13-图的基础：存储与表示

ysu_03142026-09-11 10:24

C语言数据结构系列：图基础篇

C语言数据结构系列（十三）：图的基础------存储与表示

🎯 本篇目标 ：理解图的概念，掌握邻接矩阵和邻接表两种存储方式！

📝 摘要 ：本文是 C 语言数据结构系列第十三篇，系统讲解图（Graph）的基础概念与两种核心存储方式。首先介绍图的定义、术语（顶点、边、有向/无向、带权等）与类型；随后重点剖析邻接矩阵 （二维数组，查询快 O(1)，适合稠密图）和邻接表（链表存储，节省空间 O(V+E)，适合稀疏图）的原理、图解、创建代码及优缺点；最后通过对比表格总结两者适用场景，并给出完整可运行的 C 语言示例。适合正在学习数据结构、想掌握图存储实现的读者。

一、前言

哈喽小伙伴们！👋

今天我们进入图（Graph）的世界！图是最复杂的数据结构，可以表示各种关系：

  • 🗺️ 地图导航
  • 👥 社交网络
  • 🌐 互联网
  • 📊 依赖关系

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上海
广州
深圳

让我们开始今天的学习吧！ 🚀

二、图的基本概念

2.1 什么是图？

图（Graph） 由顶点（Vertex） 和边（Edge） 组成，表示多对多的关系。

2.2 图的术语

术语 说明 示例
顶点 节点 城市
边 连接关系 道路
有向图 边有方向 微博关注
无向图 边无方向 微信好友
带权图 边有权值 距离/费用
完全图 任意两点都有边 -
连通图 任意两点可达 -

2.3 图的类型

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A
B
C
无向图
A
B
C

三、邻接矩阵

3.1 什么是邻接矩阵？

💡 用二维数组 表示图，matrix[i][j]=1 表示i和j有边

c 复制代码 
#define MAX_VERTEX 100

typedef struct {
    char vexs[MAX_VERTEX];     // 顶点表
    int arc[MAX_VERTEX][MAX_VERTEX];  // 邻接矩阵
    int vexNum, arcNum;
} MGraph;

3.2 图解

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B
C

邻接矩阵：

复制代码 
   A  B  C
A [0, 1, 1]
B [1, 0, 1]
C [1, 1, 0]

3.3 创建图

c 复制代码 
void createGraph(MGraph *G, char vexs[], int n, int edges[][2], int m) {
    G->vexNum = n;
    G->arcNum = m;
    
    // 初始化顶点
    for (int i = 0; i < n; i++) {
        G->vexs[i] = vexs[i];
    }
    
    // 初始化矩阵
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            G->arc[i][j] = 0;
        }
    }
    
    // 填充边
    for (int i = 0; i < m; i++) {
        int v1 = edges[i][0];
        int v2 = edges[i][1];
        G->arc[v1][v2] = 1;
        G->arc[v2][v1] = 1;  // 无向图
    }
}

3.4 优缺点

优点 缺点
实现简单 空间O(V²)
查询快O(1) 稀疏图浪费空间
适合稠密图 -

四、邻接表

4.1 什么是邻接表？

💡 用链表存储每个顶点的邻接点，节省稀疏图的空间

c 复制代码 
typedef struct ArcNode {
    int adjvex;
    struct ArcNode *next;
} ArcNode;

typedef struct {
    char data;
    ArcNode *first;
} VNode;

typedef struct {
    VNode vertices[MAX_VERTEX];
    int vexNum, arcNum;
} ALGraph;

4.2 图解

复制代码 
图：
A -- B
A -- C
B -- C

邻接表：
A -> B -> C -> NULL
B -> A -> C -> NULL
C -> A -> B -> NULL

4.3 创建图

c 复制代码 
void createGraph(ALGraph *G, char vexs[], int n, int edges[][2], int m) {
    G->vexNum = n;
    G->arcNum = m;
    
    for (int i = 0; i < n; i++) {
        G->vertices[i].data = vexs[i];
        G->vertices[i].first = NULL;
    }
    
    for (int i = 0; i < m; i++) {
        int v1 = edges[i][0];
        int v2 = edges[i][1];
        
        ArcNode *node = (ArcNode*)malloc(sizeof(ArcNode));
        node->adjvex = v2;
        node->next = G->vertices[v1].first;
        G->vertices[v1].first = node;
        
        node = (ArcNode*)malloc(sizeof(ArcNode));
        node->adjvex = v1;
        node->next = G->vertices[v2].first;
        G->vertices[v2].first = node;
    }
}

4.4 优缺点

优点 缺点
空间O(V+E) 查询慢O(度)
适合稀疏图 实现稍复杂
节省空间 -

五、两种存储方式对比

特性 邻接矩阵 邻接表
空间 O(V²) O(V+E)
查询边 O(1) O(度)
遍历邻接点 O(V) O(度)
适用场景 稠密图 稀疏图

六、完整示例

c 复制代码 
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>

#define MAX_VERTEX 100

// 邻接矩阵
typedef struct {
    char vexs[MAX_VERTEX];
    int arc[MAX_VERTEX][MAX_VERTEX];
    int vexNum, arcNum;
} MGraph;

// 邻接表
typedef struct ArcNode {
    int adjvex;
    struct ArcNode *next;
} ArcNode;

typedef struct {
    char data;
    ArcNode *first;
} VNode;

typedef struct {
    VNode vertices[MAX_VERTEX];
    int vexNum, arcNum;
} ALGraph;

// 查找顶点下标
int locateVex(MGraph *G, char v) {
    for (int i = 0; i < G->vexNum; i++) {
        if (G->vexs[i] == v) return i;
    }
    return -1;
}

// 创建无向图（邻接矩阵）
void createMGraph(MGraph *G) {
    printf("输入顶点数和边数: ");
    scanf("%d %d", &G->vexNum, &G->arcNum);
    
    printf("输入顶点: ");
    for (int i = 0; i < G->vexNum; i++) {
        scanf(" %c", &G->vexs[i]);
    }
    
    for (int i = 0; i < G->vexNum; i++) {
        for (int j = 0; j < G->vexNum; j++) {
            G->arc[i][j] = 0;
        }
    }
    
    printf("输入边（如 AB）:\n");
    for (int i = 0; i < G->arcNum; i++) {
        char v1, v2;
        scanf(" %c%c", &v1, &v2);
        int i1 = locateVex(G, v1);
        int i2 = locateVex(G, v2);
        G->arc[i1][i2] = 1;
        G->arc[i2][i1] = 1;
    }
}

// 打印邻接矩阵
void printMGraph(MGraph *G) {
    printf("\n邻接矩阵:\n  ");
    for (int i = 0; i < G->vexNum; i++) {
        printf("%c ", G->vexs[i]);
    }
    printf("\n");
    for (int i = 0; i < G->vexNum; i++) {
        printf("%c ", G->vexs[i]);
        for (int j = 0; j < G->vexNum; j++) {
            printf("%d ", G->arc[i][j]);
        }
        printf("\n");
    }
}

// 打印邻接表
void printALGraph(ALGraph *G) {
    printf("\n邻接表:\n");
    for (int i = 0; i < G->vexNum; i++) {
        printf("%c -> ", G->vertices[i].data);
        ArcNode *p = G->vertices[i].first;
        while (p) {
            printf("%d ", p->adjvex);
            p = p->next;
        }
        printf("\n");
    }
}

int main() {
    MGraph G;
    createMGraph(&G);
    printMGraph(&G);
    return 0;
}

七、应用

  1. 社交网络 👥

    • 好友关系

  2. 地图导航 🗺️

    • 路线规划

  3. 编译器依赖 ⚙️

    • 拓扑排序

  4. 网络路由 🌐

    • 最短路径

八、下篇预告

下一篇我们将学习 图的遍历：DFS与BFS！
#mermaid-svg-YQFaDAPxGsTRLYmt{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;fill:#333;}@keyframes edge-animation-frame{from{stroke-dashoffset:0;}}@keyframes dash{to{stroke-dashoffset:0;}}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-animation-slow{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 50s linear infinite;stroke-linecap:round;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-animation-fast{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 20s linear infinite;stroke-linecap:round;}#mermaid-svg-YQFaDAPxGsTRLYmt .error-icon{fill:#552222;}#mermaid-svg-YQFaDAPxGsTRLYmt .error-text{fill:#552222;stroke:#552222;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-thickness-normal{stroke-width:1px;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-thickness-thick{stroke-width:3.5px;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-pattern-solid{stroke-dasharray:0;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-thickness-invisible{stroke-width:0;fill:none;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-pattern-dashed{stroke-dasharray:3;}#mermaid-svg-YQFaDAPxGsTRLYmt .edge-pattern-dotted{stroke-dasharray:2;}#mermaid-svg-YQFaDAPxGsTRLYmt .marker{fill:#333333;stroke:#333333;}#mermaid-svg-YQFaDAPxGsTRLYmt .marker.cross{stroke:#333333;}#mermaid-svg-YQFaDAPxGsTRLYmt svg{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;}#mermaid-svg-YQFaDAPxGsTRLYmt p{margin:0;}#mermaid-svg-YQFaDAPxGsTRLYmt .label{font-family:"trebuchet ms",verdana,arial,sans-serif;color:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster-label text{fill:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster-label span{color:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster-label span p{background-color:transparent;}#mermaid-svg-YQFaDAPxGsTRLYmt .label text,#mermaid-svg-YQFaDAPxGsTRLYmt span{fill:#333;color:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt .node rect,#mermaid-svg-YQFaDAPxGsTRLYmt .node circle,#mermaid-svg-YQFaDAPxGsTRLYmt .node ellipse,#mermaid-svg-YQFaDAPxGsTRLYmt .node polygon,#mermaid-svg-YQFaDAPxGsTRLYmt .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#mermaid-svg-YQFaDAPxGsTRLYmt .rough-node .label text,#mermaid-svg-YQFaDAPxGsTRLYmt .node .label text,#mermaid-svg-YQFaDAPxGsTRLYmt .image-shape .label,#mermaid-svg-YQFaDAPxGsTRLYmt .icon-shape .label{text-anchor:middle;}#mermaid-svg-YQFaDAPxGsTRLYmt .node .katex path{fill:#000;stroke:#000;stroke-width:1px;}#mermaid-svg-YQFaDAPxGsTRLYmt .rough-node .label,#mermaid-svg-YQFaDAPxGsTRLYmt .node .label,#mermaid-svg-YQFaDAPxGsTRLYmt .image-shape .label,#mermaid-svg-YQFaDAPxGsTRLYmt .icon-shape .label{text-align:center;}#mermaid-svg-YQFaDAPxGsTRLYmt .node.clickable{cursor:pointer;}#mermaid-svg-YQFaDAPxGsTRLYmt .root .anchor path{fill:#333333!important;stroke-width:0;stroke:#333333;}#mermaid-svg-YQFaDAPxGsTRLYmt .arrowheadPath{fill:#333333;}#mermaid-svg-YQFaDAPxGsTRLYmt .edgePath .path{stroke:#333333;stroke-width:2.0px;}#mermaid-svg-YQFaDAPxGsTRLYmt .flowchart-link{stroke:#333333;fill:none;}#mermaid-svg-YQFaDAPxGsTRLYmt .edgeLabel{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-YQFaDAPxGsTRLYmt .edgeLabel p{background-color:rgba(232,232,232, 0.8);}#mermaid-svg-YQFaDAPxGsTRLYmt .edgeLabel rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-YQFaDAPxGsTRLYmt .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster text{fill:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt .cluster span{color:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#mermaid-svg-YQFaDAPxGsTRLYmt .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#mermaid-svg-YQFaDAPxGsTRLYmt rect.text{fill:none;stroke-width:0;}#mermaid-svg-YQFaDAPxGsTRLYmt .icon-shape,#mermaid-svg-YQFaDAPxGsTRLYmt .image-shape{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-YQFaDAPxGsTRLYmt .icon-shape p,#mermaid-svg-YQFaDAPxGsTRLYmt .image-shape p{background-color:rgba(232,232,232, 0.8);padding:2px;}#mermaid-svg-YQFaDAPxGsTRLYmt .icon-shape .label rect,#mermaid-svg-YQFaDAPxGsTRLYmt .image-shape .label rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-YQFaDAPxGsTRLYmt .label-icon{display:inline-block;height:1em;overflow:visible;vertical-align:-0.125em;}#mermaid-svg-YQFaDAPxGsTRLYmt .node .label-icon path{fill:currentColor;stroke:revert;stroke-width:revert;}#mermaid-svg-YQFaDAPxGsTRLYmt :root{--mermaid-font-family:"trebuchet ms",verdana,arial,sans-serif;} DFS: 深度优先
BFS: 广度优先

💡 学习建议：图的存储方式要根据稀疏/稠密选择！

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