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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.ArrayList;
import java.util.Comparator;
import java.util.HashSet;
import java.util.InputMismatchException;
import java.util.NoSuchElementException;
import java.util.PriorityQueue;

public class Main {

	static PrintWriter out;
	static InputReader ir;
	static boolean debug =false;
    static final long mod=(long)1e9+7;
	static void solve() {
        int n=ir.nextInt();
		int m=ir.nextInt();
        Graph[] g=new Graph[n];
        for(int i=0;i<n;i++){
            g[i]=new Graph();
        }
        int s=ir.nextInt()-1;
		int t=ir.nextInt()-1;
		for(int i=0;i<m;i++){
			int u=ir.nextInt()-1;
			int v=ir.nextInt()-1;
			int d=ir.nextInt();
			g[u].add(new long[]{v,d});
			g[v].add(new long[]{u,d});
		}
		long[] sd=dijkstra(s,g);
		long[] td=dijkstra(t,g);
		tr(sd);
		HashSet<Long> st=new HashSet<Long>();
		long[] sp=route(t,g,sd,st,1);
		long[] tp=route(s,g,td,st,0);
		tr(sp);
		long res=0;
		for(long x : st){
			res+=sp[(int)(x/n)]*tp[(int)(x%n)]*sp[(int)(x/n)]*tp[(int)(x%n)];
			res%=mod;
		}
		for(int i=0;i<n;i++){
			if(sd[i]*2==sd[t]){
				res+=sp[i]*tp[i]*sp[i]*tp[i];
				res%=mod;
			}
		}
		res=(sp[s]*tp[t])%mod-res;
		while(res<0)
			res+=mod;
		out.println(res%mod);
	}

    static class Graph extends ArrayList<long[]>{}

    static long[] dijkstra(int s,Graph[] g){
		int n=g.length;
        long[] res=new long[n];
		Arrays.fill(res,Long.MAX_VALUE/2);
		res[s]=0;
		PriorityQueue<long[]> pque=new PriorityQueue<long[]>(new Comparator<long[]>(){
			public int compare(long[] A,long[] B){
				return Long.compare(A[1],B[1]);
			}
		});
		pque.offer(new long[]{s,0});
		while(!pque.isEmpty()){
			long[] d=pque.poll();
			int from=(int)d[0];
			for(int i=0;i<g[from].size();i++){
				int to=(int)g[from].get(i)[0];
				if(res[to]>res[from]+g[from].get(i)[1]){
					res[to]=res[from]+g[from].get(i)[1];
					pque.offer(new long[]{to,res[to]});
				}
			}
		}
		return res;
    }

	static long[] route(int s,Graph[] g,long[] d,HashSet<Long> st,int f){
		int n=g.length;
		long[] res=new long[n];
		boolean[] v=new boolean[n];
		res[s]=1;
		PriorityQueue<long[]> pque=new PriorityQueue<long[]>(new Comparator<long[]>(){
			public int compare(long[] A,long[] B){
				if(A[1]!=B[1]) return -Long.compare(A[1],B[1]);
				return Long.compare(A[0],B[0]);
			}
		});
		pque.offer(new long[]{s,d[s]});
		while(!pque.isEmpty()){
			long[] e=pque.poll();
			int from=(int)e[0];
			if(v[from])
				continue;
			v[from]=true;
			for(int i=0;i<g[from].size();i++){
				int to=(int)g[from].get(i)[0];
				if(d[to]==d[from]-g[from].get(i)[1]){
					res[to]+=res[from];
					res[to]%=mod;
					if(f==1&&2*d[from]>d[s]&&2*d[to]<d[s]){
						st.add((long)from*n+to);
					}
					pque.offer(new long[]{to,d[to]});
				}
			}
		}
		return res;
	}

	public static void main(String[] args) {
		ir = new InputReader(System.in);
		out = new PrintWriter(System.out);
		solve();
		out.flush();
	}

	static class InputReader {

		private InputStream in;
		private byte[] buffer = new byte[1024];
		private int curbuf;
		private int lenbuf;

		public InputReader(InputStream in) {
			this.in = in;
			this.curbuf = this.lenbuf = 0;
		}

		public boolean hasNextByte() {
			if (curbuf >= lenbuf) {
				curbuf = 0;
				try {
					lenbuf = in.read(buffer);
				} catch (IOException e) {
					throw new InputMismatchException();
				}
				if (lenbuf <= 0)
					return false;
			}
			return true;
		}

		private int readByte() {
			if (hasNextByte())
				return buffer[curbuf++];
			else
				return -1;
		}

		private boolean isSpaceChar(int c) {
			return !(c >= 33 && c <= 126);
		}

		private void skip() {
			while (hasNextByte() && isSpaceChar(buffer[curbuf]))
				curbuf++;
		}

		public boolean hasNext() {
			skip();
			return hasNextByte();
		}

		public String next() {
			if (!hasNext())
				throw new NoSuchElementException();
			StringBuilder sb = new StringBuilder();
			int b = readByte();
			while (!isSpaceChar(b)) {
				sb.appendCodePoint(b);
				b = readByte();
			}
			return sb.toString();
		}

		public int nextInt() {
			if (!hasNext())
				throw new NoSuchElementException();
			int c = readByte();
			while (isSpaceChar(c))
				c = readByte();
			boolean minus = false;
			if (c == '-') {
				minus = true;
				c = readByte();
			}
			int res = 0;
			do {
				if (c < '0' || c > '9')
					throw new InputMismatchException();
				res = res * 10 + c - '0';
				c = readByte();
			} while (!isSpaceChar(c));
			return (minus) ? -res : res;
		}

		public long nextLong() {
			if (!hasNext())
				throw new NoSuchElementException();
			int c = readByte();
			while (isSpaceChar(c))
				c = readByte();
			boolean minus = false;
			if (c == '-') {
				minus = true;
				c = readByte();
			}
			long res = 0;
			do {
				if (c < '0' || c > '9')
					throw new InputMismatchException();
				res = res * 10 + c - '0';
				c = readByte();
			} while (!isSpaceChar(c));
			return (minus) ? -res : res;
		}

		public double nextDouble() {
			return Double.parseDouble(next());
		}

		public int[] nextIntArray(int n) {
			int[] a = new int[n];
			for (int i = 0; i < n; i++)
				a[i] = nextInt();
			return a;
		}

		public long[] nextLongArray(int n) {
			long[] a = new long[n];
			for (int i = 0; i < n; i++)
				a[i] = nextLong();
			return a;
		}

		public char[][] nextCharMap(int n, int m) {
			char[][] map = new char[n][m];
			for (int i = 0; i < n; i++)
				map[i] = next().toCharArray();
			return map;
		}
	}

	static void tr(Object... o) {
		if (debug)
			out.println(Arrays.deepToString(o));
	}
}