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: Time Limit Exceeded)

# encoding: utf-8                                                                                            

class Graph
  INF = 0x3fffffff
  def initialize(v)
    @v = v
    @edges = []
    @g = []
    @v.times.each { |idx| @g << Hash.new}
    @dists = [INF] * @v
    @prevs = [-1] * @v
  end

  def add_edge(u, v, w)
    @g[u][v] = w
  end

  def dijkstra(s)
    @dists[s] = 0
    que = { }
    que[s] = 0


    while not que.empty?
      v, d = que.min_by { |key, val| val}
      que.delete_if { |key, val| key == v and val == d}
      @g[v].each do |to, dist|
        if @dists[to] > @dists[v] + dist
          @prevs[to] = v
          @dists[to] = @dists[v] + dist
          que[to] = @dists[to]
        end
      end
    end
  end

  def solve(s)
    dijkstra(s)
    @dists.each_with_index do |dist, idx|
      puts "#{idx} #{dist}"
    end
  end

end


def input
  n = gets.to_i
  graph = Graph.new(n)
  (1..n).each do |node|
    arr = gets.rstrip.split(/\s/).map(&:to_i)
    u, k, edges = arr[0], arr[1], arr[2..-1]
    k.times { |ik| graph.add_edge(u, edges[2 * ik], edges[2 * ik + 1])}
  end
  graph
end

g = input
g.solve(0)