<circle cx="250" cy="50" r="20" fill="#0f172a" stroke="#60a5fa" stroke-width="4""/>
<circle cx="150" cy="280" r="20" fill="#0f172a" stroke="#60a5fa" stroke-width="4""/>
<circle cx="150" cy="150" r="40" fill="#0f172a" stroke="#60a5fa" stroke-width="4"/>
<text x="150" y="165" text-anchor="middle" fill="#fff" font-size="36" font-weight="bold">V</text>
</svg>
<div class="absolute bottom-6 bg-slate-900/80 px-6 py-2 rounded-full border border-slate-700">
<span class="text-slate-300 text-[32px] font-mono">Degree(V) = 3</span>
</div>
</div>
</div>
<!-- Directed Degree -->
<div class="card bg-slate-800/30 border-slate-700 p-8 flex flex-col">
<h3 class="text-white text-[40px] mb-6 border-b border-slate-700 pb-4">有向图 (Directed)</h3>
<div class="space-y-6 mb-8">
<div class="flex items-center">
<div class="w-4 h-4 rounded-full bg-blue-400 mr-4"></div>
<p class="text-[30px] m-0"><strong class="text-blue-400">入度 (In-degree)</strong>: 指向本顶线的边数量</p>
</div>
<div class="flex items-center">
<div class="w-4 h-4 rounded-full bg-purple-400 mr-4"></div>
<p class="text-[30px] m-0"><strong class="text-purple-400">出度 (Out-degree)</strong>: 从本顶点指向其他顶点的边数量</p>
</div>
<p class="text-[28px] text-slate-400 pl-8">总度数 = 入度 + 出度</p>
</div>
<div class="flex-1 flex items-center justify-center bg-black/30 rounded-2xl relative">
<svg width="400" height="300" viewBox="0 0 400 300">
<defs>
<marker id="arrow-in" markerWidth="8" markerHeight="8" refX="9" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#60a5fa" />
</marker>
<marker id="arrow-out" markerWidth="8" markerHeight="8" refX="9" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#c084fc" />
</marker>
</defs>
<!-- Incoming -->
<line x1="67" y1="61" x2="167" y2="128" stroke="#60a5fa" stroke-width="3" marker-end="url(#arrow-in)"/>
<line x1="67" y1="239" x2="167" y2="172" stroke="#60a5fa" stroke-width="3" marker-end="url(#arrow-in)"/>
<!-- Outgoing -->
<line x1="240" y1="150" x2="330" y2="150" stroke="#c084fc" stroke-width="3" marker-end="url(#arrow-out)"/>
<!-- Other Nodes (Opaque) -->
<circle cx="50" cy="50" r="20" fill="#0f172a" stroke="#94a3b8" stroke-width="4"/>
<circle cx="50" cy="250" r="20" fill="#0f172a" stroke="#94a3b8" stroke-width="4"/>
<circle cx="350" cy="150" r="20" fill="#0f172a" stroke="#94a3b8" stroke-width="4"/>
<!-- Center Node -->
<circle cx="200" cy="150" r="40" fill="#1e293b" stroke="#94a3b8" stroke-width="4"/>
<text x="200" y="165" text-anchor="middle" fill="#fff" font-size="36" font-weight="bold">V</text>
</svg>
<div class="absolute bottom-4 flex gap-4">
<div class="bg-slate-900/80 px-4 py-2 rounded-lg border border-blue-500/50">
<span class="text-blue-400 text-[24px] font-mono">In: 2</span>
</div>
<div class="bg-slate-900/80 px-4 py-2 rounded-lg border border-purple-500/50">
<span class="text-purple-400 text-[24px] font-mono">Out: 1</span>
</div>
</div>
</div>
</div>
</div>
</section>
<!-- P07: 核心概念3 - 权重 -->
<section class="slide" data-title="概念：权重">
<h2>边的权重 (Weight)</h2>
<div class="grid grid-cols-2 gap-12 h-full items-center">
<div class="flex flex-col gap-8">
<div class="card border-l-8 border-slate-600 p-8">
<h3 class="text-accent text-[40px] mb-4">定义</h3>
<p class="text-[32px] text-white leading-relaxed">
赋予边的数值，代表两个顶点之间关系的<strong class="text-accent">强度</strong>或<strong class="text-accent">成本</strong>。
</p>
</div>
<div class="card bg-slate-800/30 p-8">
<h3 class="text-slate-300 text-[32px] mb-6"><i class="fas fa-tags mr-4 text-slate-500"></i>常见含义</h3>
<ul class="space-y-4">
<li class="flex items-center text-[28px]"><span class="w-32 text-slate-400">地图:</span> <span class="text-white">距离 (km)、时间 (min)</span></li>
<li class="flex items-center text-[28px]"><span class="w-32 text-slate-400">网络:</span> <span class="text-white">带宽 (Mbps)、延迟 (ms)</span></li>
<li class="flex items-center text-[28px]"><span class="w-32 text-slate-400">社交:</span> <span class="text-white">亲密度、互动频率</span></li>
</ul>
</div>
<div class="bg-slate-800/30 border border-slate-700 p-6 rounded-xl text-center">
<p class="text-[28px] text-slate-300">
带权图 (Weighted Graph) 又称为 <strong class="text-white">网 (Network)</strong>
</p>
</div>
</div>
<!-- Visual Demo -->
<div class="card bg-black/20 flex flex-col items-center justify-center h-full relative overflow-hidden">
<div class="absolute top-4 left-6 text-slate-500 text-[24px] font-mono">Example: 城市间距离</div>
<svg width="600" height="500" viewBox="0 0 600 500">
<defs>
<marker id="arrow-weight" markerWidth="10" markerHeight="10" refX="22" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#a855f7" />
</marker>
</defs>
<!-- Edges with Weights -->
<!-- A -> B -->
<line x1="150" y1="100" x2="450" y2="100" stroke="#8b5cf6" stroke-width="4" />
<rect x="270" y="80" width="60" height="40" rx="6" fill="#1e293b" stroke="#8b5cf6"/>
<text x="300" y="108" text-anchor="middle" fill="#a855f7" font-size="20" font-weight="bold">800</text>
<!-- A -> C -->
<line x1="150" y1="100" x2="300" y2="350" stroke="#8b5cf6" stroke-width="4" />
<rect x="190" y="210" width="60" height="40" rx="6" fill="#1e293b" stroke="#8b5cf6"/>
<text x="220" y="238" text-anchor="middle" fill="#a855f7" font-size="20" font-weight="bold">500</text>
<!-- B -> C -->
<line x1="450" y1="100" x2="300" y2="350" stroke="#8b5cf6" stroke-width="4" />
<rect x="340" y="210" width="60" height="40" rx="6" fill="#1e293b" stroke="#8b5cf6"/>
<text x="370" y="238" text-anchor="middle" fill="#a855f7" font-size="20" font-weight="bold">300</text>
<!-- Nodes -->
<circle cx="150" cy="100" r="30" fill="#1e293b" stroke="#fff" stroke-width="3"/>
<text x="150" y="110" text-anchor="middle" fill="#fff" font-size="24">北京</text>
<circle cx="450" cy="100" r="30" fill="#1e293b" stroke="#fff" stroke-width="3"/>
<text x="450" y="110" text-anchor="middle" fill="#fff" font-size="24">上海</text>
<circle cx="300" cy="350" r="30" fill="#1e293b" stroke="#fff" stroke-width="3"/>
<text x="300" y="360" text-anchor="middle" fill="#fff" font-size="24">武汉</text>
</svg>
</div>
</div>
</section>
<!-- P08: 核心概念3 - 路径与环 -->
<section class="slide" data-title="概念：路径与环">
<h2>路径 (Path) & 环 (Cycle)</h2>
<div class="grid grid-cols-2 gap-12 h-full">
<div class="flex flex-col justify-center gap-8">
<div class="card border-l-8 border-slate-600 p-8">
<h3 class="text-white text-[40px] mb-4">路径 (Path)</h3>
<p class="text-[32px] text-slate-200 mb-4">从一个顶点到另一个顶点，途径的顶点序列</p>
<div class="text-[24px] text-white bg-bg-dark p-4 rounded-lg inline-block">简单路径：不重复经过顶点的路径。</div>
</div>
<div class="card border-l-8 border-slate-600 p-8">
<h3 class="text-white text-[40px] mb-4">环 (Cycle)</h3>
<p class="text-[32px] text-slate-200">起点和终点相同的路径。</p>
<p class="text-[28px] text-slate-500 mt-4">例如：A -> B -> E -> A</p>
</div>
</div>
<!-- Visual: Path & Cycle -->
<div class="card bg-black/20 flex items-center justify-center relative">
<svg width="700" height="500" viewBox="0 0 700 500">
<defs>
<marker id="arrow-gray" markerWidth="12" markerHeight="12" refX="19" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#475569" />
</marker>
</defs>
<!-- Edges Layer -->
<g>
<!-- Cross Edges -->
<line x1="250" y1="100" x2="350" y2="400" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)" /> <!-- B->E -->
<line x1="450" y1="100" x2="100" y2="250" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)" /> <!-- C->A -->
<!-- Cycle Edges (A->B->C->D->E->A) -->
<!-- A->B -->
<line x1="100" y1="250" x2="250" y2="100" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)">
<animate attributeName="stroke" values="#475569;#8b5cf6;#8b5cf6;#475569" keyTimes="0;0.1;0.9;1" dur="10s" begin="0s" repeatCount="indefinite" />
</line>
<!-- B->C -->
<line x1="250" y1="100" x2="450" y2="100" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)">
<animate attributeName="stroke" values="#475569;#475569;#8b5cf6;#8b5cf6;#475569" keyTimes="0;0.1;0.2;0.9;1" dur="10s" begin="0s" repeatCount="indefinite" />
</line>
<!-- C->D -->
<line x1="450" y1="100" x2="600" y2="250" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)">
<animate attributeName="stroke" values="#475569;#475569;#8b5cf6;#8b5cf6;#475569" keyTimes="0;0.2;0.3;0.9;1" dur="10s" begin="0s" repeatCount="indefinite" />
</line>
<!-- D->E -->
<line x1="600" y1="250" x2="350" y2="400" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)">
<animate attributeName="stroke" values="#475569;#475569;#8b5cf6;#8b5cf6;#475569" keyTimes="0;0.3;0.4;0.9;1" dur="10s" begin="0s" repeatCount="indefinite" />
</line>
<!-- E->A -->
<line x1="350" y1="400" x2="100" y2="250" stroke="#475569" stroke-width="3" marker-end="url(#arrow-gray)">
<animate attributeName="stroke" values="#475569;#475569;#8b5cf6;#8b5cf6;#475569" keyTimes="0;0.4;0.5;0.9;1" dur="10s" begin="0s" repeatCount="indefinite" />
</line>
</g>
<!-- Nodes Layer -->
<g>
<circle cx="100" cy="250" r="30" fill="#1e293b" stroke="#fff" stroke-width="2"/>
<text x="100" y="260" text-anchor="middle" fill="#fff" font-size="24">A</text>
<circle cx="250" cy="100" r="30" fill="#1e293b" stroke="#fff" stroke-width="2"/>
<text x="250" y="110" text-anchor="middle" fill="#fff" font-size="24">B</text>
<circle cx="450" cy="100" r="30" fill="#1e293b" stroke="#fff" stroke-width="2"/>
<text x="450" y="110" text-anchor="middle" fill="#fff" font-size="24">C</text>
<circle cx="600" cy="250" r="30" fill="#1e293b" stroke="#fff" stroke-width="2"/>
<text x="600" y="260" text-anchor="middle" fill="#fff" font-size="24">D</text>
<circle cx="350" cy="400" r="30" fill="#1e293b" stroke="#fff" stroke-width="2"/>
<text x="350" y="410" text-anchor="middle" fill="#fff" font-size="24">E</text>
</g>
<text x="350" y="480" text-anchor="middle" fill="#8b5cf6" font-size="28">环 (Cycle): A -> B -> C -> D -> E -> A</text>
</svg>
</div>
</div>
</section>
<!-- P09: 核心概念4 - 连通性 -->
<section class="slide" data-title="概念：连通性">
<h2>连通性 (Connectivity)</h2>
<div class="flex flex-col h-full gap-6">
<!-- Top: Concepts (Horizontal Layout) -->
<div class="grid grid-cols-3 gap-6">
<div class="card border-l-8 border-tertiary p-6 bg-slate-800/40">
<h3 class="text-white text-[32px] mb-2 font-bold">连通 (Connected)</h3>
<p class="text-[24px] text-slate-300 leading-normal">顶点之间存在路径即为连通。</p>
</div>
<div class="card border-l-8 border-blue-400 p-6 bg-slate-800/40">
<h3 class="text-white text-[32px] mb-2 font-bold">连通图 (Undirected)</h3>
<p class="text-[24px] text-slate-300 leading-normal">任意两点连通，若不连通则包含多个"连通分量"。</p>
</div>
<div class="card border-l-8 border-pink-400 p-6 bg-slate-800/40">
<h3 class="text-white text-[32px] mb-2 font-bold">强连通 (Directed)</h3>
<p class="text-[24px] text-slate-300 leading-normal">有向图中，任意两个顶点之间都可以互相到达。</p>
</div>
</div>
<!-- Bottom: Visuals (Side by Side, Larger) -->
<div class="flex-1 grid grid-cols-2 gap-8">
<!-- Left: Undirected -->
<div class="card bg-slate-800/20 border border-slate-700/50 rounded-2xl relative flex flex-col overflow-hidden">
<div class="absolute top-4 left-6 px-4 py-1 bg-slate-700/50 rounded-lg text-slate-300 text-[24px] font-bold border border-slate-600">无向图</div>
<div class="flex-1 flex items-center justify-around px-4">
<!-- Connected Graph Example -->
<div class="flex flex-col items-center gap-4">
<svg width="240" height="240" viewBox="0 0 240 240" class="overflow-visible">
<g stroke="#06b6d4" stroke-width="3">
<path d="M120 40 L50 100 L70 190 L170 190 L190 100 Z" fill="none"/>
</g>
<g fill="#0e7490" stroke="#22d3ee" stroke-width="3">
<circle cx="120" cy="40" r="16"/>
<circle cx="50" cy="100" r="16"/>
<circle cx="70" cy="190" r="16"/>
<circle cx="170" cy="190" r="16"/>
<circle cx="190" cy="100" r="16"/>
</g>
</svg>
<span class="text-cyan-400 text-[26px] font-bold">连通图</span>
</div>
<!-- Divider -->
<div class="h-[60%] w-0 border-r-2 border-dashed border-slate-600/50"></div>
<!-- Disconnected Graph Example -->
<div class="flex flex-col items-center gap-4">
<svg width="240" height="240" viewBox="0 0 240 240" class="overflow-visible">
<!-- Component 1 -->
<g stroke="#f472b6" stroke-width="3">
<path d="M70 60 L40 140 L100 140 Z" fill="none"/>
</g>
<g fill="#be185d" stroke="#f472b6" stroke-width="3">
<circle cx="70" cy="60" r="16"/>
<circle cx="40" cy="140" r="16"/>
<circle cx="100" cy="140" r="16"/>
</g>
<!-- Component 2 -->
<g stroke="#a855f7" stroke-width="3">
<line x1="160" y1="80" x2="200" y2="160" />
</g>
<g fill="#7e22ce" stroke="#a855f7" stroke-width="3">
<circle cx="160" cy="80" r="16"/>
<circle cx="200" cy="160" r="16"/>
</g>
</svg>
<span class="text-pink-400 text-[26px] font-bold">非连通图</span>
</div>
</div>
</div>
<!-- Right: Directed -->
<div class="card bg-slate-800/20 border border-slate-700/50 rounded-2xl relative flex flex-col items-center justify-center overflow-hidden">
<div class="absolute top-4 left-6 px-4 py-1 bg-slate-700/50 rounded-lg text-slate-300 text-[24px] font-bold border border-slate-600">有向图</div>
<div class="flex flex-col items-center gap-8">
<svg width="400" height="250" viewBox="0 0 400 250" class="overflow-visible">
<defs>
<marker id="arrow-strong-lg" markerWidth="12" markerHeight="12" refX="16" refY="3" orient="auto">
<path d="M0,0 L0,6 L9,3 z" fill="#c084fc" />
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<circle cx="300" cy="50" r="24"/>
<text x="300" y="58" text-anchor="middle" fill="#fff" font-size="20" stroke="none" font-weight="bold">B</text>
<circle cx="300" cy="200" r="24"/>
<text x="300" y="208" text-anchor="middle" fill="#fff" font-size="20" stroke="none" font-weight="bold">C</text>
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<div class="text-center bg-purple-900/20 px-8 py-4 rounded-xl border border-purple-500/30">
<p class="text-[32px] text-purple-400 font-bold mb-2">强连通图</p>
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<span>A ↔ B</span>
<span>B ↔ C</span>
<span>...</span>
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</div>
</div>
</div>
</div>
</div>
</section>
<!-- P11: 总结 -->
<section class="slide" data-title="小结">
<h2>本节小结</h2>
<div class="flex items-center justify-center h-full">
<div class="card p-10 border-slate-700 bg-slate-800/30 w-2/3">
<ul class="space-y-6 text-[32px]">
<li class="flex items-center">
<span class="w-12 h-12 rounded-full bg-slate-700 flex items-center justify-center text-slate-200 text-xl font-bold mr-6">1</span>
<span>图是 <strong class="text-white">G = (V, E)</strong>，表示多对多关系。</span>
</li>
<li class="flex items-center">
<span class="w-12 h-12 rounded-full bg-slate-700 flex items-center justify-center text-slate-200 text-xl font-bold mr-6">2</span>
<span>边可以有 <strong class="text-white">方向 (Directed)</strong> 和 <strong class="text-white">权重 (Weight)</strong>。</span>
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<li class="flex items-center">
<span class="w-12 h-12 rounded-full bg-slate-700 flex items-center justify-center text-slate-200 text-xl font-bold mr-6">3</span>
<span><strong class="text-white">度 (Degree)</strong> 是连接边的数量，有向图分入度和出度。</span>
</li>
<li class="flex items-center">
<span class="w-12 h-12 rounded-full bg-slate-700 flex items-center justify-center text-slate-200 text-xl font-bold mr-6">4</span>
<span><strong class="text-white">路径 (Path)</strong> 是顶点的序列，闭合的路径叫 <strong class="text-white">环 (Cycle)</strong>。</span>
</li>
<li class="flex items-center">
<span class="w-12 h-12 rounded-full bg-slate-700 flex items-center justify-center text-slate-200 text-xl font-bold mr-6">5</span>
<span><strong class="text-white">连通 (Connected)</strong> 指顶点间存在路径，有向图中有 <strong class="text-white">强连通</strong> 概念。</span>
</li>
</ul>
<div class="mt-12 text-center">
<p class="text-[28px] text-slate-500">Next : 图的存储 (邻接矩阵 & 邻接表)</p>
</div>
</div>
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