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)

class Node
  attr_reader :id, :siblings
  attr_accessor :state
 
  def initialize(id)
    @id = id
    @siblings = []
    @distances = {}
    @state = :white
  end
 
  def add(sibling, distance)
    @siblings << sibling
    @distances[sibling.id] = distance
  end
 
  def distance_from(other)
    @distances[other.id]
  end
end
 
if $PROGRAM_NAME == __FILE__
  n = gets.to_i
 
  nodes = []
 
  n.times do
    id, _, *rest = gets.chomp.split.map(&:to_i)
 
    nodes[id] ||= Node.new(id)
    node = nodes[id]
 
    rest.each_slice(2) do |sibling_id, distance|
      nodes[sibling_id] ||= Node.new(sibling_id)
      sibling = nodes[sibling_id]
      node.add(sibling, distance)
    end
  end
 
  distances = Array.new(n, Float::INFINITY)
 
  distances[0] = 0
  nodes[0].state = :gray
 
  loop do
    current = nil
    min = Float::INFINITY
 
    nodes.each do |node|
      next if node.state == :black
 
      distance = distances[node.id]
 
      if distance < min
        min = distance
        current = node
      end
    end
 
    break unless current
 
    current.state = :black
 
    current_distance = distances[current.id]
 
    current.siblings.each do |sibling|
      next if sibling.state == :black
 
      distance = current_distance + current.distance_from(sibling)
 
      if distance < distances[sibling.id]
        distances[sibling.id] = distance
        sibling.state = :gray
      end
    end
  end
 
  puts distances.map.with_index {|distance, id| [id, distance].join(' ') }.join("\n")
end