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 Extention;
using Heaps;
// optional
using IOUtil;
using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.Serialization.Formatters.Binary;


public static class Program {
    public static void Main() {
        #region SetAutoFlushIsFalse
#if !DEBUG
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
#endif
        #endregion
        
        var n = ReadLib.ReadAsInt();
        var m = new List<Node>[n];

        for (int i = 0; i < n; i++) {
            m[i] = new List<Node>();
            var a = ReadLib.ReadAsIntArray();
            for (int j = 1; j <= a[1]; j++) {
                m[i].Add(new Node(a[2 * j], a[2 * j + 1]));
            }
        }


        int[] d = Calculation.Calc(n, m);
        for (int i = 0; i < n; i++) {
            Console.WriteLine(i + " " + d[i]);
        }


        #region Flush
#if !DEBUG
        Console.Out.Flush();
        sw.Close();
#endif
        #endregion
    }
}

class Node : IComparable<Node> {
    public int Id { get; }
    public int Distance { get; }

    public Node(int key, int value) {
        Id = key;
        Distance = value;
    }

    public int CompareTo(Node other) {
        return Distance.CompareTo(other.Distance);
    }
}

class NILNode : Node {
    public NILNode() : base(int.MaxValue, int.MaxValue) {
    }
}

static class Calculation {
    enum C {
        B, G, W
    }

    internal static int[] Calc(int n, List<Node>[] nodesList) {
        // d初期化
        var d = Enumerable.Repeat(int.MaxValue, n).ToArray();
        d[0] = 0;

        var queue = new PriorityQueue<Node>(new NILNode(), IsMax: false);
        for (int i = 0; i < n; i++) {
            queue.Enqueue(new Node(i, d[i]));
        }

        var p = new int[n];
        C[] cs = Enumerable.Repeat(C.W, n).ToArray();

        while (queue.Count > 0) {
            var minNode = queue.Dequeue();
            cs[minNode.Id] = C.B;

            if (d[minNode.Id] < minNode.Distance) {
                continue;
            }

            foreach (var near in nodesList[minNode.Id]) {
                if (cs[near.Id] != C.B && near.Distance + d[minNode.Id] < d[near.Id]) {
                    d[near.Id] = near.Distance + d[minNode.Id];
                    p[near.Id] = minNode.Id;
                    queue.Enqueue(new Node(near.Id, d[near.Id]));
                }
            }
        }

        return d;
    }
}

namespace IOUtil {
    static class ReadLib {
        public static long ReadAsLong() {
            return long.Parse(Console.ReadLine());
        }

        public static int ReadAsInt() {
            return int.Parse(Console.ReadLine());
        }

        public static int[] ReadAsIntArray(int step = 0) {
            return Console.ReadLine().Split(' ').Select(s => int.Parse(s) + step).ToArray();
        }

        public static long[] ReadAsLongArray(int step = 0) {
            return Console.ReadLine().Split(' ').Select(s => long.Parse(s) + step).ToArray();
        }

        public static ulong[] ReadAsULongArray() {
            return Console.ReadLine().Split(' ').Select(s => ulong.Parse(s)).ToArray();
        }

        public static string[] ReadAsStringArray() {
            return Console.ReadLine().Split(' ');
        }

        public static int[] ReadNTimesAsInt(int n, int step = 0) {
            return Enumerable.Range(1, n).Select(s => int.Parse(Console.ReadLine()) + step).ToArray();
        }

        public static char[][] ReadNTimesAsCharArray(int n) {
            return Enumerable.Range(1, n).Select(i => Console.ReadLine().ToArray()).ToArray();
        }

        public static long[] ReadNTimesAsLong(int n, int step = 0) {
            return Enumerable.Range(1, n).Select(s => long.Parse(Console.ReadLine()) + step).ToArray();
        }

        public static int[][] ReadNTimesAsIntArray(int n, int step = 0) {
            return Enumerable.Range(1, n).Select(s => ReadAsIntArray(step)).ToArray();
        }

        public static string[] ReadNTimesAsString(int n) {
            return Enumerable.Range(1, n).Select(s => Console.ReadLine()).ToArray();
        }
    }
}

namespace Heaps {
    /// <summary>優先度付きキュー</summary>
    public class PriorityQueue<T> : IEnumerable<T>, IReadOnlyCollection<T> where T : IComparable<T> {
        public List<T> HeapArray { get; }
        public T NIL { get; }
        public bool IsMax { get; }

        public int Count => HeapArray.Count;

        /// <param name="heapArray">ヒープ</param>
        /// <param name="NIL">NILの値</param>
        /// <param name="IsMax">True時：大きい方から取り出す、False時：小さい方から取り出す</param>
        public PriorityQueue(T NIL, IEnumerable<T> heapArray = null, bool IsMax = true) {
            this.HeapArray = heapArray == null ? new List<T>() : heapArray.ToList();
            this.NIL = NIL;
            this.IsMax = IsMax;
            Update();
        }

        /// <summary>KeyのValueを取得</summary>
        public T this[int id] {
            get {
                if (id <= 0 || id > HeapArray.Count) {
                    return NIL;
                }

                return HeapArray[id - 1];
            }
        }

        class Node {
            public int Key { get; }
            public T Value => outer[Key];
            readonly PriorityQueue<T> outer;

            public Node(int key, PriorityQueue<T> outer) {
                Key = key;
                this.outer = outer;
            }

            public Node Left {
                get {
                    var leftKey = 2 * Key;
                    return new Node(leftKey, outer);
                }
            }

            public Node Right {
                get {
                    var rightkey = 2 * Key + 1;
                    return new Node(rightkey, outer);
                }
            }

            public Node Parent {
                get {
                    var parentKey = Key / 2;
                    return new Node(parentKey, outer);
                }
            }
        }

        private Node GetNode(int key) {
            return new Node(key, this);
        }

        /// <summary>max Heap木とする O(N)</summary>
        public void Update() {
            var n = HeapArray.Count() / 2;
            /// <summary>葉から順に処理を行う</summary>
            for (int i = n; i >= 1; i--) {
                Heapify(i);
            }
        }

        /// <summary>idを根とする部分木について、Heap木を維持したまま、idを下降させる</summary>
        private void Heapify(int key) {
            var current = GetNode(key);

            var left = current.Left;
            var right = current.Right;

            var largest = current;
            var sign = IsMax ? 1 : -1;
            if (left.Value.CompareTo(NIL) != 0 && left.Value.CompareTo(current.Value) * sign > 0) {
                largest = left;
            }

            if (right.Value.CompareTo(NIL) != 0 && right.Value.CompareTo(largest.Value) * sign > 0) {
                largest = right;
            }

            if (largest.Key != key) {
                HeapArray.Swap(key - 1, largest.Key - 1);   // 元と異なるときはSwapする
                Heapify(largest.Key); // largest.Key は元のkeyの場所。葉で更にMaxHeaptyを行う。
            }
        }

        /// <summary>keyの値を挿入する</summary>
        public void Enqueue(T value) {
            HeapArray.Add(value);

            var inserted = GetNode(HeapArray.Count);    // 末尾に挿入
            var parent = inserted.Parent;
            var sign = IsMax ? 1 : -1;

            // 挿入Nodeが根になる or 親 > 子になるまで親とNodeを交換
            while (inserted.Key > 1 && value.CompareTo(parent.Value) * sign > 0) {
                HeapArray.Swap(inserted.Key - 1, parent.Key - 1);
                inserted = parent;
                parent = inserted.Parent;
            }
        }

        public T Dequeue() {
            if (HeapArray.Count == 0) {
                return NIL;
            }

            var popVal = HeapArray.First();
            var newRootVal = HeapArray.Last();  // 新しい根となる。

            HeapArray.RemoveAt(HeapArray.Count - 1);    // del last

            if (HeapArray.Count > 0) {
                HeapArray[0] = newRootVal;
                Heapify(1);
            }

            return popVal;
        }

        public T Peek() {
            return HeapArray.First();
        }

        public IEnumerator<T> GetEnumerator() {
            while (HeapArray.Count > 0) {
                yield return Dequeue();
            }
        }

        IEnumerator IEnumerable.GetEnumerator() {
            return GetEnumerator();
        }

        public void Clear() {
            HeapArray.Clear();
        }

        public bool Contains(T val) {
            return HeapArray.Contains(val);
        }

        public PriorityQueue<T> Clone() {
            return new PriorityQueue<T>(NIL, HeapArray.DeepCopy());
        }
    }
}

namespace Extention {
    /// <summary>拡張メソッドs</summary>
    static class MyExtention {
        /// <summary>iとjを入れ替える</summary>
        public static void Swap<T>(this T[] arr, int i, int j) {
            var buf = arr[i];
            arr[i] = arr[j];
            arr[j] = buf;
        }

        public static void Swap<T>(this List<T> list, int i, int j) {
            var buf = list[i];
            list[i] = list[j];
            list[j] = buf;
        }

        /// <summary>
        /// partition。末尾sArr[r]を基準として、左に基準より小さい数、右に基準より大きい数が存在するように配列を並び替える。
        /// </summary>
        /// <param name="sArr"></param>
        /// <param name="p">partitionの先頭の添字</param>
        /// <param name="r">partitionの末尾の添字</param>
        /// <returns>partitionの基準が存在する添字</returns>
        public static int Partition(this int[] sArr, int p, int r) {
            var x = sArr[r];
            var i = p - 1;
            for (int j = p; j < r; j++) {
                if (sArr[j] <= x) {
                    i++;
                    sArr.Swap(i, j);
                }
            }
            sArr.Swap(i + 1, r);
            return i + 1;
        }

        /// <summary>深いコピーをする</summary>
        public static T DeepCopy<T>(this T obj) {
            T result;
            var b = new BinaryFormatter();
            using (var mem = new MemoryStream()) {
                b.Serialize(mem, obj); //  binaryFormatterによりobjをmemにserial化して書き出す
                mem.Position = 0; // memのpotisionを先頭に戻す
                result = (T)b.Deserialize(mem); // memの内容を読み取る。
            }
            return result;
        }

        public static string Left(this string str, int length) {
            return str.Substring(0, length);
        }

        public static string Right(this string str, int length) {
            return str.Substring(str.Length - length, length);
        }

        public static int Find<T>(this T[] array, T obj) {
            return Array.IndexOf(array, obj);
        }

        public static int LastFind<T>(this T[] array, T obj) {
            return Array.LastIndexOf(array, obj);
        }
    }
}

static class MyMath {
    /// <summary>階乗</summary>
    public static long Factorial(int n) {
        if (n == 0) {
            return 1;
        }

        return Permutation(n, 0);
    }

    public static long Permutation(int n, int k) {
        return Enumerable.Range(0, n - k).Select(i => n - i).Aggregate(1, (b, i) => b * i);
    }

    #region FullPower
    /// <summary>総乗、π</summary>
    public static long FullPower(IEnumerable<long> vs) {
        return vs.Aggregate(1L, (a, b) => a * b);
    }

    public static long FullPower(IEnumerable<int> vs) {
        return vs.Aggregate(1L, (a, b) => a * b);
    }
    #endregion

    /// <summary>最大公約数</summary>
    public static long Gcd(long item1, long item2) {
        var a = Math.Max(item1, item2);
        var b = Math.Min(item1, item2);

        while (true) {
            long mod;
            Math.DivRem(a, b, out mod);
            if (mod == 0) {
                break;
            }

            a = b;
            b = mod;
        }
        return b;
    }

    /// <summary>約分。最大公約数で割ったものを返す。</summary>
    public static Tuple<long, long> DeviedByGcd(long item1, long item2) {
        var gcd = Gcd(item1, item2);
        return new Tuple<long, long>(item1 / gcd, item2 / gcd);
    }

    public static long Lcm(int item1, int item2) {
        return Math.BigMul(item1, item2) / Gcd(item1, item2);
    }

    public static ModCombination ModCombination(int maxN, int mod = 1000000007) {
        return new ModCombination(maxN, mod);
    }

    /// <summary>a^n mod</summary>
    public static int ModPow(long a, long n, int mod = 1000000007) {
        long res = 1;
        long pow = a;

        while (n > 0) {
            if ((n & 1) == 1) {
                res = (pow * res) % mod;    // resにaをかける
            }

            pow = (pow * pow) % mod;    // 次のaはa^2
            n >>= 1;
        }

        return (int)res;
    }
}

static class MyMethod {
    /// <summary>数え上げて、key, 出現回数に変換</summary>
    public static Dictionary<T, int> CountUp<T>(IEnumerable<T> str) {
        return
            str
            .GroupBy(c => c)
            .Select(g => new KeyValuePair<T, int>(g.Key, g.Count()))
            .ToDictionary(e => e.Key, e => e.Value);
    }

    /// <summary>ある要素を持つindexをすべて返す</summary>
    public static int[] InducesOfValue<T>(this T[] arr, T searchValue) {
        return arr
            .Select((v, i) => new { v, i })
            .Where(o => searchValue.Equals(o.v))
            .Select(o => o.i)
            .ToArray();
    }

    /// <summary>ある要素を持つindexをすべて返す</summary>
    public static int[] Induces<T>(this IEnumerable<T> arr, T searchValue) {
        return arr.ToArray().InducesOfValue(searchValue);
    }
}

/// <summary>組み合わせ</summary>
class ModCombination {
    private int mod;
    private int[] fac;
    private int[] finv;
    private int[] inv;

    public ModCombination(int max, int mod = 1000000007) {
        this.mod = mod;
        fac = new int[max]; // modが法のn!の列

        inv = new int[max]; // indexの逆元の列
        finv = new int[max]; // modを法とした、invの累積の累積

        fac[0] = 1; fac[1] = 1;
        finv[0] = 1; finv[1] = 1;
        inv[1] = 1;

        for (int i = 2; i < max; i++) {
            fac[i] = (int)(Math.BigMul(fac[i - 1], i) % mod);  // factorを求める
            var s = Math.BigMul(inv[mod % i], mod / i) % mod;
            inv[i] = (int)(mod - s);　 // iに対する逆元を求める
            finv[i] = (int)(Math.BigMul(finv[i - 1], inv[i]) % mod);
        }
    }
    /// <summary>nCk</summary>
    public int Calculation(int n, int k) {
        if (n < k) return 0;
        if (n < 0 || k < 0) return 0;

        return (int)(Math.BigMul(fac[n], (int)(Math.BigMul(finv[k], finv[n - k]) % mod)) % mod);
    }
}

/// <summary>あまりを計算するためのintのラッパー</summary>
struct ModInt {
    int Value { get; }
    int Mod { get; }

    public override string ToString() {
        return Value.ToString();
    }

    public static explicit operator ModInt(long value) {
        return new ModInt(value);
    }

    public ModInt(long value, int mod = 1000000007) {
        this.Value = (int)(value % mod);
        this.Mod = mod;
    }

    public static explicit operator int(ModInt modInt) {
        return modInt.Value;
    }

    private static void CheckNotEqualMod(ModInt item1, ModInt item2) {
        if (item1.Mod != item2.Mod) {
            throw new Exception("mod not equal");
        }
    }

    public static ModInt operator +(ModInt item1, ModInt item2) {
        CheckNotEqualMod(item1, item2);
        long value = item1.Value + item2.Value;
        return new ModInt(value, item1.Mod);
    }

    public static ModInt operator *(ModInt item1, ModInt item2) {
        CheckNotEqualMod(item1, item2);
        long value = Math.BigMul(item1.Value, item2.Value);
        return new ModInt(value, item1.Mod);
    }

    public static ModInt operator -(ModInt item1, ModInt item2) {
        CheckNotEqualMod(item1, item2);
        long value = item1.Value - item2.Value;
        if (value < 0) {
            value += item1.Mod;
        }
        return new ModInt(value, item1.Mod);
    }

    public static ModInt operator +(ModInt item1, int item2) {
        return item1 + new ModInt(item2, item1.Mod);
    }

    public static ModInt operator *(ModInt item1, int item2) {
        return item1 * new ModInt(item2, item1.Mod);
    }

    public static ModInt operator -(ModInt item1, int item2) {
        return item1 - new ModInt(item2, item1.Mod);
    }


    public override int GetHashCode() {
        return this.Value;
    }

    public override bool Equals(object obj) {
        if (obj == null || this.GetType() != obj.GetType()) {
            return false;
        }

        var modIntObj = (ModInt)obj;
        return modIntObj.Value == this.Value;
    }

    public static bool operator ==(ModInt item1, ModInt item2) {
        return item1.Value == item2.Value;
    }

    public static bool operator !=(ModInt item1, ModInt item2) {
        return !(item1 == item2);
    }

    public static bool operator ==(ModInt item1, int item2) {
        return item1.Value == item2;
    }

    public static bool operator !=(ModInt item1, int item2) {
        return !(item1.Value == item2);
    }

    public static bool operator >=(ModInt item1, ModInt item2) {
        return item1.Value >= item2.Value;
    }

    public static bool operator <=(ModInt item1, ModInt item2) {
        return item1.Value <= item2.Value;
    }

    public static bool operator >=(ModInt item1, int item2) {
        return item1.Value >= item2;
    }

    public static bool operator <=(ModInt item1, int item2) {
        return item1.Value <= item2;
    }

    public static bool operator >(ModInt item1, ModInt item2) {
        return item1.Value > item2.Value;
    }

    public static bool operator <(ModInt item1, ModInt item2) {
        return item1.Value < item2.Value;
    }

    public static bool operator >(ModInt item1, int item2) {
        return item1.Value >= item2;
    }

    public static bool operator <(ModInt item1, int item2) {
        return item1.Value <= item2;
    }
}