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)

N = gets.to_i
@a = N.times.map{gets.split.map(&:to_i)}

INFTY = 1<<21
WHITE = 0
GRAY  = 1
BLACK = 2

@adj = Array.new(N){[]}
@pq = []
@h = 0

def max_heapify(i)
  l = 2 * i
  r = 2 * i + 1

  if l <= @h && (@pq[l] <=> @pq[i]) == 1
    largest = l
  else
    largest = i
  end

  if r <= @h && (@pq[r] <=> @pq[largest]) == 1
    largest = r
  end

  if largest != i
    @pq[i], @pq[largest] = @pq[largest], @pq[i]
    max_heapify(largest)
  end
end

def extract
  return -INFTY if @h < 1
  maxv = @pq[1]

  @pq[1] = @pq[@h]
  @h -= 1

  max_heapify(1)
  maxv
end

def increase_key(i, key)
  return if (key <=> @pq[i]) == -1

  @pq[i] = key
  while i > 1 && (@pq[i / 2] <=> @pq[i]) == -1
    @pq[i], @pq[i / 2] = @pq[i / 2], @pq[i]
    i = i / 2
  end
end

def insert(key)
  @h += 1
  @pq[@h] = -INFTY
  increase_key(@h, key)
end

def dijkstra
  insert([0, 0])

  d = Array.new(N){INFTY}
  d[0] = 0
  color = Array.new(N){WHITE}
  color[0] = GRAY

  loop do
    f = extract
    break if f == -INFTY

    u = f[1]
    color[u] = BLACK
    next if d[u] < f[0] * -1

    (0...@adj[u].size).each do |j|
      v = @adj[u][j][0]
      next if color[v] == BLACK

      if d[v] > d[u] + @adj[u][j][1]
        d[v] = d[u] + @adj[u][j][1]
        insert([d[v] * -1, v])

        color[v] = GRAY
      end
    end
  end

  (0...N).each do |i|
    puts "#{i} #{d[i] == INFTY ? -1 : d[i]}"
  end
end

(0...N).each do |i|
  u = @a[i][0]
  k = @a[i][1]

  (2...(k * 2 + 2)).each do |j|
    next if j.odd?
    v = @a[i][j]
    c = @a[i][j + 1]
    @adj[u] = [] if @adj[u].nil?
    @adj[u].push([v, c])
  end
end
dijkstra