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 Ruby (judge verdict: Accepted)

def dijkstra(g, s)
  def penq(pq, val)
    pq << val
    cd = pq.size
    pr = cd / 2
    until pr.zero? || (pq[pr - 1] <=> pq[cd - 1]) < 0
      pq[pr - 1], pq[cd - 1] = pq[cd - 1], pq[pr - 1]
      cd = pr
      pr = cd / 2
    end
  end
  def pdeq(pq)
    pq[0], pq[-1] = pq[-1], pq[0]
    rs = pq.delete_at(-1)
    unless pq.empty?
      pr = 1
      cd = [pr * 2, pr * 2 + 1].select { |i| pq[i - 1] }.min_by { |i| pq[i - 1] }
      until cd.nil? || (pq[pr - 1] <=> pq[cd - 1]) < 0
        pq[pr - 1], pq[cd - 1] = pq[cd - 1], pq[pr - 1]
        pr = cd
        cd = [pr * 2, pr * 2 + 1].select { |i| pq[i - 1] }.min_by { |i| pq[i - 1] }
      end
    end
    rs
  end

  pq = []
  d = Array.new(g.size) { Float::INFINITY }
  d[s] = 0
  penq(pq, [0, s])
  until pq.empty?
    v_cost, v = pdeq(pq)
    next if d[v] < v_cost
    g[v].each do |e|
      if d[e[:to]] > d[v] + e[:cost]
        d[e[:to]] = d[v] + e[:cost]
        penq(pq, [d[e[:to]], e[:to]])
      end
    end
  end
  d
end

v, e, r = gets.split.map(&:to_i)
g = Array.new(v) { [] }
e.times do
  s, t, d = gets.split.map(&:to_i)
  g[s] << {to: t, cost: d}
end

dijkstra(g, r).each do |sum|
  puts (sum == Float::INFINITY ? 'INF' : sum)
end