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: Time Limit Exceeded)

class Dijkstra
  #????????????????????°???????????????????????????????????????????????????
  def initialize graph
    @graph = graph
    #?¢?????????????
    @confirms = {}
    @edges = {}
    @min = []
  end

  def exec start
    @edges[start] = 0
    @min = [start, 0]
    until @edges.empty?
      edge = @min
      confirm_node(edge[0])
    end
    return @confirms
  end

  def confirm_node node
    @confirms[node] = @edges[node]
    @edges.delete(node)
    @min = [node, 200000]
    @graph[node].each do |(to, length)|
      next if @confirms.key?(to)
      candidate = @confirms[node] + length
      if !@edges[to] || @edges[to] > candidate
        @edges[to] = candidate
        @min = [to, candidate] if candidate < @min[1]
      end
    end
  end
end

nodes,edges,start = gets.split(" ").map(&:to_i)
graph = Hash.new{|h,k|h[k] = []}
edges.times do
  s,t,d = gets.split(" ").map(&:to_i)
  graph[s] << [t,d]
end
dijkstra = Dijkstra.new(graph)
confirms = dijkstra.exec(start)
nodes.times do |i|
  puts confirms[i] ? confirms[i] : "INF"
end