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)

require 'set'

class PriorityQueue < SortedSet
  @@id = 0
  def << obj
    add obj
  end

  def add obj
    super([obj,@@id])
    @@id+=1
  end

  def push obj
    add obj
  end

  def top
    self.min[0]
  end

  def pop
    self.delete(self.min)
  end
end

class State
  attr_accessor :v, :d
  def initialize v,d
    self.v = v
    self.d = d
  end

  def <=> other
    self.d <=> other.d
  end
end

INF = 2**30

def dijkstra g, s
  dist = Array.new(g.length, INF)
  dist[s] = 0
  q = PriorityQueue.new
  q << State.new(s,0)
  while q.size > 0
    s = q.top
    q.pop
    v = s.v
    next if dist[v] < s.d
    g[v].each do |e|
      if dist[e.dst] > dist[v] + e.weight
        dist[e.dst] = dist[v] + e.weight
        q << State.new(e.dst, dist[e.dst])
      end
    end
  end
  dist
end

class Edge
  attr_accessor :src, :dst, :weight
  def initialize src, dst, weight
    self.src = src
    self.dst = dst
    self.weight = weight
  end
end

V,E,r = gets.chomp.split.map(&:to_i)

graph = Array.new(V).map{ Array.new }

E.times do
  a,b,c = gets.chomp.split.map(&:to_i)
  graph[a] << Edge.new(a,b,c)
end

d = dijkstra graph,r

for ans in d
  if ans == INF
    puts "INF"
  else
    puts ans
  end
end