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.*;
import java.util.*;
import java.math.*;
// import java.awt.Point;
 
public class Main {
    InputStream is;
    PrintWriter out;
    String INPUT = "";
 
    long mod = 1_000_000_007;
    long inf = Long.MAX_VALUE;

    boolean[][] used;
    void solve(){
        int n = ni();
        int m = ni();
        int s = ni()-1;
        int t = ni()-1;
        Dijkstra dijk = new Dijkstra(n);
        long[][] edge = new long[2*m][3];
        for(int i = 0; i < m; i++){
            int from = ni()-1;
            int to = ni()-1;
            long d = nl();
            dijk.addDirectedEdge(from ,to, d);
            dijk.addDirectedEdge(to ,from, d);
            edge[2*i][0] = from;
            edge[2*i][1] = to;
            edge[2*i][2] = d;
            edge[2*i+1][0] = to;
            edge[2*i+1][1] = from;
            edge[2*i+1][2] = d;
        }
        long[][] dp1 = dijk.getDistMinPath(s);
        long[][] dp2 = dijk.getDistMinPath(t);
        long dist = dp1[t][0];
        //総数
        long total = dp1[t][1]%mod*dp2[s][1]%mod;
        //頂点でぶつかる場合
        for(int i = 0; i < n; i++){
            if(2*dp1[i][0]==dist){
                total -= dp1[i][1]%mod*dp1[i][1]%mod*dp2[i][1]%mod*dp2[i][1]%mod;
                total = (total+mod)%mod;
            }
        }
        //辺でぶつかる場合
        for(int i = 0; i < 2*m; i++){
            int from = (int)edge[i][0];
            int to = (int)edge[i][1];
            long cost = edge[i][2];
            if(2*dp1[from][0]<dist && 2*dp1[to][0]>dist && dp1[from][0]+cost+dp2[to][0]==dist){
                total -= dp1[from][1]%mod*dp2[to][1]%mod*dp1[from][1]%mod*dp2[to][1]%mod;
                total = (total+mod)%mod;
            }
        }
        out.println(total);
    }

    static class Dijkstra {
        int n;
        ArrayList<Pair>[] G;
        long INF = Long.MAX_VALUE / 3;
        long mod = 1000000007;
        public Dijkstra(int n) {
            this.n = n;
            G = new ArrayList[n];
            for (int i = 0; i < n; i++) {
                G[i] = new ArrayList<>();
            }
        }

        public void addDirectedEdge(int from, int to, long cost) {
            G[from].add(new Pair(to, cost));
        }

        public long[] getDist(int s) {
            PriorityQueue<Pair> Q = new PriorityQueue<>();
            Q.add(new Pair(s, 0));
            long[] dist = new long[n];
            Arrays.fill(dist, INF);
            boolean[] used = new boolean[n];
            while (!Q.isEmpty()) {
                Pair p = Q.poll();
                if (used[p.x]) continue;
                used[p.x] = true;
                dist[p.x] = p.y;

                for (Pair edge : G[p.x]) {
                    Q.add(new Pair(edge.x, p.y + edge.y));
                }
            }
            return dist;
        }
        // dp[i][0]:iまでの最短距離
        // dp[i][1]:iまでの最短ルートの経路数
        public long[][] getDistMinPath(int s) {
            PriorityQueue<Pair> Q = new PriorityQueue<>();
            Q.add(new Pair(s, 0));
            long[][] dp = new long[n][2];
            for(int i = 0; i < n; i++){
                dp[i][0] = INF;
                dp[i][1] = 0;
            }
            dp[s][0] = 0;
            dp[s][1] = 1;
            boolean[] used = new boolean[n];
            while (!Q.isEmpty()) {
                Pair p = Q.poll();
                if (used[p.x]) continue;
                used[p.x] = true;
                // dp[p.x][0] = p.y%mod;
                for (Pair edge : G[p.x]) {
                    long dist = dp[p.x][0] + edge.y;
                    long root = dp[p.x][1];
                    if(dist==dp[edge.x][0]){
                        dp[edge.x][1] += root;
                        // dp[edge.x][1] %= mod;
                    }
                    if(dist<dp[edge.x][0]){
                        dp[edge.x][0] = dist;
                        dp[edge.x][1] = root;
                        // dp[edge.x][0] %= mod;
                        // dp[edge.x][1] %= mod;
                    }
                    Q.add(new Pair(edge.x, dp[edge.x][0]));
                }
            }
            return dp;
        }

        class Pair implements Comparable<Pair> {
            int x;
            long y;

            Pair(int x, long y) {
                this.x = x;
                this.y = y;
            }

            public int compareTo(Pair p) {
                return Long.compare(y, p.y);
            }

        }
    }



    void run() throws Exception
    {
        is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
        out = new PrintWriter(System.out);
        long s = System.currentTimeMillis();
        solve();
        out.flush();
        if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
    }
    
    public static void main(String[] args) throws Exception { new Main().run(); }
    
    private byte[] inbuf = new byte[1024];
    private int lenbuf = 0, ptrbuf = 0;
    
    private int readByte()
    {
        if(lenbuf == -1)throw new InputMismatchException();
        if(ptrbuf >= lenbuf){
            ptrbuf = 0;
            try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
            if(lenbuf <= 0)return -1;
        }
        return inbuf[ptrbuf++];
    }
    
    private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
    private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
    
    private double nd() { return Double.parseDouble(ns()); }
    private char nc() { return (char)skip(); }
    
    private String ns()
    {
        int b = skip();
        StringBuilder sb = new StringBuilder();
        while(!(isSpaceChar(b) && b != ' ')){
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }
    
    private char[] ns(int n)
    {
        char[] buf = new char[n];
        int b = skip(), p = 0;
        while(p < n && !(isSpaceChar(b))){
            buf[p++] = (char)b;
            b = readByte();
        }
        return n == p ? buf : Arrays.copyOf(buf, p);
    }
    
    private char[][] nm(int n, int m)
    {
        char[][] map = new char[n][];
        for(int i = 0;i < n;i++)map[i] = ns(m);
        return map;
    }
    
    private int[] na(int n)
    {
        int[] a = new int[n];
        for(int i = 0;i < n;i++)a[i] = ni();
        return a;
    }
    
    private int ni()
    {
        int num = 0, b;
        boolean minus = false;
        while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
        if(b == '-'){
            minus = true;
            b = readByte();
        }
        
        while(true){
            if(b >= '0' && b <= '9'){
                num = num * 10 + (b - '0');
            }else{
                return minus ? -num : num;
            }
            b = readByte();
        }
    }
    
    private long nl()
    {
        long num = 0;
        int b;
        boolean minus = false;
        while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
        if(b == '-'){
            minus = true;
            b = readByte();
        }
        
        while(true){
            if(b >= '0' && b <= '9'){
                num = num * 10 + (b - '0');
            }else{
                return minus ? -num : num;
            }
            b = readByte();
        }
    }
    
    private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
 
}