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 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 = []
    @g = []
    (0..@v - 1).each { @g << Array.new}
    @dist = [0x3fffffff] * v
    @prev = [-1] * v
  end

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

  def dijkstra(s)
    @dist[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}
      next if (@dist[v] < d)
      @g[v].each do |node|
        to, weight = node[:to], node[:weight]
        if @dist[to] > @dist[v] + weight
          @prev[to] = v
          @dist[to] = @dist[v] + weight
          que[to] = @dist[to]
        end
      end
    end

    (0..@v - 1).each do |i|
      puts "#{i} #{@dist[i]}"
    end
  end
end

def main
  n = gets.rstrip.to_i
  g = Graph.new(n)
  (0..n - 1).each do |row|
    info = gets.rstrip.split(/\s/).map(&:to_i)
    idx, kk = info[0], info[1]
    edges = info[2..-1]
    (0..kk - 1).each do |k|
      v, c = edges[2 * k], edges[2 * k + 1]
      g.add_edge(row, v, c)
    end
  end
  g.dijkstra(0)
end

main