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: Runtime Error)

package AtCoder.Indeed10;

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Comparator;
import java.util.HashSet;
import java.util.List;
import java.util.PriorityQueue;
import java.util.StringTokenizer;

public class Main {

    static int n,m,s,t;
    static List[] graph;

    static class nodeE {
        int node; int weight;
        nodeE(int node, int weight) {
            this.node = node; this.weight = weight;
        }
    }

    static class pathDis {
        long[] dis;
        long[] paths;
        pathDis(long[] dis, long[] paths) {
            this.dis = dis; this .paths = paths;
        }
    }

    static pathDis shortest(int node) {
        long[] pathlen = new long[n]; long[] ways = new long[n];

        Arrays.fill(pathlen, Integer.MAX_VALUE);
        Arrays.fill(ways, 0);

        PriorityQueue<nodeE> pq = new PriorityQueue<>(Comparator.comparingInt(o -> o.weight));
        HashSet<Integer> coveredNodes = new HashSet<>();

        pq.add(new nodeE(node, 0));

        while(pq.size() > 0) {
            nodeE topNode = pq.poll();
            if (coveredNodes.contains(topNode.node)) continue;

            pathlen[topNode.node] = topNode.weight;
            coveredNodes.add(topNode.node);
            long totalways = 0;
            for (nodeE neighbour: (List<nodeE>) graph[topNode.node]) {
                if (coveredNodes.contains(neighbour.node) && pathlen[neighbour.node] + neighbour.weight == topNode.weight) totalways += ways[neighbour.node];
                if (pathlen[neighbour.node] > topNode.weight + neighbour.weight) {
                    pq.add(new nodeE(neighbour.node, topNode.weight + neighbour.weight));
                }
            }
            if (totalways == 0) { // source node
                totalways++;
            }

            ways[topNode.node] = totalways;
        }

        return new pathDis(pathlen, ways);
    }

    public static void main(String[] args) {
        n = in.NI(); m = in.NI(); s = in.NI()-1; t = in.NI()-1;
        graph = new List[n];
        for (int i=0;i<n;i++) graph[i] = new ArrayList<nodeE>();
        for (int i=0;i<m;i++) {
            int u = in.NI()-1; int v = in.NI()-1; int w = in.NI();
            graph[u].add(new nodeE(v,w));
            graph[v].add(new nodeE(u,w));
        }

        pathDis destPathDis = shortest(t);
        pathDis sourcePathDis = shortest(s);

        long totans = sourcePathDis.paths[t] * sourcePathDis.paths[t];
        long shortestPathDis = sourcePathDis.dis[t];

        for (int i=0;i<n;i++) {
            if (sourcePathDis.dis[i] == shortestPathDis/2 && destPathDis.dis[i] == shortestPathDis/2) totans--;
        }

        long tsub = 0;
/*        for (int i=0;i<n;i++) {
            for (nodeE neigh: (List<nodeE>) graph[i]) {
                int n1 = i; int n2 = neigh.node;
                if (Math.abs(sourcePathDis.dis[n2] - sourcePathDis.dis[n1]) != (double)neigh.weight) continue;

                double s1 = sourcePathDis.dis[n1]; double s2 = sourcePathDis.dis[n2];
                double d1 = destPathDis.dis[n1]; double d2 = destPathDis.dis[n2];

                long sourcepath = 0; long destpath = 0;
                if (s1<s2) {
                    sourcepath = sourcePathDis.paths[n1];
                } else {
                    sourcepath = sourcePathDis.paths[n2];
                }

                if (d1<d2) {
                    destpath = destPathDis.paths[n1];
                } else {
                    destpath = destPathDis.paths[n2];
                }

                double ss = Math.min(s1,s2) + 0.001; double se = Math.max(s1,s2) - 0.001;
                double ds = Math.min(d1,d2) + 0.001; double de = Math.max(d1,d2) - 0.001;
                //if ((ds>ss && ds<se) || (de>ss && de<se) || (ss>ds && ss<de) || (se>ds && se<de)) tsub++;
                if (de<ss || ds>se) continue;
                tsub += sourcepath*destpath;
            }
        }*/

        for (int i=0;i<n;i++) {
            for (nodeE neigh: (List<nodeE>) graph[i]) {
                long d1 = sourcePathDis.dis[i]; long d2 =  destPathDis.dis[neigh.node];
                long w = neigh.weight;

                if (d1<shortestPathDis/2 && d2<shortestPathDis/2 && d1+d2+w==shortestPathDis) {
                    totans-=(sourcePathDis.paths[i]*destPathDis.paths[neigh.node]);
                }
            }
        }
        tsub/=2;
        totans-=tsub;
        out.println(totans);

        out.close();
    }

    static InputStream inputStream = System.in;
    static OutputStream outputStream = System.out;
    static InputReader in = new InputReader(inputStream);
    static PrintWriter out = new PrintWriter(outputStream);

    static class InputReader {
        public BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        public int NI() {
            return Integer.parseInt(next());
        }

        public long NL() {
            return Long.parseLong(next());
        }

    }
}