[C ++] single-source shortest path: Dijkstra (the Dijkstra) algorithm (greedy algorithm)

1 Dijkstra algorithm

1.1 algorithm basic information

• Solve the problem / proposed background
  • Single-source shortest path (in the weighted directed graph, each vertex find the shortest path remaining from a vertex to)
• ALGORITHM
  • how are you
    • Ascending order according to path length, the shortest path algorithm for sequentially generating
  • The scope of the Dijkstra algorithm applies only to FIG model [] weight is positive in
• time complexity
  • O (n ^ 3)
• Supplement
  • It is also applicable to multi-source shortest path [] (recommended: Floyd algorithm (dynamic programming, O (n ^ 3)))

    • Dijkstra time complexity: O (n ^ 3)

      1.2 Algorithm Description

• 1.2.1 solving process (specific ideas)

• 1.2.2 Example

1.2 Programming reproduction

• 1> FIG defined model (the adjacency matrix notation) of the basic memory structure []

# define MaxInt 32767 // 表示极大值 即 ∞ (无穷大)
# define MVNum 100 // 最大顶点数 

typedef int VertexType; // 假设顶点的数据类型为整型

typedef int ArcType; // 假设Vi与Vj之边的权值类型为整型 

typedef struct {
    VertexType vexs[MVNum]; // 顶点表 (存储顶点信息)
    ArcType arcs[MVNum][MVNum]; // 邻接矩阵
    int vexnum,arcnum; // 图的当前顶点数与边数 
}AMGraph; // Adjacent Matrix Graph 邻接矩阵图 

• 2> Dijkstra algorithm defined auxiliary data structure []

bool S[MVNum]; // S[i] 记录从源点V0到终点Vi是否已被确定为最短路径长度  【划分确定与未确定: 跟贪心算法的适用范围(不可取消性)有直接联系】
               // true：表已确定；false：表尚未确定
ArcType D[MVNum]; // D[i] 记录从源点V0到终点Vi的【当前】最短路径【长度】 
int Path[MVNum];  // Path[i] 记录从源点V0到终点Vi的【当前】最短路径上【Vi的[直接前驱]的顶点序号】 

• 3> Initialization (the adjacency matrix) with a weighted graph model of the FIG.

void InitAMGraph(AMGraph &G){
    cout<<"Please Input Vertexs Number:";
    cin>>G.vexnum;
    cout<<"\nPlease Directed Edges Number:";
    cin>>G.arcnum;
    
    for(int i=0;i<MVNum;i++){
        for(int j=0;j<MVNum;j++){
            if(i!=j){ // 【易错】 初始化<Vi, Vj>时: <Vi,Vj> 路径长度无穷大 (i!=j) 
                G.arcs[i][j] = MaxInt;
            } else { //  【易错】 初始化<Vi, Vj>时: <Vi,Vi>【自回环】路径长度为0 (i==i) 
                G.arcs[i][j] = 0;
            }
        }
    }
    for(int i=0;i<G.vexnum;i++){
        G.vexs[i] = i;
    }
    cout<<"\nPlease Input All Directed Edges and their Weight now:";
    cout<<"\nDirected Edges(i,j,weight): "<<endl;
    int i,j;
    int weight;
    for(int k=0;k<G.arcnum;k++){
//      cout<<"("<<(k+1)<<") ";
        cin>>i;cin>>j;cin>>weight;
        G.arcs[i][j] = weight;
    }
    cout<<endl;
}

• 4> Dijkstra algorithm: Solving single source shortest path

void ShortestPath_Dijkstra(AMGraph G, int V0){
    //step1 n个顶点依次初始化
    int n =G.vexnum;  
    for(int v=0;v<n;v++){
        S[v] = false;
        D[v] = G.arcs[V0][v];
        if(D[v]<MaxInt){
            Path[v] = V0;
        } else {
            Path[v] = -1;
        }
    }
    //step2 将源点V0划入已确定集合S中 
    S[V0] = true;
    D[V0] = 0; // 源点V0到源点V0的最短路径长度必然为0
    //step3 贪心算法策略：
    //          3.1 循环遍历所有结点：
    //              3.2 先确定当前最短路径的终点v；
    //              3.3 然后，将v划入已确定集合S中；
    //              3.4 最后，以利用结点v更新所有尚未确定的结点的最短路径
    int v;
    int min;
    D[G.vexnum] = MaxInt;
    for(int i=1;i<n;i++){//3.1循环遍历所有结点 （即 求从源点V0到图中每一顶点(共计n-1个顶点)的最短路径） 
        //3.2 确定当前最短路径的终点v；
        min = MaxInt;
        for(int w=0;w<n;w++){
            if(S[w]==false && D[w]<min){//比本轮循环中，已知的最短路径还短 【易错/易漏】 S[w]==false ： 必须满足当前结点 Vw 属于尚未确定的结点 
                v = w;
                min = D[w];
            }
        }
        //3.3 然后，将v划入已确定集合S中；
        S[v] = true;
        //3.4 最后，以利用结点v更新所有尚未确定的结点的最短路径
        for(int w=0;w<n;w++){
            //↓更新Vw结点的最短路径长度为 D[v] + G.arcs[v][w] 
            //cout<<"S["<<w<<"]:"<<S[w]<<"D["<<v<<"]"<<D[v]<<"G.arcs["<<v<<"]["<<w<<"]"<<"D["<<w<<"]"<<D[w]<<endl; 
            if(S[w]==false && (D[v] + G.arcs[v][w] < D[w])){//【易错/易漏】 S[w]==false ： 必须满足当前结点 Vw 属于尚未确定的结点 
                D[w] = D[v] + G.arcs[v][w];
                Path[w] = v; // 更新 结点Vw的前驱为 v 
            }
        }
        v = G.vexnum;
    } 
}

• 5> output D [i], Path [j]

void OutputD(AMGraph G, int V0){
    cout<<"Shortest Distance Weight of the Pair of Directed Vertices("<<V0<<", j):"<<endl; 
    for(int j=0;j<G.vexnum;j++){
        cout<<D[j]<<"\t"; 
    }
    cout<<endl;
}

void OutputPath(AMGraph G,int V0){
    cout<<"Shortest Distance Path("<<V0<<",j) of the Pair of Directed Vertices:"<<endl; 
    for(int j=0;j<G.vexnum;j++){
        cout<<Path[j]<<"\t"; 
    }
    cout<<endl;
}

• 6> execute: Main function

int main(){
    int V0; //源点V0的下标 
    AMGraph G;
    InitAMGraph(G);
    
    cout<<"Please Input the Index of Source Node 'V0':";
    cin>>V0;
    ShortestPath_Dijkstra(G, V0);
    OutputD(G, V0);
    OutputPath(G, V0);
    return 0;
}

• 7> Test: Output of Main

Please Input Vertexs Number:6

Please Directed Edges Number:8

Please Input All Directed Edges and their Weight now:
Directed Edges(i,j,weight):
1 2 5
0 2 10
3 5 10
4 3 20
0 4 30
2 3 50
4 5 60
0 5 100

Please Input the Index of Source Node 'V0':0

Shortest Distance Weight of the Pair of Directed Vertices(0, j):
0       32767   10      50      30      60

Shortest Distance Path(0,j) of the Pair of Directed Vertices:
0       -1      0       4       0       3

Reference 2

• "Data Structure (C language / Dongmei Yan Wei Min Wu Weimin eds)"