6-11 Shortest Path [1]（25 分）

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#无权有向图单源最短路径 #BFT

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本文介绍了一种用于计算有向图中从任意顶点到指定源顶点的无权最短路径的算法。该算法通过广度优先搜索实现，并详细展示了如何在遍历过程中更新最短距离。

Write a program to find the unweighted shortest distances from any vertex to a given source vertex in a digraph.

Format of functions:

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

where LGraph is defined as the following:

typedef struct AdjVNode *PtrToAdjVNode; 
struct AdjVNode{
    Vertex AdjV;
    PtrToAdjVNode Next;
};

typedef struct Vnode{
    PtrToAdjVNode FirstEdge;
} AdjList[MaxVertexNum];

typedef struct GNode *PtrToGNode;
struct GNode{  
    int Nv;
    int Ne;
    AdjList G;
};
typedef PtrToGNode LGraph;

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.

Sample program of judge:

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

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

typedef struct AdjVNode *PtrToAdjVNode; 
struct AdjVNode{
    Vertex AdjV;
    PtrToAdjVNode Next;
};

typedef struct Vnode{
    PtrToAdjVNode FirstEdge;
} AdjList[MaxVertexNum];

typedef struct GNode *PtrToGNode;
struct GNode{  
    int Nv;
    int Ne;
    AdjList G;
};
typedef PtrToGNode LGraph;

LGraph ReadG(); /* details omitted */

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

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

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

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

    return 0;
}

/* Your function will be put here */

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

Sample Output:

-1 1 0 3 2 2 1

/* Your function will be put here */
void ShortestDist( LGraph Graph, int dist[], Vertex S )
{//不需要为了确保每个元素只进队一次而使用visited[]数组，因为dist[]初值为-1，随后由于BFT其值只被改变一次（再次访问某顶点时，当时的距离值level必然更大）
    for(int i = 0; i < Graph->Nv; ++i)
        dist[i] = -1;
    dist[S] = 0;
    int Q[Graph->Nv];//输入顶点数有可能大于MaxVertexNum
    int first = 0, rear = 0;
    Q[rear++] = S;
    PtrToAdjVNode current;
    int Qhead;
    int level = 1;//level其实是距离
    //level和level_end的设定类似于求二叉树的深度
    int level_end = 0;//相当于二叉树每层最后一个结点的位置
    //level_end一开始用值标记，最后还是觉得用位置标记更准确
    while(first < rear)//BFT
    {
        Qhead = Q[first++];
        current = Graph->G[Qhead].FirstEdge;
        while(current != NULL)
        {
	    //入队语句一开始的位置
            if(dist[current->AdjV] == -1)
            {
                Q[rear++] = current->AdjV;
                //入队这句，一开始放在if的上面了，这将导致环路顶点二次入队使队列数组越界
                dist[current->AdjV] = level;
            }
            current = current->Next;
        }
        if(first - 1 == level_end)
        {
            level_end = rear - 1;
            ++level;
        }
    }
}