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)

ToNode = Struct.new(:id, :weight)

_ = gets

$graph       = {}
$open_list   = []
$closed_list = []

while (line = gets)
  node_id, _, *nodes = line.split(' ').map(&:to_i)
  $graph[node_id] = nodes.each_slice(2).map{|id, weight| ToNode.new(id, weight)}
  $open_list.push(node_id)
end

$visited_nodes = []



$start_node_id = 0
# Min Distance '0' to `key`
$weights = {$start_node_id => 0}
$visited_nodes.push($start_node_id)


loop do
  # Node which should be visited
  next_visit_node =
    $visited_nodes
      .flat_map{|node_id|
        to_nodes = $graph[node_id]
        w        = $weights[node_id]
        to_nodes.map{|n| {id: n.id, weight: w + n.weight} }
      }
      .reject{|e| $visited_nodes.include?(e[:id])}
      .min_by{|e| e[:weight]}

  if next_visit_node.nil?
    break
  end

  $visited_nodes.push(next_visit_node[:id])
  $weights[next_visit_node[:id]] = next_visit_node[:weight]
end

$weights
  .sort_by{|k, v| k}
  .each{|node, weight|
    puts("#{node} #{weight}")
  }