Problem:
Score : 400 points

Problem Statement

You are given an undirected connected weighted graph with N vertices and M edges that contains neither self-loops nor double edges.

The i-th (1≤i≤M) edge connects vertex a_i and vertex b_i with a distance of c_i.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

A connected graph is a graph where there is a path between every pair of different vertices.

Find the number of the edges that are not contained in any shortest path between any pair of different vertices.

Constraints

2≤N≤100

N-1≤M≤min(N(N-1)/2,1000)

1≤a_i,b_i≤N

1≤c_i≤1000

c_i is an integer.

The given graph contains neither self-loops nor double edges.

The given graph is connected.

Input

The input is given from Standard Input in the following format:

N M
a_1 b_1 c_1
a_2 b_2 c_2
:
a_M b_M c_M

Output

Print the number of the edges in the graph that are not contained in any shortest path between any pair of different vertices.

Sample Input 1

3 3
1 2 1
1 3 1
2 3 3

Sample Output 1

1

In the given graph, the shortest paths between all pairs of different vertices are as follows:

The shortest path from vertex 1 to vertex 2 is: vertex 1 → vertex 2, with the length of 1.

The shortest path from vertex 1 to vertex 3 is: vertex 1 → vertex 3, with the length of 1.

The shortest path from vertex 2 to vertex 1 is: vertex 2 → vertex 1, with the length of 1.

The shortest path from vertex 2 to vertex 3 is: vertex 2 → vertex 1 → vertex 3, with the length of 2.

The shortest path from vertex 3 to vertex 1 is: vertex 3 → vertex 1, with the length of 1.

The shortest path from vertex 3 to vertex 2 is: vertex 3 → vertex 1 → vertex 2, with the length of 2.

Thus, the only edge that is not contained in any shortest path, is the edge of length 3 connecting vertex 2 and vertex 3, hence the output should be 1.

Sample Input 2

3 2
1 2 1
2 3 1

Sample Output 2

0

Every edge is contained in some shortest path between some pair of different vertices.

Solution in C# (Mono 4.6.2.0) (judge verdict: Runtime Error)

// optional
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 {
    static List<KeyValue>[] nearedList;
    static int[][] dmap;
    static int[][] rawDMap;
    static int n;
    public static void Main() {
        #region SetAutoFlushIsFalse
#if !DEBUG
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
#endif
        #endregion
        try {
            var nm = ReadLib.ReadAsIntArray();
            dmap = Enumerable.Range(0, nm[0])
                .Select(j =>
                    Enumerable
                    .Range(0, nm[0])
                    .Select(i => int.MaxValue).ToArray()
                    )
                .ToArray();

            rawDMap = dmap.DeepCopy();
            n = nm[0];

            // 距離は0
            for (int i = 0; i <= nm[0]; i++) {
                dmap[i][i] = 0;
            }
            // 隣接グラフ生成
            nearedList = new List<KeyValue>[nm[0] + 1];
            for (int i = 0; i <= nm[0]; i++) {
                nearedList[i] = new List<KeyValue>();
            }

            for (int i = 0; i < nm[1]; i++) {
                var abc = ReadLib.ReadAsIntArray();
                nearedList[abc[0]].Add(new KeyValue() { Key = abc[1], Value = abc[2] });
                nearedList[abc[1]].Add(new KeyValue() { Key = abc[0], Value = abc[2] });
            }

            for (int i = 1; i <= nm[0]; i++) {
                Calc(i);
            }

            var score = 0;
            for (int i = 1; i <= n; i++) {
                for (int j = 1; j <= n; j++) {
                    if (rawDMap[i][j] == int.MaxValue) {
                        continue;
                    }

                    if (dmap[i][j] == rawDMap[i][j]) {
                        score++;
                    }
                }
            }
            
            Console.WriteLine(score);

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

    enum C {
        B, G, W
    }

    private static void Calc(int idx) {
        var d = dmap[idx];
        var colors = new C[n + 1];
        for (int i = 0; i <= n; i++) {
            colors[i] = C.W;
        }

        var heap = new Heaps.PriorityQueue<KeyValue>
            (new KeyValue() { Key = -1, Value = -1 }, IsMax: false);

        for (int i = 1; i < d.Length; i++) {
            heap.Enqueue(new KeyValue() { Key = i, Value = d[i] });
        }

        heap.Update();

        while (heap.Count > 0) {
            var kv = heap.Dequeue();
            colors[kv.Key] = C.B;

            if (d[kv.Key] < kv.Value) {
                continue;
            }

            var list = nearedList[kv.Key];
            for (int i = 0; i < nearedList.Length; i++) {
                var e = list[i];
                if (colors[e.Key] == C.B) {
                    continue;
                }

                if (d[e.Key] > d[kv.Key] + e.Value) {
                    heap.Enqueue(new KeyValue() {
                        Key = e.Key,
                        Value = d[kv.Key] + e.Value }
                    );
                }
            }
        }

        for (int i = 0; i < d.Length; i++) {
            dmap[idx][i] = d[i];
        }
    }

    struct KeyValue : IComparable<KeyValue> {
        public int Key { get; set; }
        public int Value { get; set; }

        public int CompareTo(KeyValue other) {
            return this.Value.CompareTo(other.Value);
        }
    }

}

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();
    }

    public static char[,] ReadMapAsChar(int hSize, int wSize) {
        var map = new char[hSize, wSize];
        for (int h = 0; h < hSize; h++) {
            var a = Console.ReadLine();
            for (int w = 0; w < wSize; w++) {
                map[h, w] = a[w];
            }
        }
        return map;
    }
}

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());
        }
    }
}

/// <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);
    }

    /// <summary>同一の値で埋める</summary>
    public static void Initialize<T>(this T[,] array, T value) {
        for (int i = 0; i < array.GetLength(0); i++) {
            for (int j = 0; j < array.GetLength(1); j++) {
                array[i, j] = value;
            }
        }
    }
}

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;
    }

    /// <summary>約数を列挙</summary>
    public static long[] Divisors(long n) {
        var ret = new List<long>();
        var sqrt = Math.Ceiling(Math.Sqrt(n));
        for (long i = 1; i <= sqrt; i++) {
            if (i * i > n) {
                continue;
            }

            if (n % i != 0) {
                continue;
            }

            ret.Add(i);

            if (n / i == i) {
                continue;
            }

            ret.Add(n / i);
        }

        return ret.ToArray();
    }
}

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>
///  * Combination, nCk mod p を求める
/// https://ja.wikipedia.org/wiki/%E3%83%A2%E3%82%B8%E3%83%A5%E3%83%A9%E9%80%86%E6%95%B0
/// http://drken1215.hatenablog.com/entry/2018/06/08/210000
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（モジュラ逆数）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
            // mod % i < i
            // modの倍数を加えても逆元である
            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;
    }
}