Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

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)

# encoding: utf-8

class UFSet
  def initialize(sz)
    @sz = sz
    @par = [0] * sz
    (0..sz - 1).each { |idx| @par[idx] = idx}
  end

  def find(x)
    if @par[x] == x
      x
    else
      @par[x] = find(@par[x])
    end
  end

  def same(x, y)
    find(x) == find(y)
  end

  def unite(x, y)
    fx = find(x)
    fy = find(y)
    @par[fx] = fy if fx != fy
  end
end

class Graph
  def initialize(v)
    @v = v
    @edge = []
  end

  def add_edge(u, v, w)
    @edge << { from: u, to: v, weight: w}
  end

  def mst_kruskal
    ans = 0
    @edge = @edge.sort_by { |x| x[:weight]}
    ufset = UFSet.new(@v)
    @edge.each do |ee|
      u, v, c = ee[:from], ee[:to], ee[:weight]
      if not ufset.same(u, v)
        ans += c
        ufset.unite(u, v)
      else
        next
      end
    end
    ans
  end
end

def main
  n = gets.rstrip.to_i
  g = Graph.new(n)
  (0..n - 1).each do |row|
    r = gets.lstrip.rstrip.split(/\s/).map(&:to_i)
    r.each_with_index do |node, idx|
      g.add_edge(row, idx, node) if node >= 0
    end
  end
  puts g.mst_kruskal
end

main