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: Wrong Answer)

# 無限大の距離に利用する.
INF = 2**25

# 重み付き隣接行列に変換する.
# edgeがない場合は-1が入っている.
def toAdj(edges, n)
  a = Array.new(n).map{ Array.new(n, -1) }
  edges.each do |i, j, c|
    a[i][j] = c
  end
  a
end

# Bellman-Ford法.
# dsにsからの距離を格納して返す.
# 到達不可能の場合、INFが入っている.
def bellmanFord(adj, n, m, s)
  ds = Array.new(n, INF)
  ds[s] = 0
  n.times do
    adj.each_with_index do |row, i|
      row.each_with_index do |c, j|
        if c > 0
          ds[j] = ds[j] < ds[i] + c ? ds[j] : ds[i] + c
        end
      end
    end
  end
  ds
end

if __FILE__ == $0
  n, m, r = gets.split.map(&:to_i)

  edges = m.times.map{ gets.split.map(&:to_i) }
  adjMatrix = toAdj(edges, n)
  
  bellmanFord(adjMatrix, n, m, r).each do |c|
    puts c == INF ? "INF" : c.to_s
  end
end