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++11 (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <set>
#include <vector>
#include <utility>
#include <queue>

using namespace std;

int V, E, r;
vector<pair<int, int> > edges[100010];
priority_queue <pair<int, int>, vector<pair<int, int> >, greater<pair<int, int> > > Q;
pair<int, int> Cost[100010];
const int Inf = 10000*100000 + 100;
bool t[100010], q[100010];
set<int> s;
int SizeQ;

int main() {
  cin >> V >> E >> r;
  for(int i = 0; i < E; i++) {
    int t;
    pair<int, int> e;
    cin >> e.first >> t >> e.second;
    edges[t].push_back(e);
  }
  for(int i = 0; i < V; i++) {
    Cost[i] = make_pair(Inf, i);
    t[i] = false;
    q[i] = false;
    s.insert(i);
  }
  Cost[r].first = 0;
  Q.push(Cost[r]);
  SizeQ = 1;
  q[r] = true;
  while(SizeQ) {
    pair<int, int> u = Q.top();
    Q.pop();
    if(!t[u.second]) {
      SizeQ--;
      t[u.second] = true;
      s.erase(u.second);
      for(auto itr = s.begin(); itr != s.end(); itr++) {
	bool update = false;
	for(int i = 0; i < edges[*itr].size(); i++) {
	  if(Cost[edges[*itr][i].first].first != Inf && Cost[*itr].first > Cost[edges[*itr][i].first].first+edges[*itr][i].second) {
	    Cost[*itr].first = Cost[edges[*itr][i].first].first+edges[*itr][i].second;
	    update = true;
	  }
	}
	if(update) {
	  Q.push(Cost[*itr]);
	  if(!q[*itr]) {
	    SizeQ++;
	    q[*itr] = true;
	  }
	}
      }
    }
  }
  for(int i = 0; i < V; i++){
    if(Cost[i].first != Inf) cout << Cost[i].first << endl;
    else cout << "INF" << endl;
  }
  return 0;
}