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 Rust (1.15.1) (judge verdict: Wrong Answer)

fn getline() -> String {
    let mut res = String::new();
    std::io::stdin().read_line(&mut res).ok();
    res
}

#[allow(unused_macros)]
macro_rules! readl {
    ($t: ty) => {
        {
            let s = getline();
            s.trim().parse::<$t>().unwrap()
        }
    };
    ($( $t: ty),+ ) => {
        {
            let s = getline();
            let mut iter = s.trim().split(' ');
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
}

#[allow(unused_macros)]
macro_rules! readlvec {
    ($t: ty) => {
        {
            let s = getline();
            let iter = s.trim().split(' ');
            iter.map(|x| x.parse().unwrap()).collect::<Vec<$t>>()
        }
    }
}

#[allow(unused_macros)]
macro_rules! debug {
    ($x: expr) => {
        println!("{}: {:?}", stringify!($x), $x)
    }
}

fn printiter<'a, T>(v: &'a T)
where
    &'a T: std::iter::IntoIterator,
    <&'a T as std::iter::IntoIterator>::Item: std::fmt::Display,
{
    for (i, e) in v.into_iter().enumerate() {
        if i != 0 {
            print!(" ");
        }
        print!("{}", e);
    }
    println!("");
}


#[allow(unused_imports)]
use std::cmp::{min, max};
#[allow(unused_imports)]
use std::collections::{BinaryHeap, BTreeMap, HashMap, HashSet};
#[allow(unused_imports)]
use std::collections::btree_map::Entry::{Occupied, Vacant};

use std::cmp::Ordering;
use std::clone::Clone;
#[derive(Clone, Debug, Hash, Eq, PartialEq)]
struct Edge<Cost>
where
    Cost: std::clone::Clone,
{
    to: usize,
    cost: Cost,
}

impl<Cost> Ord for Edge<Cost>
where
    Cost: Ord + std::clone::Clone,
{
    fn cmp(&self, other: &Edge<Cost>) -> Ordering {
        (self.cost).cmp(&(other.cost)).reverse()
    }
}

impl<Cost> PartialOrd for Edge<Cost>
where
    Cost: std::clone::Clone,
    Edge<Cost>: Ord,
{
    fn partial_cmp(&self, other: &Edge<Cost>) -> Option<std::cmp::Ordering> {
        Some(self.cmp(other))
    }
}

trait Graph
where
    Self::Cost: Clone
{
    type Cost;
    fn len(&self) -> usize;
    fn get<'a>(&'a self, p: usize) -> std::slice::Iter<'a, Edge<Self::Cost>>;
    fn get_order(&self, p: usize) -> Vec<usize>;
}

#[derive(Clone, Debug, Hash)]
struct AdjList<Cost: Clone> {
    adj: Vec<Vec<Edge<Cost>>>,
}

impl<Cost: Clone> std::ops::Index<usize> for AdjList<Cost> {
    type Output = Vec<Edge<Cost>>;

    fn index(&self, i: usize) -> &Vec<Edge<Cost>> {
        &self.adj[i]
    }
}

impl<Cost: Clone> std::ops::IndexMut<usize> for AdjList<Cost> {
    fn index_mut<'a>(&'a mut self, i: usize) -> &'a mut Vec<Edge<Cost>> {
        &mut self.adj[i]
    }
}

impl<Cost: Clone> AdjList<Cost> {
    fn new(n: usize) -> AdjList<Cost> {
        AdjList {
            adj: vec![vec![]; n],
        }
    }

    fn add_edge(&mut self, from: usize, to_: usize, cost_: Cost) {
        self.adj[from].push(Edge {
            to: to_,
            cost: cost_,
        });
    }

    fn add_uedge(&mut self, from: usize, to_: usize, cost_: Cost) {
        self.add_edge(from, to_, cost_.clone());
        self.add_edge(to_, from, cost_);
    }
}

impl<Cost> Graph for AdjList<Cost>
where
    Cost: Clone
{
    type Cost = Cost;

    fn len(&self) -> usize {
        self.adj.len()
    }

    fn get<'a>(&'a self, p: usize) -> std::slice::Iter<'a, Edge<Cost>> {
        self.adj[p].iter()
    }

    fn get_order(&self, s: usize) -> Vec<usize> {
        let mut res = Vec::with_capacity(self.len());
        let mut deg = vec![0; self.len()];
        for from in 0..self.len() {
            for e in self.get(from) {
                deg[e.to] += 1;
            }
        }
        let mut stack = Vec::new();
        stack.push(s);
        while let Some(p) = stack.pop() {
            res.push(p);
            for e in self.get(p) {
                deg[e.to] -= 1;
                if deg[e.to] == 0 {
                    stack.push(e.to);
                }
            }
        }
        // debug!(deg);
        res
    }
}

trait Zero {
    fn zero() -> Self;
}

impl Zero for i64 {
    fn zero() -> i64 {
        0
    }
}

trait Max {
    fn max_value() -> Self;
}

impl Max for i64 {
    fn max_value() -> i64 {
        std::i64::MAX
    }
}

trait GraphHavingOrderedCost: Graph
where
    Self::Cost: Ord + Add<Self::Cost, Output = Self::Cost> + Zero + Max + Copy,
{
    fn get_shortest_path(&self, from: usize) -> (Vec<Self::Cost>, Vec<usize>);
    fn get_shortest_path_graph(&self, from: usize, dist: &Vec<Self::Cost>) -> AdjList<Self::Cost>;
}

use std::ops::Add;
impl<T> GraphHavingOrderedCost for T
where
    T: Graph,
    T::Cost: Ord + Add<T::Cost, Output = T::Cost> + Zero + Max + Copy,
    Edge<T::Cost>: Ord,
{
    fn get_shortest_path(&self, from: usize) -> (Vec<Self::Cost>, Vec<usize>) {
        let n = self.len();

        let mut dist = vec![Self::Cost::max_value(); n];
        let mut pre = vec![n as usize; n];
        let mut heap = std::collections::BinaryHeap::new();
        heap.push(Edge {
            to: from,
            cost: Self::Cost::zero(),
        });
        dist[from] = Self::Cost::zero();
        while let Some(p) = heap.pop() {
            for e in self.get(p.to) {
                if p.cost + e.cost >= dist[e.to] {
                    continue;
                }
                heap.push(Edge {
                    to: e.to,
                    cost: p.cost + e.cost,
                });
                dist[e.to] = p.cost + e.cost;
                pre[e.to] = p.to;
            }
        }
        (dist, pre)
    }

    fn get_shortest_path_graph(&self, from: usize, dist: &Vec<Self::Cost>) -> AdjList<Self::Cost> {
        let n = self.len();

        let mut res = AdjList::new(n);
        for from in 0..n {
            for e in self.get(from) {
                if dist[from] + e.cost == dist[e.to] {
                    res.add_edge(from, e.to, e.cost);
                }
            }
        }
        res
    }
}


macro_rules! make_modint {
    ($MOD: expr, $name: ident) => {
        #[derive(Ord, Hash, Eq, PartialOrd, PartialEq)]
        struct $name {
            val: i64,
        }

        impl $name {
            fn new(x: i64) -> $name {
                let x = x%$MOD;
                $name{val: if x < 0 { x+$MOD } else { x }}
            }

            fn pow(&self, x: i64) -> $name {
                let mut res = $name::new(1);
                let mut tmp = x;
                let mut p = *self;
                while tmp != 0 {
                    if tmp&1 == 1 {
                        res *= p;
                    }
                    tmp = tmp>>1;
                    p = p*p;
                }
                res
            }

            fn inv(&self) -> $name {
                assert!(self.val != 0);
                let mut a = self.val;
                let mut b = $MOD;
                let mut u = 1;
                let mut v = 0;
                use std::mem::swap;
                while b != 0 {
                    let t = a/b;
                    a -= t*b;
                    swap(&mut a, &mut b);
                    u -= t*v;
                    swap(&mut u, &mut v);
                }
                $name::new(u)
            }
        }

        impl std::clone::Clone for $name {
            fn clone(&self) -> $name {
                $name{ val: self.val }
            }
        }

        impl std::marker::Copy for $name { }

        impl std::ops::Add for $name {
            type Output = $name;
            fn add(self, y: $name) -> $name {
                let tmp = self.val+y.val;
                $name{val: if tmp >= $MOD {tmp-$MOD} else {tmp}}
            }
        }

        impl std::ops::Neg for $name {
            type Output = $name;
            fn neg(self) -> $name {
                $name::new(self.val)
            }
        }

        impl std::ops::Sub for $name {
            type Output = $name;
            fn sub(self, other: $name) -> $name{
                let tmp = self.val-other.val;
                $name{val: if tmp < 0 {tmp+$MOD} else {tmp}}
            }
        }

        impl std::ops::Mul for $name {
            type Output = $name;
            fn mul(self, y: $name) -> $name {
                $name{val: (self.val*y.val)%$MOD}
            }
        }

        impl std::ops::Div for $name {
            type Output = $name;
            fn div(self, other: $name) -> $name {
                self*other.inv()
            }
        }

        impl std::fmt::Display for $name {
            fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
                write!(f, "{}", self.val)
            }
        }

        impl std::ops::AddAssign for $name {
        fn add_assign(&mut self, other: $name) {
            *self = *self+other;
        }
        }

        impl std::ops::SubAssign for $name {
            fn sub_assign(&mut self, other: $name) {
                *self = *self-other;
            }
        }

        impl std::ops::MulAssign for $name {
            fn mul_assign(&mut self, other: $name) {
                *self = *self*other;
            }
        }

        impl std::ops::DivAssign for $name {
            fn div_assign(&mut self, other: $name) {
                *self = *self*other.inv();
            }
        }

        impl std::fmt::Debug for $name {
            fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
                write!(f, "{}", self.val)
            }
        }
    }
}

make_modint!(1_000_000_000 + 7, ModInt);

fn mint(x: i64) -> ModInt {
    ModInt::new(x)
}


use std::io::prelude::*;
impl<Cost: Clone> AdjList<Cost> {
    fn vizualize(&self, name: &str) {
        let path = std::path::Path::new("tmp.dot");
        {
            let mut file = std::fs::File::create(path).unwrap();
            let n = self.len();
            file.write_all("digraph graph_name {\n".as_bytes()).unwrap();
            for p in 0..n {
                let s = ["    ".to_string(), p.to_string(), "[];\n".to_string()].connect("");
                file.write_all(s.as_bytes()).unwrap();
            }
            for p in 0..n {
                for e in &self[p] {
                    file.write_all(b"    ").unwrap();
                    file.write_all(p.to_string().as_bytes()).unwrap();
                    file.write_all(b" -> ").unwrap();
                    file.write_all(e.to.to_string().as_bytes()).unwrap();
                    file.write_all(b"\n").unwrap();
                }
            }
            file.write(b"}\n").unwrap();
        }
        let mut cmd = std::process::Command::new("dot");
        cmd.arg("-T");
        cmd.arg("png");
        cmd.arg("tmp.dot");
        cmd.arg("-o");
        cmd.arg(name);
        cmd.status().expect("commend failed to start");
    }
}


fn main() {
    use std::io::Write;
    let out = std::io::stdout();
    let mut out = std::io::BufWriter::new(out.lock());
    macro_rules! printf { ($($arg:tt)*) => (write!(out, $($arg)*).unwrap()); }
    macro_rules! printfln { () => (writeln!(out).unwrap()); ($($arg:tt)*) => (writeln!(out, $($arg)*).unwrap()); }
    let (n, m) = readl!(usize, usize);
    let (s, t) = readl!(usize, usize);
    let s = s-1;
    let t = t-1;
    let mut g = AdjList::new(n);
    for _ in 0..m {
        let (u, v, d) = readl!(usize, usize, i64);
        g.add_uedge(u-1, v-1, d);
    }
    let g = g;
    let (ds, _) = g.get_shortest_path(s);
    let gs = g.get_shortest_path_graph(s, &ds);
    

    let (dt, _) = g.get_shortest_path(t);
    let gt = g.get_shortest_path_graph(t, &dt);
    
    let mut dps = vec![mint(0); n];
    dps[s] = mint(1);
    for p in gs.get_order(s) {
        for e in gs.get(p) {
            dps[e.to] += dps[p];
        }
    }
    // gs.vizualize("2.png");
    let dps = dps;
    // debug!(dps);

    let mut dpt = vec![mint(0); n];
    dpt[t] = mint(1);
    for p in gt.get_order(t) {
        // debug!(p);
        for e in gt.get(p) {
            dpt[e.to] += dpt[p];
        }
    }
    // gt.vizualize("3.png");
    let dpt = dpt;
    // debug!(dpt);

    let mut ans = dps[t]*dps[t];
    let d = ds[t];
    for from in 0..n {
        for e in gs.get(from) {
            if 2*ds[from] < d && d < 2*(d-dt[e.to]) {
                let t = dps[from]*dpt[e.to];
                ans -= t*t;
            }
        }
    }
    for v in 0..n {
        if ds[v] == dt[v] {
            let t = dps[v]*dpt[v];
            ans -= t*t;
        }
    }
    printfln!("{}", ans);
}