Graph learning, depth and breadth traversal

1. What is a graph

Represents a "many-to-many" relationship
including:

• A set of vertices: V (Vertex) is usually used to represent the set of vertices
• A set of edges: usually E (Edge) represents the set of edges
  • Edges are pairs of vertices: (v, w) ∈ E, where v, w ∈ V
  • A directed edge <v, w> represents an edge from v to w (single line)
  • Multiple edges and self-loops are not considered

2. Abstract data type definition

• Type name: Graph
• Data object set: G(V, E) consists of a non-empty finite vertex set v and a finite edge set E.
• Operation set: for any graph G ∈ Graph, and v ∈ V, e ∈ E
  • Graph Create(): Create and return an empty graph;
  • Graph InsertVertex(Graph G, Vertex v): Insert v into G;
  • Graph InsertEdge(Graph G, Edge e): Insert e into G;
  • void DFS(Graph G, Vertex v): Depth-first traverse graph G starting from vertex v;
  • void BFS(Graph G, Vertex v): trigger breadth-first traversal of graph G from vertex v;
  • void ShortestPath(Graph G, Vertex v, int Dist[]): Calculate the shortest distance from vertex v to any other vertex in graph G;
  • void MST(Graph G): Calculate the minimum spanning tree of graph G;
  • …
• The definition of sparse graph in the data structure is: a graph with few edges or arcs (the number of edges |E| is much smaller than |V|²) is called a sparse graph (sparse graph), otherwise the number of edges |E |close to |V|², called a dense graph (dense graph).

How to represent graphs:

/* 图的邻接矩阵表示法 */
  
 #define MaxVertexNum 100    /* 最大顶点数设为100 */
 #define INFINITY 65535        /* ∞设为双字节无符号整数的最大值65535*/
 typedef int Vertex;         /* 用顶点下标表示顶点,为整型 */
 typedef int WeightType;        /* 边的权值设为整型 */
 typedef char DataType;        /* 顶点存储的数据类型设为字符型 */
  
 /* 边的定义 */
 typedef struct ENode *PtrToENode;
 struct ENode{
    
    
     Vertex V1, V2;      /* 有向边<V1, V2> */
     WeightType Weight;  /* 权重 */
 };
 typedef PtrToENode Edge;
         
 /* 图结点的定义 */
 typedef struct GNode *PtrToGNode;
 struct GNode{
    
    
     int Nv;  /* 顶点数 */
     int Ne;  /* 边数   */
     WeightType G[MaxVertexNum][MaxVertexNum]; /* 邻接矩阵 */
     DataType Data[MaxVertexNum];      /* 存顶点的数据 */
     /* 注意：很多情况下，顶点无数据，此时Data[]可以不用出现 */
 };
 typedef PtrToGNode MGraph; /* 以邻接矩阵存储的图类型 */
  
  
  
 MGraph CreateGraph( int VertexNum )
 {
    
     /* 初始化一个有VertexNum个顶点但没有边的图 */
     Vertex V, W;
     MGraph Graph;
      
     Graph = (MGraph)malloc(sizeof(struct GNode)); /* 建立图 */
     Graph->Nv = VertexNum;
     Graph->Ne = 0;
     /* 初始化邻接矩阵 */
     /* 注意：这里默认顶点编号从0开始，到(Graph->Nv - 1) */
     for (V=0; V<Graph->Nv; V++)
         for (W=0; W<Graph->Nv; W++)  
             Graph->G[V][W] = INFINITY;
              
     return Graph; 
 }
         
 void InsertEdge( MGraph Graph, Edge E )
 {
    
    
      /* 插入边 <V1, V2> */
      Graph->G[E->V1][E->V2] = E->Weight;    
      /* 若是无向图，还要插入边<V2, V1> */
      Graph->G[E->V2][E->V1] = E->Weight;
 }
  
 MGraph BuildGraph()
 {
    
    
     MGraph Graph;
     Edge E;
     Vertex V;
     int Nv, i;
      
     scanf("%d", &Nv);   /* 读入顶点个数 */
     Graph = CreateGraph(Nv); /* 初始化有Nv个顶点但没有边的图 */ 
      
     scanf("%d", &(Graph->Ne));   /* 读入边数 */
     if ( Graph->Ne != 0 ) {
    
     /* 如果有边 */ 
         E = (Edge)malloc(sizeof(struct ENode)); /* 建立边结点 */ 
         /* 读入边，格式为"起点 终点 权重"，插入邻接矩阵 */
         for (i=0; i<Graph->Ne; i++) {
    
    
             scanf("%d %d %d", &E->V1, &E->V2, &E->Weight); 
             /* 注意：如果权重不是整型，Weight的读入格式要改 */
             InsertEdge( Graph, E );
         }
     } 
  
     /* 如果顶点有数据的话，读入数据 */
     for (V=0; V<Graph->Nv; V++) 
         scanf(" %c", &(Graph->Data[V]));
  
     return Graph;
 }

Link table: G[N] is an array of pointers, corresponding to a linked list for each row of the matrix, and only non-zero elements are stored.

For a network, the domain in the structure to add weight to.

   /* 图的邻接表表示法 */
      
     #define MaxVertexNum 100    /* 最大顶点数设为100 */
     typedef int Vertex;         /* 用顶点下标表示顶点,为整型 */
     typedef int WeightType;        /* 边的权值设为整型 */
     typedef char DataType;        /* 顶点存储的数据类型设为字符型 */
      
     /* 边的定义 */
     typedef struct ENode *PtrToENode;
     struct ENode{
    
    
         Vertex V1, V2;      /* 有向边<V1, V2> */
         WeightType Weight;  /* 权重 */
     };
     typedef PtrToENode Edge;
      
     /* 邻接点的定义 */
     typedef struct AdjVNode *PtrToAdjVNode; 
     struct AdjVNode{
    
    
         Vertex AdjV;        /* 邻接点下标 */
         WeightType Weight;  /* 边权重 */
         PtrToAdjVNode Next;    /* 指向下一个邻接点的指针 */
     };
      
     /* 顶点表头结点的定义 */
     typedef struct Vnode{
    
    
         PtrToAdjVNode FirstEdge;/* 边表头指针 */
         DataType Data;            /* 存顶点的数据 */
         /* 注意：很多情况下，顶点无数据，此时Data可以不用出现 */
     } AdjList[MaxVertexNum];    /* AdjList是邻接表类型 */
      
     /* 图结点的定义 */
     typedef struct GNode *PtrToGNode;
     struct GNode{
    
      
         int Nv;     /* 顶点数 */
         int Ne;     /* 边数   */
         AdjList G;  /* 邻接表 */
     };
     typedef PtrToGNode LGraph; /* 以邻接表方式存储的图类型 */
      
      
      
     LGraph CreateGraph( int VertexNum )
     {
    
     /* 初始化一个有VertexNum个顶点但没有边的图 */
         Vertex V;
         LGraph Graph;
          
         Graph = (LGraph)malloc( sizeof(struct GNode) ); /* 建立图 */
         Graph->Nv = VertexNum;
         Graph->Ne = 0;
         /* 初始化邻接表头指针 */
         /* 注意：这里默认顶点编号从0开始，到(Graph->Nv - 1) */
            for (V=0; V<Graph->Nv; V++)
             Graph->G[V].FirstEdge = NULL;
                  
         return Graph; 
     }
             
     void InsertEdge( LGraph Graph, Edge E )
     {
    
    
         PtrToAdjVNode NewNode;
          
         /* 插入边 <V1, V2> */
         /* 为V2建立新的邻接点 */
         NewNode = (PtrToAdjVNode)malloc(sizeof(struct AdjVNode));
         NewNode->AdjV = E->V2;
         NewNode->Weight = E->Weight;
         /* 将V2插入V1的表头 */
         NewNode->Next = Graph->G[E->V1].FirstEdge;
         Graph->G[E->V1].FirstEdge = NewNode;
              
         /* 若是无向图，还要插入边 <V2, V1> */
         /* 为V1建立新的邻接点 */
         NewNode = (PtrToAdjVNode)malloc(sizeof(struct AdjVNode));
         NewNode->AdjV = E->V1;
         NewNode->Weight = E->Weight;
         /* 将V1插入V2的表头 */
         NewNode->Next = Graph->G[E->V2].FirstEdge;
         Graph->G[E->V2].FirstEdge = NewNode;
     }
      
     LGraph BuildGraph()
     {
    
    
         LGraph Graph;
         Edge E;
         Vertex V;
         int Nv, i;
          
         scanf("%d", &Nv);   /* 读入顶点个数 */
         Graph = CreateGraph(Nv); /* 初始化有Nv个顶点但没有边的图 */ 
          
         scanf("%d", &(Graph->Ne));   /* 读入边数 */
         if ( Graph->Ne != 0 ) {
    
     /* 如果有边 */ 
             E = (Edge)malloc( sizeof(struct ENode) ); /* 建立边结点 */ 
             /* 读入边，格式为"起点 终点 权重"，插入邻接矩阵 */
             for (i=0; i<Graph->Ne; i++) {
    
    
                 scanf("%d %d %d", &E->V1, &E->V2, &E->Weight); 
                 /* 注意：如果权重不是整型，Weight的读入格式要改 */
                 InsertEdge( Graph, E );
             }
         } 
      
         /* 如果顶点有数据的话，读入数据 */
         for (V=0; V<Graph->Nv; V++) 
             scanf(" %c", &(Graph->G[V].Data));
      
         return Graph;
     }

in

typedef struct Vnode{
    
    
    PtrToAdjVNode FirstEdge;/* 边表头指针 */
    DataType Data;            /* 存顶点的数据 */
    /* 注意：很多情况下，顶点无数据，此时Data可以不用出现 */
} AdjList[MaxVertexNum];    /* AdjList是邻接表类型 */

//AdjList是一个Vnode为元素的数组的别名

The degree of a graph is the number of edges associated with a vertex

Three, graph traversal

3.1 Depth-first algorithm

adjacency list

/* 邻接表存储的图 - DFS*/

void Visit(Vertex V)
{
    
    
    printf("Now visit Vertex %d\n", V);
}

/* Visited[]为全局变量，已经初始化false */
void DFS(LGraph Graph, Vertex V, void (*Visit)(Vertex))
{
    
       /* 以V为出发点对邻接表存储的图Graph进行DFS搜索 */
    PtrToAdjVNode W;

    Visit(V);   /* 访问第V个顶点 */
    Visited[V] = true;  /* 标记V已访问 */

    for(W=Graph->G[V].FirstEdge;W;W=W->Next)    /* 对V的每个邻接点W->AdjV */
        if(!Visited[W->AdjV])       /* 若W->AdjV未被访问 */
            DFS(Graph, W->AdjV, Visit);     /* 则递归访问之 */
}

adjacency matrix

void Visit(Vertex V)
{
    
    
    printf("Now visit Vertex %d\n", V);
}

void DFS(MGraph Graph, Vertex V, int *Visited)
{
    
    
    Vertex W;

    Visit(V);
    Visited[V] = 1;         //已访问

    for(W=0;W<Graph->Nv;W++)
        if(Graph->G[V][W]==1 && Visited[W]==0)
            DFS(Graph, W, Visited);
}

3.2 Breadth-first algorithm

adjacency matrix

/* 邻接矩阵存储的图 - BFS */

/* IsEdge(Graph, V, W)检查<V, W>是否图Graph中的一条边，即W是否V的邻接点 */
/* 此函数根据图的不同类型要做不同的实现，关键取决于对不存在的边的表示方法  */
/* 例如对有权图，如果不存在的边被初始化为INFINITY，则函数实现如下：       */
bool IsEdge(MGraph Graph, Vertex V, Vertex W)
{
    
    
    return Graph->G[V][W]<INFINITY?true:false;
}

/* Visited[]为全局变量，已经初始化为false */
void BFS(MGraph Graph, Vertex S, void(*Visit)(Vertex))
{
    
       /* 以S为出发点对邻接矩阵存储的图Graph进行BFS搜索 */
    Queue Q;
    Vertex V, W;

    Q = CreateQueue(MaxSize);   /* 创建空队列，MaxSize为外部定义的常数 */
    /* 访问顶点S:此处可根据具体访问需要改写 */
    Visit(S);
    Visited[S] = true;  /* 标记S已访问 */
    AddQ(Q, S);          /* S入对列 */

    while(!IsEmpty(Q)) {
    
    
        V = DeleteQ(Q);     /* 弹出V */
        for(W=0;W<Graph->Nv;W++)    /* 对图中的每个顶点W */
            /* 若W是V的邻接点并且未访问过 */
            if(!Visited[W] && IsEdge(Graph, V, W)) {
    
    
                /* 访问顶点W */
                Visist(W);
                Visited[W] = true;  /* 标记W已访问 */
                AddQ(Q, W);         /* W入队列 */
            }
    } /* while结束 */
}