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 2100000000
#define MIN(a, b) ((a) < (b) ? (a) : (b))

typedef struct edge{
	int to_node;
	int cost;
	struct edge *next;
}edge;

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

int V;
node N[LARGE];
edge* E[LARGE];
node *H[LARGE];


edge * Make_Edge(int t,int d){
	edge *new=malloc(sizeof(edge));
	new->to_node=t;
	new->cost=d;
	new->next=NULL;
	return new;
}

void Insert(int s,edge* new){
	edge **p=&E[s];

	while((*p)!=NULL){
		p=&((*p)->next);
	}
	(*p)=new;
	
	return;
}

void swap(int a,int b){
	node *temp; int TEMP;
	temp=H[a];
	H[a]=H[b];
	H[b]=temp;
	
	TEMP=H[a]->heapid;
	H[a]->heapid=H[b]->heapid;
	H[b]->heapid=TEMP;
	return;
}


void heap_reconst(int x){
	int i;
	i=x;
	for(i=x;i>0;i--){
		if(i%2==0){
			if(H[i]->cost < H[i/2-1]->cost){
				swap(i,i/2-1);
				
			}
			
		}
		else {
			if(H[i]->cost < H[i/2]->cost){
				swap(i,i/2);
				
			}
			
		}
	}

	return;
}


void Dijkstra(int start,int rest){
	int min_idx=-1;
	int i;
	edge *p;

	if(rest==0)return;
	
	if(rest==V){
		H[0]->cost=0;
		H[0]->flg=1;
	}
	swap(0,rest-1);
	
	
	p=E[start];
	
	while(p!=NULL){
		if(N[p->to_node].flg==0){
			if(N[p->to_node].cost > N[start].cost+(p->cost)){
				N[p->to_node].cost=N[start].cost+(p->cost);
				
			}
		}
		p=p->next;
	}
	heap_reconst(rest-2);
	min_idx=H[0]->id;
	
	N[min_idx].flg=1;
	
	Dijkstra(min_idx,rest-1);
	
	return;
	
}


int main(){
	int start;
	int s,t,d;
	int n,e;
	int i,j;
	int rest;
	edge* new;
	
	scanf("%d %d %d",&V,&e,&start);
	
	/*初期化*/
	for(i=0;i<V;i++){
		E[i]=NULL;
		N[i].cost=INF;
		N[i].id=i;
		N[i].heapid=i;
		N[i].flg=0;
		H[i]=&N[i];
	}
	
	/*隣接リスト作成*/
	for(i=0;i<e;i++){
		scanf("%d %d %d",&s,&t,&d);
		new=Make_Edge(t,d);
		Insert(s,new);
	}
	if(e==0)N[start].cost=0;
	
	swap(start,0);
	rest=V;
	Dijkstra(start,rest);
	
	for(i=0;i<V;i++){
		if(N[i].cost==INF)printf("INF\n");
		else printf("%d\n",N[i].cost);
	}
	
	return 0;
}