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.Generic;
using System.Linq;
using System.Text;

namespace ConsoleApplication1
{
    class Node
    {
        internal int cost;
        internal bool isVisit;
        internal List<Edge> routes;

        public Node()
        {
            cost = int.MaxValue;
            isVisit = false;
            routes = new List<Edge>();
        }
    }

    class Edge : IComparable
    {
        internal int target;
        internal int cost;

        public Edge(int target, int cost)
        {
            this.target = target;
            this.cost = cost;
        }

        public int CompareTo(object obj)
        {
            Edge temp = obj as Edge;
            return this.cost - temp.cost;
        }
    }

    class Program
    {
        static void Main(string[] args)
        {
            int n = int.Parse(Console.ReadLine());

            Node[] nodeList = new Node[n];
            for (int i = 0; i < n; i++) nodeList[i] = new Node();

            for (int i = 0; i < n; i++)
            {
                string[] input = Console.ReadLine().Split(' ');
                int u = int.Parse(input[0]);
                int k = int.Parse(input[1]);

                for (int j = 0; j < k; j++)
                {
                    int v = int.Parse(input[2 + j * 2]);
                    int c = int.Parse(input[3 + j * 2]);

                    nodeList[u].routes.Add(new Edge(v, c));
                }
            }

            Dijkstra(nodeList, n);
        }

        static void Dijkstra(Node[] nodeList, int n)
        {
            PriorityQueue<Edge> pq = new PriorityQueue<Edge>();

            nodeList[0].cost = 0;
            pq.Enqueue(new Edge(0, 0));

            while (pq.Count() > 0)
            {
                Edge f = pq.Dequeue();
                int u = f.target;
                nodeList[u].isVisit = true;

                if (nodeList[u].cost < -f.cost) continue;

                for (int j = 0; j < nodeList[u].routes.Count; j++)
                {
                    int v = nodeList[u].routes[j].target;

                    if (nodeList[v].isVisit == true) continue;

                    if (nodeList[v].cost > nodeList[u].cost + nodeList[u].routes[j].cost)
                    {
                        nodeList[v].cost = nodeList[u].cost + nodeList[u].routes[j].cost;
                        pq.Enqueue(new Edge(v, -nodeList[v].cost));
                    }
                }
            }

            StringBuilder sb = new StringBuilder();
            for (int i = 0; i < n; i++)
            {
                sb.AppendLine(i + " " + nodeList[i].cost);
            }
            Console.Write(sb);
        }
    }

    class PriorityQueue<T> where T : IComparable
    {
        List<T> heap;
        int size;

        public PriorityQueue()
        {
            heap = new List<T>();
        }

        public void Enqueue(T x)
        {
            heap.Add(x);
            int i = size++;

            while (i > 0)
            {
                int p = (i - 1) / 2;

                if (Compare(heap[p], x) <= 0) break;

                heap[i] = heap[p];
                i = p;
            }
            heap[i] = x;
        }

        public T Dequeue()
        {
            T ret = heap[0];
            T x = heap[--size];

            int i = 0;
            while (i * 2 + 1 < size)
            {
                int a = i * 2 + 1;
                int b = i * 2 + 2;

                if (b < size && Compare(heap[b], heap[a]) < 0)
                    a = b;
                if (Compare(heap[a], x) >= 0)
                    break;

                heap[i] = heap[a];
                i = a;
            }
            heap[i] = x;
            heap.RemoveAt(size);
            return ret;
        }

        public int Count()
        {
            return size;
        }

        private int Compare(T x, T y)
        {
            return y.CompareTo(x);
        }
    }
}