Shortest Path [4] PTA 6-22

两种答案，先说思路，第一段代码即答案，第二段也是正确答案，第一个答案更符合陈越老师教材，第二种答案是对第一种的改良，但是两种思路一样，题目在最后：

建立在Dijkstra算法的基础上，多加了一个数组PathNeS来记录从S到每个点经过的最少边数，对于和S相邻的边，我们初始化其PathNeS为1；

在while的每次循环里确定v后，在for循环里，

我们先判断S通过v到达i的距离(dist[v]+Graph->G[v][i])是否等于dist[i],若等于，则判断S通过v到达i经过的最短边数(PathNeS[v]+1)是否小于原先记录的最短边数(PathNeS[i])若满足，则更新PathNeS[i]和path[i],使其通过v来到达；（注意这道题要求到这个点最短时边数最少的路）

再判断是否收集i和通过v到i是否会比原先记录的短，若满足，更新dist和path时也更新PathNeS[i]=PathNeS[v]+1;因为通过v到i的边数即再v边数基础上加一；（v已被收集，则说明dist[v]已是最短）；

答案代码如下：

void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S ){
	int v,i,min;
	static int collected[9999],PathNeS[9999];
	for(i=0;i<Graph->Nv;i++){
		path[i]=-1;
		dist[i]=Graph->G[S][i];
		if(dist[i]<INFINITY){
			path[i]=S;
			PathNeS[i]=1;//和S相邻的点的边数初始就是1 
		}
		else PathNeS[i]=INFINITY;//和S不想-相连的初始化为最大值 
	}
	collected[S]=1;//收集S 
	dist[S]=0;
	while(1){
		min=INFINITY;
		v=-1;
		for(i=0;i<Graph->Nv;i++){
			if(!collected[i]&&dist[i]<min){
				v=i;
				min=dist[v];
			}
		}
		if(v==-1) break;
		collected[v]=1;
		for(i=0;i<Graph->Nv;i++){
			if(dist[i]==dist[v]+Graph->G[v][i]){//连个if写在一起用&&连接也可以，判断是否有更少边数的路径 
				if(PathNeS[v]+1<PathNeS[i]){
					PathNeS[i]=PathNeS[v]+1;
					path[i]=v;
				}
			}
			if(!collected[i]&&dist[i]>dist[v]+Graph->G[v][i]){
				dist[i]=dist[v]+Graph->G[v][i];
				path[i]=v;
				PathNeS[i]=PathNeS[v]+1;//更新最短边数 
			}
		}
	}
	for(i=0;i<Graph->Nv;i++){
		if(dist[i]==INFINITY) dist[i]=-1;
	}
	return;
} 

中间再插一个改良版，陈越老师的教材中的算法（也就是进入while之前就收集S）更符合第一种答案，但是进入while以后再收集S可以让代码更简洁，但是思路是一样的，以下即这种思路的答案：

void ShortestDist( MGraph Graph, int dist[], int path[], Vertex S ) {
    static int collected[9999],PathNeS[9999];
    for(int i = 0;i < Graph -> Nv;i ++) {
        dist[i] = INFINITY;
        path[i] = -1;
    }
    dist[S] = 0;
    path[S] = -1;
    PathNeS[S] = 1;
    while(1) {
        int v= -1,m = INFINITY;
        for(int i = 0;i < Graph -> Nv;i ++) {
            if(!collected[i] && m > dist[i]) m = dist[i],v = i;
        }
        if(v == -1) break;
        collected[v] = 1;
        for(int i = 0;i < Graph -> Nv;i ++) {
            if(collected[i] || Graph -> G[v][i] == INFINITY) continue;
            if(dist[v] + Graph -> G[v][i] < dist[i]) {
                dist[i] = dist[v] + Graph -> G[v][i];
                path[i] = v;
                PathNeS[i] =PathNeS[v] + 1;
            }
            else if(dist[v] + Graph -> G[v][i] == dist[i] && PathNeS[i] > PathNeS[v] + 1) {
              PathNeS[i] = PathNeS[v] + 1;
                path[i] = v;
            }
        }
    }
    for(int i = 0;i < Graph -> Nv;i ++) {
        if(dist[i] == INFINITY) dist[i] = -1;
    }
}

题目：

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

    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