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 C# (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
public static class P
{
    public static void Main()
    {
        int n = int.Parse(Console.ReadLine());

        Tuple<int, long>[][] childs = new Tuple<int, long>[n][];
        for (int i = 0; i < n; i++)
        {
            var data = Console.ReadLine().Split().Select(int.Parse).ToArray();
            var id = data[0];
            childs[id] = new Tuple<int, long>[data[1]];
            for (int j = 2; j < data.Length; j += 2)
            {
                childs[id][j / 2 - 1] = new Tuple<int, long>(data[j], data[j + 1]);
            }
        }

        long[] dists = Enumerable.Repeat(-1L, n).ToArray();
        PriorityQueue<Tuple<int, long>, long> pqueue = new PriorityQueue<Tuple<int, long>, long>(x => x.Item2);
        pqueue.Push(new Tuple<int, long>(0, 0));
        while (pqueue.Count > 0)
        {
            var elem = pqueue.Pop();
            var node = elem.Item1;
            var dist = elem.Item2;
            if (dists[node] != -1) continue;
            dists[node] = dist;
            foreach (var item in childs[node])
            {
                if (dists[item.Item1] != -1) continue;
                pqueue.Push(new Tuple<int, long>(item.Item1, dist + item.Item2));
            }
        }
        for (int i = 0; i < dists.Length; i++)
        {
            Console.WriteLine($"{i} {dists[i]}");
        }
    }
}



class PriorityQueue<TValue, TKey> where TKey : IComparable<TKey>
{
    public int Count { get; private set; }
    private Func<TValue, TKey> KeySelector;
    private bool Descendance;
    private TValue[] data = new TValue[65536];
    private TKey[] keys = new TKey[65536];
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public PriorityQueue(Func<TValue, TKey> keySelector, bool descendance = false) { KeySelector = keySelector; Descendance = descendance; }
    public TValue Top
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        get { ValidateNonEmpty(); return data[1]; }
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public TValue Pop()
    {
        var top = Top;
        var item = data[Count];
        var key = keys[Count--];
        int index = 1;
        while (true)
        {
            if ((index << 1) >= Count)
            {
                if (index << 1 > Count) break;
                if (key.CompareTo(keys[index << 1]) > 0 ^ Descendance)
                { data[index] = data[index << 1]; keys[index] = keys[index << 1]; index <<= 1; }
                else break;
            }
            else
            {
                var nextIndex = keys[index << 1].CompareTo(keys[(index << 1) + 1]) <= 0 ^ Descendance ? (index << 1) : (index << 1) + 1;
                if (key.CompareTo(keys[nextIndex]) > 0 ^ Descendance)
                { data[index] = data[nextIndex]; keys[index] = keys[nextIndex]; index = nextIndex; }
                else break;
            }
        }
        data[index] = item;
        keys[index] = key;
        return top;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public void Push(TValue item)
    {
        var key = KeySelector(item);
        int index = ++Count;
        if (data.Length == Count) Extend(data.Length * 2);
        while ((index >> 1) != 0)
        {
            if (keys[index >> 1].CompareTo(key) > 0 ^ Descendance)
            { data[index] = data[index >> 1]; keys[index] = keys[index >> 1]; index >>= 1; }
            else break;
        }
        data[index] = item;
        keys[index] = key;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    private void Extend(int newSize)
    {
        TValue[] newData = new TValue[newSize];
        TKey[] newKeys = new TKey[newSize];
        data.CopyTo(newData, 0);
        keys.CopyTo(newKeys, 0);
        data = newData;
        keys = newKeys;
    }
    private void ValidateNonEmpty() { if (Count == 0) throw new IndexOutOfRangeException(); }
}