Problem:
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Single Source Shortest Path II

For a given weighted graph $G = (V, E)$, find the shortest path from a source to each vertex. For each vertex $u$, print the total weight of edges on the shortest path from vertex $0$ to $u$.

Input

In the first line, an integer $n$ denoting the number of vertices in $G$ is given. In the following $n$ lines, adjacency lists for each vertex $u$ are respectively given in the following format:

$u$ $k$ $v_1$ $c_1$ $v_2$ $c_2$ ... $v_k$ $c_k$

Vertices in $G$ are named with IDs $0, 1, ..., n-1$. $u$ is ID of the target vertex and $k$ denotes its degree. $v_i (i = 1, 2, ... k)$ denote IDs of vertices adjacent to $u$ and $c_i$ denotes the weight of a directed edge connecting $u$ and $v_i$ (from $u$ to $v_i$).

Output

For each vertex, print its ID and the distance separated by a space character in a line respectively. Print in order of vertex IDs.

Constraints

$1 \leq n \leq 10,000$

$0 \leq c_i \leq 100,000$

$|E| < 500,000$

All vertices are reachable from vertex $0$

Sample Input 1

5
0 3 2 3 3 1 1 2
1 2 0 2 3 4
2 3 0 3 3 1 4 1
3 4 2 1 0 1 1 4 4 3
4 2 2 1 3 3

Sample Output 1

0 0
1 2
2 2
3 1
4 3

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in C (judge verdict: Accepted)


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

#define NIL_VERTEX -1
#define DENOTE_INF_COST -1
#define COMP_INF_COST 100008

// translate 1-origin into 0-origin
int i0O(int index1Origin) {
    return index1Origin-1;
}


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

#define MAX_NUM_STACK_VALUES 100
#define STR_MAX_LENGTH 100

typedef enum {
    WHITE,
    GRAY,
    BLACK
} Color;

typedef struct {
    struct Vtx * vertex;
//    int iVertex;
    int weight;
} ArrowTo;

int isNullArrow(ArrowTo a) {
//    return iVertex == NIL_VERTEX;
    return a.vertex == NULL;
}

typedef struct Vtx {
    ArrowTo * adjList;
    Color color;
    int discoveryCost;
    struct Vtx * discoverFrom;
    int key;
} Vertex;

void 
dijkstra(
        Vertex vertices[], 
        int nVertices,
        int startingVertexIndex
) {
    int mincost, u, v, i;
    
    for (i = 0; i < nVertices; i++) {
        vertices[i].discoveryCost = COMP_INF_COST;
        vertices[i].discoverFrom = NULL;
        vertices[i].color = WHITE;
    }
    
    vertices[startingVertexIndex].discoveryCost = 0;
    
    while (1) {
        mincost = COMP_INF_COST;
        
        for (i = 0; i < nVertices; i++) // for each vertex
            if (vertices[i].color != BLACK && vertices[i].discoveryCost < mincost)
                mincost = vertices[u = i].discoveryCost;
        
        if (mincost == COMP_INF_COST) break;
        
        vertices[u].color = BLACK;
        
        for (v = 0; !isNullArrow(vertices[u].adjList[v]); v++) {// for each in adjList
            i = vertices[u].discoveryCost + vertices[u].adjList[v].weight;

            if (vertices[u].adjList[v].vertex->color != BLACK && i < vertices[u].adjList[v].vertex->discoveryCost) {
                vertices[u].adjList[v].vertex->discoverFrom = &vertices[u];
                vertices[u].adjList[v].vertex->discoveryCost = i;
            }
        }
        
    }
}

int main(int argc, char** argv) {
    int nVertices, i, j;
    
    // scan number of already existed events
    scanf("%d", &nVertices); 
    
//    // adjMatrix ----------------------------
//    int ** adjMatrix;
//    
//    adjMatrix = (int **)malloc(nVertices * sizeof(int *));
//    
//    for (i = 0; i < nVertices; i++) {
//        adjMatrix[i] = (int *)malloc(nVertices * sizeof(int));
//        
//        for (j = 0; j < nVertices; j++)
//            adjMatrix[i][j] = COMP_INF_COST;
//    }
    
    // scan --------------------------------
    Vertex vertices[nVertices];
    int iVertexFrom, iVertexTo, nArrow;
    int weight;
    
    for (i = 0; i < nVertices; i++) {
        vertices[i].key = i;
        scanf("%d %d", &iVertexFrom, &nArrow); 
        
        vertices[iVertexFrom].adjList = (ArrowTo *)malloc((nArrow + 1) * sizeof(ArrowTo));
        
        for (j = 0; j < nArrow; j++) {
            scanf("%d %d", &iVertexTo, &weight); 
            vertices[iVertexFrom].adjList[j].vertex = &vertices[iVertexTo];
            vertices[iVertexFrom].adjList[j].weight = weight;
        }
        
        vertices[iVertexFrom].adjList[nArrow].vertex = NULL;
    }
    
//    // scan result
//    for (i = 0; i < nVertices; i++) {
//        printf("%d:", i);
//        for (j = 0; !isNullArrow(vertices[i].adjList[j]); j++) {
//            printf(" %d %d",
//                    vertices[i].adjList[j].vertex->key,
//                    vertices[i].adjList[j].weight);
//        }
//        printf("\n");
//    }
    
    dijkstra(vertices, nVertices, 0);
    
    for (i = 0; i < nVertices; i++)
        printf("%d %d\n", i, vertices[i].discoveryCost);

    return (EXIT_SUCCESS);
}