Problem:
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Minimum Spanning Tree

For a given weighted graph $G = (V, E)$, find the minimum spanning tree (MST) of $G$ and print total weight of edges belong to the MST.

Input

In the first line, an integer $n$ denoting the number of vertices in $G$ is given. In the following $n$ lines, a $n \times n$ adjacency matrix $A$ which represents $G$ is given.
$a_{ij}$ denotes the weight of edge connecting vertex $i$ and vertex $j$. If there is no edge between $i$ and $j$, $a_{ij}$ is given by -1.

Output

Print the total weight of the minimum spanning tree of $G$.

Constraints

$1 \leq n \leq 100$

$0 \leq a_{ij} \leq 2,000$ (if $a_{ij} \neq -1$)

$a_{ij} = a_{ji}$

$G$ is a connected graph

Sample Input 1

5
-1 2 3 1 -1
2 -1 -1 4 -1
3 -1 -1 1 1
1 4 1 -1 3
-1 -1 1 3 -1

Sample Output 1

5

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, :parent

  def initialize(id)
    @id = id
    @siblings = []
    @distances = {}
    @state = :white
    @parent = nil
  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|
    nodes[id] ||= Node.new(id)
    node = nodes[id]

    distances = gets.chomp.split.map(&:to_i)

    distances.each_with_index do |distance, sibling_id|
      next if distance == -1

      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

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

      distance_from_current = sibling.distance_from(current)

      if distance_from_current < distances[sibling.id]
        distances[sibling.id] = distance_from_current
        sibling.parent = current
        sibling.state = :gray
      end
    end
  end

  puts nodes.select(&:parent).inject(0) {|sum, node| sum + node.distance_from(node.parent) }
end