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 INF (1<<30)

struct node{
	int ID,cost;
	struct node *next;
};
typedef struct node* nP;

nP List[10000],NIL;
int n,size;
struct node minHeap[500050],empty;

//{{{ Functions
void MinHeapify(int pos){
	int L=pos*2, R=pos*2 +1,min=pos;
	struct node tmp;

	if(L<=size && minHeap[L].cost<minHeap[pos].cost){
		min=L;
	}
	if(R<=size && minHeap[min].cost>minHeap[R].cost){
		min=R;
	}

	if(min!=pos){
		//swap
		tmp=minHeap[pos];
		minHeap[pos]=minHeap[min];
		minHeap[min]=tmp;

		MinHeapify(min);
	}

	return;
}

void Push(int id, int c){
	int pos;
	struct node data,tmp;
	data.ID=id;	data.cost=c;

	minHeap[++size]=data;
	pos=size;

	while(pos/2>0 && minHeap[pos].cost<minHeap[pos/2].cost){
		// swap
		tmp=minHeap[pos];
		minHeap[pos]=minHeap[pos/2];
		minHeap[pos/2]=tmp;

		pos/=2;
	}

	return;
}

struct node Pop(void){
	if(size==0)return *NIL;
	struct node ret=minHeap[1];
	minHeap[1]=minHeap[size];	--size;
	MinHeapify(1);

	return ret;
}
//{{{
void Insert(nP head, int v, int c){
	nP data=(nP)malloc(sizeof(struct node));
	data->ID=v;	data->cost=c;

	data->next=head->next;
	head->next=data;

	return;
}
//}}}

void Dijkstra(void){
	int i,minCost;
	struct node u;
	nP v;
	int c[10000],visited[10000];

	for(i=0;i<n;i++){
		c[i]=INF;	visited[i]=0;
	}
	c[0]=0;
	Push(0,0);

	while(1){
		u=Pop();	if(u.ID==-1)break;
		if(c[u.ID]<u.cost)continue;
		visited[u.ID]=1;

		//printf("%d: ",u.ID);
		for(v=List[u.ID]->next; v!=NIL; v=v->next){
			//printf("%d->",v->ID);
			if(visited[v->ID]==1)continue;

			if(c[v->ID]>c[u.ID]+v->cost){
				c[v->ID]=c[u.ID]+v->cost;
				Push(v->ID, c[v->ID]);
			}
		}
		//printf("\n");
	}

	for(i=0;i<n;i++)printf("%d %d\n",i,c[i]);
}

//}}}
int main(){
	int i,j,u,k,v,c;
	scanf("%d",&n);

	NIL=(nP)malloc(sizeof(struct node));
	NIL->ID=-1;
	for(i=0;i<n;i++){
		List[i]=(nP)malloc(sizeof(struct node));
		List[i]->next=NIL;
	}

	for(i=0;i<n;i++){
		scanf("%d%d",&u,&k);
		for(j=0;j<k;j++){
			scanf("%d%d",&v,&c);
			Insert(List[u],v,c);
		}
	}

	Dijkstra();

	return 0;
}