6-17 Shortest Path [4] (25分)

Write a program to find the weighted shortest distances from any vertex to a given source vertex in a digraph. If there is more than one minimum path from v to w, a path with the fewest number of edges is chosen. It is guaranteed that all the weights are positive and such a path is unique for any vertex.

Format of functions:

void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S );

 

where MGraph is defined as the following:

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

 

The shortest distance from V to the source S is supposed to be stored in dist[V]. If V cannot be reached from S, store -1 instead. If W is the vertex being visited right before V along the shortest path from S to V, then path[V]=W. If V cannot be reached from S, path[V]=-1, and we have path[S]=-1.

Sample program of judge:

#include <stdio.h>
#include <stdlib.h>

typedef enum {false, true} bool;
#define INFINITY 1000000
#define MaxVertexNum 10  /* maximum number of vertices */
typedef int Vertex;      /* vertices are numbered from 0 to MaxVertexNum-1 */
typedef int WeightType;

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

MGraph ReadG(); /* details omitted */

void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S );

int main()
{
    int dist[MaxVertexNum], path[MaxVertexNum];
    Vertex S, V;
    MGraph G = ReadG();

    scanf("%d", &S);
    ShortestDist( G, dist, path, S );

    for ( V=0; V<G->Nv; V++ )
        printf("%d ", dist[V]);
    printf("
");
    for ( V=0; V<G->Nv; V++ )
        printf("%d ", path[V]);
    printf("
");

    return 0;
}

/* Your function will be put here */

 

Sample Input (for the graph shown in the figure):

8 11
0 4 5
0 7 10
1 7 40
3 0 40
3 1 20
3 2 100
3 7 70
4 7 5
6 2 1
7 5 3
7 2 50
3

 

Sample Output:

40 20 100 0 45 53 -1 50 
3 3 3 -1 0 7 -1 0 

代码：

#include <stdio.h>
#include <stdlib.h>

typedef enum {false, true} bool;
#define INFINITY 1000000
#define MaxVertexNum 10  /* maximum number of vertices */
typedef int Vertex;      /* vertices are numbered from 0 to MaxVertexNum-1 */
typedef int WeightType;

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

MGraph ReadG() { /* details omitted */
    MGraph mg = (MGraph)malloc(sizeof(struct GNode));
    int a,b,c;
    scanf("%d%d",&mg -> Nv,&mg -> Ne);
    for(int i = 0;i < mg -> Nv;i ++) {
        for(int j = 0;j < mg -> Nv;j ++) {
            mg -> G[i][j] = INFINITY;
        }
    }
    for(int i = 0;i < mg -> Ne;i ++) {
        scanf("%d%d%d",&a,&b,&c);
        mg -> G[a][b] = c;
    }
    return mg;
}


void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S );

int main()
{
    int dist[MaxVertexNum], path[MaxVertexNum];
    Vertex S, V;
    MGraph G = ReadG();

    scanf("%d", &S);
    ShortestDist( G, dist, path, S );

    for ( V=0; V<G->Nv; V++ )
        printf("%d ", dist[V]);
    printf("
");
    for ( V=0; V<G->Nv; V++ )
        printf("%d ", path[V]);
    printf("
");

    return 0;
}

/* Your function will be put here */
void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S ) {
    int vis[MaxVertexNum] = {0},num[MaxVertexNum];
    for(int i = 0;i < Graph -> Nv;i ++) {
        dist[i] = INFINITY;
        path[i] = -1;
    }
    dist[S] = 0;
    path[S] = -1;
    num[S] = 1;
    while(1) {
        int t = -1,m = INFINITY;
        for(int i = 0;i < Graph -> Nv;i ++) {
            if(!vis[i] && m > dist[i]) m = dist[i],t = i;
        }
        if(t == -1) break;
        vis[t] = 1;
        for(int i = 0;i < Graph -> Nv;i ++) {
            if(vis[i] || Graph -> G[t][i] == INFINITY) continue;
            if(dist[t] + Graph -> G[t][i] < dist[i]) {
                dist[i] = dist[t] + Graph -> G[t][i];
                path[i] = t;
                num[i] = num[t] + 1;
            }
            else if(dist[t] + Graph -> G[t][i] == dist[i] && num[i] > num[t] + 1) {
                num[i] = num[t] + 1;
                path[i] = t;
            }
        }
    }
    for(int i = 0;i < Graph -> Nv;i ++) {
        if(dist[i] == INFINITY) dist[i] = -1;
    }
}