Problem:
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Single Source Shortest Path II

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

$1 \leq n \leq 10,000$

$0 \leq c_i \leq 100,000$

$|E| < 500,000$

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 PriorityQueue
  attr_reader :values

  def initialize
    @values = [nil]
    @pos = 0
  end

  def push(value)
    @pos += 1
    @values[@pos] = value

    i = @pos

    while 2 <= i && @values[i] < @values[i / 2]
      @values[i], @values[i / 2] = @values[i / 2], @values[i]
      i /= 2
    end
  end

  def pop
    return nil if empty?

    ret = @values[1]
    @values[1] = @values[@pos]
    @pos -= 1

    i = 1

    loop do
      l = i * 2
      r = l + 1

      smallest = l <= @pos && @values[l] < @values[i] ? l : i
      smallest = r <= @pos && @values[r] < @values[smallest] ? r : smallest

      break if smallest == i

      @values[i], @values[smallest] = @values[smallest], @values[i]

      i = smallest
    end

    ret
  end

  def empty?
    @pos <= 0
  end
end

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_to(sibling)
    @distances[sibling.id]
  end
end

Pair = Struct.new(:distance, :id) do
  include Comparable

  def <=>(other)
    distance <=> other.distance
  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

  queue = PriorityQueue.new
  queue.push(Pair.new(0, 0))

  until queue.empty?
    pair = queue.pop
    current = nodes[pair.id]
    current.state = :black

    next if distances[current.id] < pair.distance

    current.siblings.each do |sibling|
      next if sibling.state == :black

      distance = distances[current.id] + current.distance_to(sibling)

      if distance < distances[sibling.id]
        distances[sibling.id] = distance
        sibling.state = :gray
        queue.push(Pair.new(distance, sibling.id))
      end
    end
  end

  puts distances.map.with_index {|distance, id| [id, distance].join(' ') }.join("\n")
end