Problem:
Score : 700 points

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i.
The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

1 \leq N \leq 100 000

1 \leq M \leq 200 000

1 \leq S, T \leq N

S \neq T

1 \leq U_i, V_i \leq N (1 \leq i \leq M)

1 \leq D_i \leq 10^9 (1 \leq i \leq M)

If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).

U_i \neq V_i (1 \leq i \leq M)

D_i are integers.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
S T
U_1 V_1 D_1
U_2 V_2 D_2
:
U_M V_M D_M

Output

Print the answer.

Sample Input 1

4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.

Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

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

class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	
	const long mod = (long) 1e9 + 7;
	
	public void Solve(){
		
		List<int>[] E = new List<int>[N];
		for(int i=0;i<N;i++) E[i] = new List<int>();
		for(int i=0;i<M;i++){
			E[U[i]].Add(i);
			E[V[i]].Add(i);
		}
		
		long Inf = (long) 1e18;
		long[][] min = new long[2][];
		for(int i=0;i<2;i++){
			min[i] = new long[M];
			for(int j=0;j<N;j++) min[i][j] = Inf;
		}
		
		int[] strt = new int[]{S, T};
		for(int t=0;t<2;t++){
			var SH = new SkewHeap<Pair>();
			bool[] used = new bool[N];
			SH.Push(new Pair(strt[t],0));
			min[t][strt[t]] = 0;
			while(SH.Count > 0){
				
				var p = SH.Top; SH.Pop();
				var now = p.Pos;
				var cost = p.Cost;
				if(used[now]) continue;
				used[now] = true;
				min[t][now] = cost;
				foreach(var ne in E[now]){
					long nc = cost + D[ne];
					int nv = U[ne] == now ? V[ne] : U[ne];
					if(min[t][nv] > nc){
						min[t][nv] = nc;
						SH.Push(new Pair(nv, nc));
					}
				}
			}
		}
		
		long[][] comb = new long[2][];
		HashSet<int> minedge = new HashSet<int>();
		for(int t=0;t<2;t++){
			int fr = strt[t];
			int to = strt[t^1];
			List<Pair> L = new List<Pair>();
			for(int i=0;i<N;i++){
				L.Add(new Pair(i,min[t][i]));
			}
			L.Sort((p,q) => p.Cost.CompareTo(q.Cost));
			
			comb[t] = new long[N];
			comb[t][fr] = 1;
			long dmin = min[t][to];
			
			foreach(var pp in L){
				int now = pp.Pos;
				if(min[t][now] >= dmin) break;
				foreach(var ne in E[now]){
					long cost = D[ne];
					int ttt = U[ne] == now ? V[ne] : U[ne];
					if(min[t^1][ttt] + cost == min[t^1][now]){
						minedge.Add(ne);
						comb[t][ttt] += comb[t][now]; if(comb[t][now] >= mod) comb[t][now] -= mod;
					}
				}
			}
		}
		
		long dmax = min[0][T];
		
		long total = (comb[0][T] * comb[1][S]) % mod;
		bool[] uused = new bool[N];
		long remv = 0;
		for(int i=0;i<N;i++){
			if(min[0][i] == dmax - min[0][i]){
				long c = comb[0][i] * comb[1][i] % mod;
				remv += c; if(remv >= mod) remv -= mod;
			}
		}
		//Console.WriteLine(total);
		//Console.WriteLine(remv);
		
		foreach(var e in minedge){
			int u = U[e];
			int v = V[e];
			long di = D[e];
			
			if(min[0][u] < min[0][v]){
				if(2 * min[0][u] < dmax && 2 * min[0][v] > dmax ){
					long c = comb[0][u] * comb[1][v] % mod;
					remv += c; if(remv >= mod) remv -= mod;
				}
			} else if(min[0][u] > min[0][v]){
				if(2 * min[0][v] < dmax && 2 * min[0][u] > dmax ){
					long c = comb[0][v] * comb[1][u] % mod;
					remv += c; if(remv >= mod) remv -= mod;
				}
			}
		}
		
		long ans = (total + mod - remv) % mod;
		Console.WriteLine(ans);
		
	}
	int N,M;
	int S,T;
	int[] U,V;
	long[] D;
	
	public Sol(){
		var d = ria();
		N = d[0]; M = d[1];
		d = ria();
		S = d[0] - 1; T = d[1] - 1;
		U = new int[M];
		V = new int[M];
		D = new long[M];
		for(int i=0;i<M;i++){
			d = ria();
			U[i] = d[0] - 1;
			V[i] = d[1] - 1;
			D[i] = d[2];
			
		}
		
		
		
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}


class Pair : IComparable<Pair> {
	public int Pos;
	public long Cost;
	public Pair(int p, long c) {
		Pos = p; Cost = c;
	}
	public int CompareTo(Pair t) {
		//return this.Cost > t.Cost ? -1 : this.Cost < t.Cost ? 1 : 0;
		return this.Cost > t.Cost ? 1 : this.Cost < t.Cost ? -1 : 0;
	}
}


class SkewHeap<T> where T:IComparable<T>{
	public int Count{
		get{return cnt;}
		private set{cnt=value;}
	}
	
	public SkewHeap(){
		root=null;
		this.Count=0;
	}
	
	public void Push(T v){
		NodeSH<T> p=new NodeSH<T>(v);
		root=NodeSH<T>.Meld(root,p);
		this.Count++;
	}
	
	public void Pop(){
		if(root==null)return;
		root=NodeSH<T>.Meld(root.L,root.R);
		this.Count--;
	}
	
	public T Top{
		get{return root.Val;}
	}
	
	int cnt;
	NodeSH<T> root;
	
	class NodeSH<S> where S : IComparable<S> {
		public NodeSH<S> L, R;
		public S Val;

		public NodeSH(S v){
			Val = v;
			L = null; R = null;
		}
		public static NodeSH<S> Meld(NodeSH<S> a, NodeSH<S> b){
			if(a == null)return b;
			if (b == null) return a;
			if (a.Val.CompareTo(b.Val) > 0) swap(ref a, ref b);
			a.R = Meld(a.R, b);
			swap(ref a.L, ref a.R);
			return a;
		}

		static void swap<U>(ref U x, ref U y) {
			U t = x; x = y; y = t;
		}
	}
}