广度优先搜索算法

1.基本概念

BFS即从图中的某个顶点出发，将与该顶点相邻接且未被访问过的结点全部访问完，然后按照之前的访问顺序依次再访问它们自己的邻接结点，直到图中的所有已经被访问道的邻接结点都被访问到。如果此时还有结点没有被访问到，则选择该结点继续按照之前的方式进行访问。

如下图所示：

该广度优先搜索算法的遍历过程如下：

首先访问结点A，然后找到A的邻接结点B和C，访问这两个结点。然后从B出发，找到B的未被访问的结点邻接结点D和E，然后访问这两个结点，然后从结点C出发，找到C的未被访问的邻接结点F和G，然后访问这两个结点，然后从结点D出发，找到D的违背访问的邻接结点H，访问这个结点。然后从结点E出发…依此类推。

2.非递归实现

由于广度优先搜索需要记录某结点的所有未被访问的邻接结点，并且需要按照这些邻接结点的访问顺序获得邻接结点，因此在实现时需要用到队列。

其实现如下所示：

void BFSTraverse(ALGraph G){//非递归形式宽度优先访问树for(int i = 0; i<G.vertexNum; i++) visited[i] = false;queue<int> Q;for(int v = 0; v<G.vertexNum; v++){if(!visited[v]){visited[v] = true;cout<<"node: "<<G.vertices[v].data<<" ";Q.push(v);while(!Q.empty()){int u = Q.front();Q.pop();for(int w = FirstAdjVex(G,u); w>= 0; w = NextAdjVex(G, u, w)){if(!visited[w]){visited[w] = true;cout<<"node: "<<G.vertices[w].data<<" ";Q.push(w);}}}}}}

测试全部代码如下所示：

// BFS.cpp : Defines the entry point for the console application.//#include "stdafx.h"#include <iostream>#include <string>#include <queue>#include <stack>#define MAX_VERTEX_NUM 20using namespace std;typedef int infoType;//弧信息typedef char vertexType;//顶点保存字符信息typedef int Status;typedef struct ArcNode{int adjvex;struct ArcNode *nextArc;infoType *info;}ArcNode;typedef struct VertexNode{vertexType data;ArcNode *firstArc;}VertexNode, AdjList[MAX_VERTEX_NUM];typedef struct ALGraph{AdjList vertices;int vertexNum, arcNum;string kind;}ALGraph;int locateNode(ALGraph &G, VertexNode node){for(int i=0; i<G.vertexNum;i++){if(node.data == G.vertices[i].data)return i;}return -1;}void insertArcAction(ALGraph &G, int nodeNum1, int nodeNum2);Status insertArc(ALGraph &G, VertexNode node1, VertexNode node2);Status createGraph(ALGraph &G, string kind, int vertexNum, int arcNum){//采用邻接表构造图（有向或无向图）G.kind = kind;G.vertexNum = vertexNum;G.arcNum = arcNum;//初始化顶点信息for(int i = 0; i<G.vertexNum; i++){cin>>G.vertices[i].data;G.vertices[i].firstArc = NULL;}cout<<"Try to input arcs info"<<endl;for(int j = 0; j<G.arcNum; j++){cout<<"please input two nodes of "<<j+1<<"-th arc"<<endl;VertexNode node1, node2;cin>>node1.data>>node2.data;insertArc(G, node1, node2);}return 1;}void insertArcAction(ALGraph &G, int nodeNum1, int nodeNum2){ArcNode *p;ArcNode *arc;arc = new ArcNode[1];arc->adjvex = nodeNum2;p = G.vertices[nodeNum1].firstArc;//相当于链表的插入if(!p){G.vertices[nodeNum1].firstArc = arc;arc->nextArc = NULL;}else{G.vertices[nodeNum1].firstArc = arc;arc->nextArc = p;}}Status insertArc(ALGraph &G, VertexNode node1, VertexNode node2){int nodeNum1 = locateNode(G, node1);//i和j表示AdjList[MAX_VERTEX_NUM]中的位置int nodeNum2 = locateNode(G, node2);if(nodeNum1<0 || nodeNum2<0) exit(-1);if(G.kind == "DG")insertArcAction(G, nodeNum1, nodeNum2);else{insertArcAction(G, nodeNum1, nodeNum2);insertArcAction(G, nodeNum2, nodeNum1);}return 1;}Status printALGraph(ALGraph &G){for(int i = 0; i<G.vertexNum; i++){cout<<i<<" "<<G.vertices[i].data;ArcNode *arc = G.vertices[i].firstArc;while(arc){cout<<"-->"<<arc->adjvex;arc = arc->nextArc;}cout<<"-->NULL"<<endl;}return 1;}bool visited[MAX_VERTEX_NUM];int FirstAdjVex(ALGraph G, int j){ArcNode *firstArc = G.vertices[j].firstArc;if(firstArc)return (firstArc->adjvex);else return -1;}int NextAdjVex(ALGraph G, int j, int k){ArcNode *firstArc = G.vertices[j].firstArc;ArcNode *cur = firstArc;if(!firstArc)return -1;else{while(cur){if(cur->adjvex == k)break;cur = cur->nextArc;}}if(cur->nextArc)return (cur->nextArc->adjvex);elsereturn -1;}void BFSTraverse(ALGraph G){//非递归形式宽度优先访问树for(int i = 0; i<G.vertexNum; i++) visited[i] = false;queue<int> Q;for(int v = 0; v<G.vertexNum; v++){if(!visited[v]){visited[v] = true;cout<<"node: "<<G.vertices[v].data<<" ";Q.push(v);while(!Q.empty()){int u = Q.front();Q.pop();for(int w = FirstAdjVex(G,u); w>= 0; w = NextAdjVex(G, u, w)){if(!visited[w]){visited[w] = true;cout<<"node: "<<G.vertices[w].data<<" ";Q.push(w);}}}}}}int _tmain(int argc, _TCHAR* argv[]){ALGraph G;string kind = "UDG";int vertexNum = 9;int arcNum = 9;cout<<"Try to create a Adjacency list Graph ..."<<endl;createGraph(G, kind, vertexNum, arcNum);cout<<"Try to print a Adjacence list Graph ..."<<endl;printALGraph(G);cout<<"Try to traverse a undigraph in a regular form..."<<endl;//非递归BFSTraverse(G);cout<<endl;system("pause");return 0;}