Problem:
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Single Source Shortest Path

For a given weighted graph $G = (V, E)$, find the shortest path from a source to each vertex. For each vertex $u$, print the total weight of edges on the shortest path from vertex $0$ to $u$.

Input

In the first line, an integer $n$ denoting the number of vertices in $G$ is given. In the following $n$ lines, adjacency lists for each vertex $u$ are respectively given in the following format:

$u$ $k$ $v_1$ $c_1$ $v_2$ $c_2$ ... $v_k$ $c_k$

Vertices in $G$ are named with IDs $0, 1, ..., n-1$. $u$ is ID of the target vertex and $k$ denotes its degree. $v_i (i = 1, 2, ... k)$ denote IDs of vertices adjacent to $u$ and $c_i$ denotes the weight of a directed edge connecting $u$ and $v_i$ (from $u$ to $v_i$).

Output

For each vertex, print its ID and the distance separated by a space character in a line respectively. Print in order of vertex IDs.

Constraints

$1 \leq n \leq 100$

$0 \leq c_i \leq 100,000$

$|E| \leq 10,000$

All vertices are reachable from vertex $0$

Sample Input 1

5
0 3 2 3 3 1 1 2
1 2 0 2 3 4
2 3 0 3 3 1 4 1
3 4 2 1 0 1 1 4 4 3
4 2 2 1 3 3

Sample Output 1

0 0
1 2
2 2
3 1
4 3

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in Ruby (judge verdict: Accepted)

MAX_COST = Float::INFINITY

def calc_min_costs_from(v_start, costs)
  vert_struct = Struct.new :id, :cost_from_start do
    define_method :cost_to do |v_to|
      costs[id][v_to.id]
    end

    def connected_to?(v_to)
      0 <= cost_to(v_to)
    end

    def cost_from_start_through(other_v)
      other_v.cost_from_start + other_v.cost_to(self)
    end
  end

  n_vertexes = costs.size
  verts = n_vertexes.times.map do |id|
    vert_struct.new(id, MAX_COST)
  end

  verts[v_start].cost_from_start = 0
  visiting = {}
  visiting[v_start] = verts[v_start]

  while visiting.any?
    v_from = visiting.min_by { |_, v| v.cost_from_start }[1]
    visiting.delete v_from.id

    verts.each do |v_to|
      next unless v_from != v_to && v_from.connected_to?(v_to)
      through_cost = v_to.cost_from_start_through v_from
      if through_cost < v_to.cost_from_start
        v_to.cost_from_start = through_cost
        visiting[v_to.id] = v_to
      end
    end
  end

  verts.map { |v| v.cost_from_start }
end

n_vertexes = gets.to_i
costs      = Array.new(n_vertexes) { Array.new(n_vertexes, -1) }
n_vertexes.times do
  v_from, n_degrees, *inputs = gets.split.map &:to_i
  1.step(n_degrees * 2, 2) do |i|
    v_to, cost = inputs[i - 1], inputs[i]
    costs[v_from][v_to] = cost
  end
end

min_costs = calc_min_costs_from(0, costs)
puts min_costs.map.with_index{ |cost, v| "#{v} #{cost}" }