Problem:
Single Source Shortest Path

For a given weighted graph G(V, E) and a source r, find the source shortest path to each vertex from the source (SSSP: Single Source Shortest Path).

Input

An edge-weighted graph G (V, E) and the source r.

|V| |E| r
s0 t0 d0
s1 t1 d1
:
s|E|-1 t|E|-1 d|E|-1

|V| is the number of vertices and |E| is the number of edges in G. The graph vertices are named with the numbers 0, 1,..., |V|-1 respectively. r is the source of the graph.

si and ti represent source and target vertices of i-th edge (directed) and di represents the cost of the i-th edge.

Output

Print the costs of SSSP in the following format.

c0
c1
:
c|V|-1

The output consists of |V| lines. Print the cost of the shortest path from the source r to each vertex 0, 1, ... |V|-1 in order. If there is no path from the source to a vertex, print INF.

Constraints

1 ≤ |V| ≤ 100000

0 ≤ di ≤ 10000

0 ≤ |E| ≤ 500000

There are no parallel edges

There are no self-loops

Sample Input 1

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

Sample Output 1

0
1
3
4

Sample Input 2

4 6 1
0 1 1
0 2 4
2 0 1
1 2 2
3 1 1
3 2 5

Sample Output 2

3
0
2
INF

Solution in C (judge verdict: Time Limit Exceeded)

#include<stdio.h>
#include<stdlib.h>
#define LARGE 100000
#define INF 2000000000
#define MIN(a, b) ((a) < (b) ? (a) : (b))

int V,E;

int heap_id[LARGE];
int id_heap[LARGE];

typedef struct list{
	int id;
	int cost;
	struct list *next;
}list;

typedef struct node{
	int id;
	int cost;
}node;

void Make_AdjList(list* L[],int s,int t,int d){
	list *new=malloc(sizeof(list));
	list **p=&L[s];
	/*card作成*/
	new->id=t;	new->cost=d;  new->next=NULL;
	
	while((*p)!=NULL)p=&((*p)->next);
	(*p)=new;
	
	return;
}

void swap(int x,int y){
	int temp;

	temp=heap_id[x];
	heap_id[x]=heap_id[y];
	heap_id[y]=temp;
	
	temp=id_heap[heap_id[x]];
	id_heap[heap_id[x]]=id_heap[heap_id[y]];
	id_heap[heap_id[y]]=temp;
	
	return;
}

int check(int i,int num){
	if(num%2==1){
		if((num-1)/2<=i && i<num)return 0;
	}
	else {
		if(num/2 <=i && i<num)return 0;
	}
	
	return 1;
}

void heap_updata(node N[],int i,int num){
	
	
	while(i>0){
	
		if(i%2==0){
			if(N[heap_id[i]].cost < N[heap_id[i/2-1]].cost){
				swap(i,i/2-1);
				if(check(i,num))i=MIN(i*2+2,num-1);
			}
			else i--;
		}else {
			if(N[heap_id[i]].cost < N[heap_id[i/2]].cost){
				swap(i,i/2);
				if(check(i,num))i=MIN(i*2+2,num-1);
			}
			else i--;
		}
	}
	
	return;

}


int Dijkstra(list* L[],node N[],int num){
	int u=heap_id[0];
	list *p=L[u];
	
	swap(0,num-1);
	num--;
	heap_updata(N,num-1,num);
	
	while(p!=NULL){
		if(N[p->id].cost > N[u].cost+(p->cost)){
			N[p->id].cost=N[u].cost+(p->cost);
			heap_updata(N,id_heap[p->id],num);
		}
		p=p->next;
	}
	
	return num;
	
}

int main(){
	int i,j;
	int s,t,d;
	int num,start;
	list *L[LARGE];
	node N[LARGE];
	
	scanf("%d %d %d",&V,&E,&start);
	num=V;
	
	for(i=0;i<V;i++){
		L[i]=NULL;
		N[i].id=i;
		N[i].cost=INF;
		heap_id[i]=i;
		id_heap[i]=i;
	}
	N[start].cost=0;
	swap(0,start);
	
	for(i=0;i<E;i++){
		scanf("%d %d %d",&s,&t,&d);
		Make_AdjList(L,s,t,d);
	}
	
	while(num!=0){
		num=Dijkstra(L,N,num);
	}
	
	for(i=0;i<V;i++){
		if(N[i].cost==INF)printf("INF\n");
		else printf("%d\n",N[i].cost);
	}
	
	return 0;
	
}