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)

# ALDS1_12_B: Single Source Shortest Path 1

class Graph
  def initialize(n_v)
    @n_v   = n_v
    @costs = Array.new(n_v) { Hash.new }
  end
 
  def connect(v_from, v_to, cost)
    @costs[v_from][v_to] = cost
  end
 
  def calc_min_costs_from(v_start)
    costs_from_start = Array.new @n_v, Float::INFINITY
    visiting = {}
    costs_from_start[v_start] = 0
    visiting[v_start] = 0
 
    while visiting.any?
      v_from, _ = visiting.min_by do |v, _|
        costs_from_start[v]
      end
      visiting.delete v_from

      @costs[v_from].each do |v_to, cost|
        through_cost = costs_from_start[v_from] + cost
        if through_cost < costs_from_start[v_to]
          costs_from_start[v_to] = through_cost
          visiting[v_to] = true
        end
      end
    end
    
    costs_from_start
  end
end
 
# Read input.
n_v   = gets.to_i
graph = Graph.new(n_v)
n_v.times do
  v_from, n_degs, *inputs = gets.split.map &:to_i
  1.step(n_degs * 2, 2) do |i|
    v_to, cost = inputs[i - 1], inputs[i]
    graph.connect v_from, v_to, cost
  end
end

graph.calc_min_costs_from(0).each_with_index do |cost, idx|
  puts "#{idx} #{cost}"
end