The shortest path of the data structure refers to the path with the smallest sum of weights on the edges between two nodes in the network graph.
Understanding of Dijkstra's algorithm
It is similar to the previous minimum spanning tree.
From the origin to the end point,
then start with the source point V0, and
define three arrays
final[MAX]: it is initialized to 0 for judgment.
ShortPathTable[MAX]: It is used to store the sum of weights (according to the array subscript).
It is updated every time the loop is based on the current node position and the weights of the following nodes and the smallest path found last time The
updated part of the sum of values min is: for example, I go from V0 to V8 node. There is a V5 node in between, and min is the smallest path from V0 to V5, then the elements of the SPT array will be updated to the weight of the edge related to V5 + min
Patharc[MAX]: Store the shortest path of the previous node record of the subscript position of the array
#include<iostream>
#include <iomanip>
using namespace std;
typedef char Vertextype;//顶点类型
typedef int Edgetype;//边缘权值
typedef int Status;
#define MAXVER 9//最大顶点数
#define INF 65535//代表无穷
#define NULL 0
//定义Dijktre算法的相关数组
typedef int Patharc[MAXVER];//存储最短路径数组下标
typedef int ShortPathTable[MAXVER];//用于存储路径之间的权值最小和
//定义图——邻接矩阵
typedef struct Graph
{
Vertextype Ver[MAXVER];
Edgetype Arc[MAXVER][MAXVER];
int NumVer, NumEdg;
}MGraph;
//生成图——邻接矩阵
Status CreatGraph(MGraph &G)
{
int i, j, w;
cout << "Please enter the number of verticesof the graph ： " << endl;
cin >> G.NumVer;
cout << "Please enter the number of edges the graph ： " << endl;
cin >> G.NumEdg;
cout << "Please enter the name of vex : " << endl;
for (i = 0; i < G.NumVer; i++){
//cout << "Please enter the NO." << i + 1 << "%d name of vex : " << endl;
cin >> G.Ver[i];
}
cout << "Diagonal infinity ..." << endl;
for (i = 0; i < G.NumVer; i++)
for (j = 0; j < G.NumVer; j++)
{
G.Arc[i][j] = INF;//简单图 不循环
//cout << G.arc[i][j] << endl;//不理解为啥是1
}
cout << "...Diagonal infinity" << endl;
for (int k = 0; k < G.NumEdg; k++){
//因为具体哪条边存在不一定 所以选择性输入边
cout << "Enter the subscripts and weights from vertex vi to vertex vj : " << endl;
cin >> i >> j >> w;
/*cout << "Please enter the subscript j of the edge : " << endl;
cin >> j;
cout << "Please enter the weight from vertex "<<i<<" to vertex "<<j<<" : " << endl;
cin >> w;*/
G.Arc[i][j] = w;
G.Arc[j][i] = G.Arc[i][j];//无向图 边的信息 是对称的 //有向图的话 无需设置
}
return 0;
}
Status Shortpath_Dijkstra(MGraph &G, int V0, Patharc &P, ShortPathTable &D)
{
int v, w, k, min;
int final[MAXVER];
//初始化过程
//final数组为判决数组 初始化为0 若走过 则改 1 原始位置为1
//Patharc数组记录最小路径的位置 （下标对应 该位置的上一结点）
//ShortPathTable数组 记录两个结点之间最小路径的权值之和 会根据结点所在位置不断更新
for (v = 0; v < G.NumVer; v++)
{
final[v] = 0;
(D)[v] = G.Arc[V0][v];
(P)[v] = 0;
}
(D)[V0] = 0;
final[V0] = 1;
//一个一个的开始找
for (v = 1; v < G.NumVer; v++)
{
min = INF;//找出某结点边的最小权值 初始化为一个最大值
//寻找出某结点边权值的最小个 且包括源点到它的总权值
for (w = 0; w < G.NumVer; w++)
{
if (!final[w] && (D)[w] < min)
{
k = w;
min = (D)[w];
}
}
//当找出了某结点的最小权值之和
final[k] = 1;//标记此结点
for (w = 0; w < G.NumVer; w++)
{
if (!final[w] && min + G.Arc[k][w] < D[w])//判断结点是否走过 和 选择相关边
{
//从某点 找到 该点 最小权值对应结点K 然后将D P 更新 分别保存D为对应w位置值为前min+对应点边值p记录位置
(D)[w] = min + G.Arc[k][w];
(P)[w] = k;
}
}
}
return 0;
}
Status DispPath(Patharc &P, ShortPathTable &D)
{
for (int i = 0; i < MAXVER; i++)
{
cout << P[i] << endl;
}
for (int i = 0; i < MAXVER; i++)
{
cout << D[i] << endl;
}
return 0;
}
int main()
{
MGraph G;
Patharc P;
ShortPathTable D;
CreatGraph(G);
int V = 0;
Shortpath_Dijkstra(G, V, P, D);
DispPath(P,D);
system("pause");
return 0;
}
Here we must understand the meaning of Patharc[MAXVER] and ShortPathTable[MAXVER]. The
final result of the former is the distance between the subscript of the array and the source point. The distance between the point and the point must be subtracted from the distance including the source point. The
latter is similar to a
hand animation. Can deepen understanding