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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using dbl = double;
using pii = pair<int, int>;
using pl4 = pair<ll, ll>;
using vi = vector<int>;
using vvi = vector<vi>;
using vs = vector<string>;
using vvs = vector<vs>;
using vll = vector<ll>;
using vvll = vector<vll>;
using vvvll = vector<vvll>;
using vpii = vector<pii>;
using vvpii = vector<vpii>;
using vpl4 = vector<pl4>;
using vvpl4 = vector<vpl4>;
using vd = vector<dbl>;
using vvd = vector<vd>;

#define fi first
#define se second
#define mp make_pair
#define pb push_back
#define pob pop_back()
#define pf push_front
#define pof pop_front()
#define sz size()
#define bgn begin()
#define en end()
#define asn assign
#define emp empty()
#define fr front()
#define bk back()
#define clr clear()
#define ins insert
#define ers erase
#define res resize
#define tp top()

#define p_q priority_queue

#define FOR(i,a,b) for(int i=(a);i<=(b);i++)
#define rFOR(i,a,b) for(int i=(b);i>=(a);i--)
#define REP(i,a) FOR((i),0,(a)-1)
#define REP0(i,a) FOR((i),0,(a))
#define REP1(i,a) FOR((i),1,(a))
#define rREP(i,a) rFOR((i),0,(a)-1)
#define rREP0(i,a) rFOR((i),0,(a))
#define rREP1(i,a) rFOR((i),1,(a))
#define ROR(v,i) for(auto &(i):(v))
#define IOTA(a,n) iota((a).bgn,(a).en,(n))
#define SORT(a) sort((a).bgn,(a).en)
#define rSORT(a) sort((a).rbegin(),(a).rend())
#define UNIQUE(a) (a).erase(unique((a).bgn,(a).en),(a).en)
#define PREVP(a) prev_permutation((a).bgn,(a).en)
#define NEXTP(a) next_permutation((a).bgn,(a).en)
#define BINS(a,b) binary_search((a).bgn,(a).en,(b))
#define LOWB(a,b) (lower_bound((a).bgn,(a).en,(b))-(a).bgn)
#define UPB(a,b) (upper_bound((a).bgn,(a).en,(b))-(a).bgn)
#define CNT(a,b) count((a).bgn,(a).en,b)
#define SUM(a) accumulate((a).bgn,(a).en,0)
#define REV(a) reverse((a).bgn,(a).en)
#define REGS(a,b) regex_search((a),regex(b))
#define REGM(a,b) regex_match((a),regex(b))
#define yn(a) cout <<((a)?"yes":"no")<<endl;
#define Yn(a) cout <<((a)?"Yes":"No")<<endl;
#define YN(a) cout <<((a)?"YES":"NO")<<endl;
#define Imp(a) cout <<((a)?"Possible":"Impossible")<<endl;
#define IMP(a) cout <<((a)?"POSSIBLE":"IMPOSSIBLE")<<endl;
#define say(a) cout <<(a);
#define sal(a) cout <<(a)<<endl;
#define sak cout <<endl;
#define sas cout <<" ";
#define sat cout <<"\t";
#define dbg(a) cout <<(#a)<<": "<<(a)<<endl;
#define c2l(a) ((ll)(a-48))
#define a2l(a) ((ll)(a-97))
#define A2l(a) ((ll)(a-65))
#define l2c(a) ((char)(a+48))
#define l2a(a) ((char)(a+97))
#define l2A(a) ((char)(a+65))
#define DigN2(a) ((llabs(a)==0)?(1):((ll)(log2(double(llabs(a))))+1))
#define DigN10(a) ((llabs(a)==0)?(1):((ll)(log10(double(llabs(a))))+1))
#define Dig2(a,b) (((a)>>(b))&1)
#define Dig10(a,b) (ll)(((a)/((ll)(pow(10.0,(double)(b)))))%10)
#define Pow2(a) (1<<(a))
#define Pow10(a) ((ll)(pow(10.0,double(a))))
#define LSB(a) ((a)&(-(a)))
#define llin(a) ll (a);cin >>(a);
#define stin(a) string (a);cin >>(a);
#define vin(v) ROR((v),(i)){cin >>(i);};
#define vllin(v,N) vll (v)((N));vin(v);
#define vsin(v,N) vs (v)((N));vin(v);
#define rdn(a,b) ((a)/(b))
#define rou(a,b) ((((double(a)/double(b))-((a)/(b)))<0.5)?((a)/(b)):(((a)/(b))+1))
#define rup(a,b) ((((a)%(b))==0)?((a)/(b)):(((a)/(b))+1))
#define min(a,b) ((a<b)?(a):(b))
#define max(a,b) ((a>b)?(a):(b))
#define powll(a,b) (ll)(pow((double)(a),(double)(b)))
#define Triangle(x1,y1,x2,y2,x3,y3) (((x1)-(x2))*((y1)-(y3))-((x1)-(x3))*((y1)-(y2)))

#define int ll

const ll MOD = 1e9 + 7;
//const ll MOD = 998244353;
//const ll MOD = 9007199254740881;
const ll INF = 1LL << 60;
const double PI = acos(-1.0);
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";

ll Bset(ll a, ll b, ll c) {
	if (c) a |= b;
	else a &= ~b;
	return a;
}


class UFT {

public:
	ll size;
	ll mode;
	vll root;
	vll rank;
	UFT(ll sizeget, ll modeget = 0){
		size = sizeget;
		mode = modeget;
		root.res(size, 0);
		if (!mode) rank.res(size, 0);
		IOTA(root, 0);
	}
	ll find(ll x) {
		return root[x] == x ? x : root[x] = find(root[x]);
	}
	bool same(ll x, ll y) {
		return find(x) == find(y);
	}
	void unite(ll x, ll y) {
		x = find(x);
		y = find(y);
		if (x == y) return;
		if (mode) root[y] = x;
		else {
			if (rank[x] < rank[y]) {
				root[x] = y;
			}
			else {
				root[y] = x;
				if (rank[x] == rank[y]) rank[x]++;
			}
		}
	}
};

template <typename valtype> class SegT {

public:

	ll size;
	vector<valtype> v;

	valtype initv;
	function<valtype(valtype x, valtype y)> calc;

	SegT(ll sizeget, ll modeget = 0) {
		sizeset(sizeget);
		modeset(modeget);
		init();
	}
	SegT(vector<valtype> cpyvec, ll modeget = 0) {
		sizeset(cpyvec.sz);
		modeset(modeget);
		init();
		copy(cpyvec);
	}
	SegT(ll sizeget, valtype initvget, function<valtype(valtype x, valtype y)> calcget) {
		sizeset(sizeget);
		initv = initvget;
		calc = calcget;
		init();
	}
	SegT(vector<valtype> cpyvec, valtype initvget, function<valtype(valtype x, valtype y)> calcget) {
		sizeset(cpyvec.sz);
		initv = initvget;
		calc = calcget;
		init();
		copy(cpyvec);
	}
	void sizeset(ll rsize) {
		size = DigN2(rsize);
		if (rsize == Pow2(size - 1)) size--;
		return;
	}
	void modeset(ll mode) {
		switch (mode) {
		case 0:
			initv = 0;
			calc = [](valtype x, valtype y) {return x + y; };
			break;
		case 1:
			initv = INF;
			calc = [](valtype x, valtype y) {return min(x, y); };
			break;
		case 2:
			initv = -INF;
			calc = [](valtype x, valtype y) {return max(x, y); };
			break;
		}
		return;
	}
	void init() {
		v.asn(Pow2(size + 1) - 1, initv);
	}
	void copy(vector<valtype> cpyvec) {
		REP(i, min(cpyvec.sz, Pow2(size))) set(i, cpyvec[i]);
	}
	ll i2v(ll i) const{
		if (i < 0 || i >= Pow2(size)) return -1;
		return Pow2(size) + i - 1;
	}
	ll top(ll i) const{
		if (i == 0) return -1;
		return (i - 1) / 2;
	}
	pl4 bot(ll i) const{
		if (i + 1 >= Pow2(size)) return mp(-1, -1);
		return mp(2 * i + 1, 2 * i + 2);
	}
	void set(ll i, valtype x) {
		i = i2v(i);
		v[i] = x;
		while (i > 0) {
			i = top(i);
			v[i] = calc(v[bot(i).fi], v[bot(i).se]);
		}
		return;
	}
	void add(ll i, valtype x) {
		set(i, v[i2v(i)] + x);
		return;
	}
	valtype operator[](const ll &i) const {
		return v[i2v(i)];
	}
//	valtype que(ll a = 0, ll b = Pow2(size) - 1) {
	valtype que(ll a, ll b) {
		if (a == b) return v[i2v(a)];
		if (a > b) swap(a, b);
		valtype ans = initv;
		ll ai = i2v(a);
		ll bi = i2v(b);
		FOR(i, 1, size + 1) {
			if (a > b) break;
			if (a%Pow2(i)) {
				ans = calc(ans, v[ai]);
				a += Pow2(i - 1);
				ai = top(ai) + 1;
			}
			else {
				ai = top(ai);
			}
			if (a > b) break;
			if ((b + 1) % Pow2(i)) {
				ans = calc(ans, v[bi]);
				b -= Pow2(i - 1);
				bi = top(bi) - 1;
			}
			else {
				bi = top(bi);
			}
			if (a > b) break;
		}
		return ans;
	}


};

template <typename lentype> class Graph {

public:
	
	ll size;
	ll mode;
	ll mapmode;
	lentype zlen;
	lentype ilen;
	vector<map<ll, ll>> v2n;
	vvll n2v;
	vector<vector<lentype>> Edge;
	vector<pair<lentype, ll>> Primresult;

	Graph(ll sizeget = 0, ll modeget = 0, ll mapmodeget = 0, lentype zlenget = 0, lentype ilenget = INF) {
		size = sizeget;
		mode = modeget;
		mapmode = mapmodeget;
		zlen = zlenget;
		ilen = ilenget;
		init();
	}
	void init() {
		Edge.res(size);
		v2n.res(size);
		n2v.res(size);
		Primresult.asn(size, mp(ilen, -1));
	}
	lentype lenplus(lentype l, lentype r) {
		return l + r;
	}
	lentype lenequal(lentype l, lentype r) {
		return l == r;
	}
	lentype lenlcr(lentype l, lentype r) {
		return l < r;
	}
	ll addV(ll vs) {
		size += vs;
		init();
		return size;
	}
	void caddE(ll x, ll y, lentype c) {
		if (!mapmode) v2n[x][y] = Edge[x].sz;
		Edge[x].pb(c);
		n2v[x].pb(y);
	}
	void csetE(ll x, ll y, lentype c) {
		if (!mapmode) Edge[x][v2n[x][y]] = c;
	}
	void cersE(ll x, ll y) {
		if (!mapmode) {
			ll n = v2n[x][y];
			Edge[x][n] = ilen;
			n2v[x][n] = -1;
			v2n[x].ers(y);
		}
	}
	void addE(ll x, ll y, lentype c) {
		caddE(x, y, c);
		if (!mode) caddE(y, x, c);
	}
	void setE(ll x, ll y, lentype c) {
		csetE(x, y, c);
		if (!mode) csetE(y, x, c);
	}
	void ersE(ll x, ll y, lentype c) {
		cersE(x, y, c);
		if (!mode) cersE(y, x, c);
	}
	lentype getE(ll x, ll y) {
		if (!mapmode) {
			if (v2n[x].count(y)) {
				return Edge[x][v2n[x][y]];
			}
		}
		return ilen;
	}
	ll getVsz(ll x) {
		return Edge[x].sz;
	}
	pair<lentype, ll> getV(ll x, ll n) {
		if (n >= getVsz(x)) return mp(ilen, -1);
		return mp(Edge[x][n], n2v[x][n]);
	}

	vector<pair<lentype, vll>> Dijk(ll x) {
		vector<pair<lentype, vll>> result(size);
		REP(i, size) {
			result[i].fi = ilen;
			result[i].se = { -1 };
		}
		vll stat(size, 0);
		pair<lentype, ll> now;
		pair<lentype, ll> nowlv;
		SegT<pair<lentype, ll>> Q(size, mp(ilen, -1), [](pair<lentype, ll> l, pair<lentype, ll> r) {
			if (l.se == -1) return r;
			if (r.se == -1) return l;
			return l.fi < r.fi ? l : r;
		});
		Q.set(x, mp(zlen, x));
		result[x].fi = zlen;
		result[x].se = { -1 };
		while (Q.que(0, size - 1).se != -1) {
			now = Q.que(0, size - 1);
			Q.set(now.se, mp(ilen, -1));
			stat[now.se] = 1;
			REP(i, getVsz(now.se)) {
				nowlv = getV(now.se, i);
				if (stat[nowlv.se]) continue;
				if (Q[nowlv.se].se == -1) {
					result[nowlv.se].fi = lenplus(result[now.se].fi, nowlv.fi);
					result[nowlv.se].se = { now.se };
					Q.set(nowlv.se, mp(result[nowlv.se].fi, nowlv.se));
				}
				else {
					if (lenlcr(lenplus(result[now.se].fi, nowlv.fi), result[nowlv.se].fi)) {
						result[nowlv.se].fi = lenplus(result[now.se].fi, nowlv.fi);
						result[nowlv.se].se = { now.se };
						Q.set(nowlv.se, mp(result[nowlv.se].fi, nowlv.se));
					}
					else if (lenequal(lenplus(result[now.se].fi, nowlv.fi), result[nowlv.se].fi)) {
						result[nowlv.se].se.pb(now.se);
					}
				}
			}
		}
		return result;
	}

	lentype Prim(ll x = 0) {
		lentype ans = zlen;
		vll stat(size, 0);
		pair<lentype, ll> now;
		pair<lentype, ll> nowlv;
		SegT<pair<lentype, ll>> Q(size, mp(ilen, -1), [](pair<lentype, ll> l, pair<lentype, ll> r) {
			if (l.se == -1) return r;
			if (r.se == -1) return l;
			return l.fi < r.fi ? l : r;
		});
		Q.set(x, mp(zlen, x));
		Primresult[x] = mp(zlen, -1);
		while (Q.que(0, size - 1).se != -1) {
			now = Q.que(0, size - 1);
			Q.set(now.se, mp(ilen, -1));
			stat[now.se] = 1;
			ans = lenplus(ans, Primresult[now.se].fi);
			REP(i, getVsz(now.se)) {
				nowlv = getV(now.se, i);
				if (stat[nowlv.se]) continue;
				if (Q[nowlv.se].se == -1) {
					Primresult[nowlv.se] = mp(nowlv.fi, now.se);
					Q.set(nowlv.se, mp(Primresult[nowlv.se].fi, nowlv.se));
				}
				else {
					if (lenlcr(nowlv.fi, Primresult[nowlv.se].fi)) {
						Primresult[nowlv.se] = mp(nowlv.fi, now.se);
						Q.set(nowlv.se, mp(Primresult[nowlv.se].fi, nowlv.se));
					}
				}
			}
		}
		return ans;
	}

};


ll gcd(ll a, ll b) { if (b == 0)return a; return gcd(b, a%b); }
ll lcm(ll a, ll b) { return a / gcd(a, b)*b; }

pl4 Bezout(ll a, ll b) {
	if (b != 0) {
		pl4 xy;
		xy = Bezout(b, a%b);
		return mp(xy.se, xy.fi - ((a / b)*xy.se));
	}
	if (b == 0) {
		return mp(1, 0);
	}
}
pl4 Bez(ll a, ll b, ll c) {
	pl4 xy;
	ll x, y, z, gc;
	xy = Bezout(a, b);
	gc = gcd(a, b);
	if (c%gc != 0) return mp(-1, -1);
	x = xy.fi*(c / gc); y = xy.se*(c / gc);
	if (x < 0) z = rup(-x, (b / gc));
	if (x >= 0) z = -x / (b / gc);
	x += z * (b / gc);
	y -= z * (a / gc);
	return mp(x, y);
}

void salv(vll v) {
	say("{");
	FOR(i, 0, v.sz - 1) {
		say(v[i]);
		if (i != v.sz - 1) say(",");
	}
	sal("}")
}

ll DigS10(ll n) {
	ll m = 0;
	FOR(i, 0, DigN10(n) - 1) {
		m += (ll)((llabs(n) % (ll)(pow(10.0, (double)(i + 1)))) / (ll)(pow(10.0, (double)i)));
	}
	return m;
}

ll isP(ll n) {
	if (n <= 1) return 0;
	FOR(i, 2, (ll)sqrt(n)) {
		if (n%i == 0) return 0;
	}
	return 1;
}

vll FactM(1, 1);
vll FactMI(1, 1);

ll PowM(ll a, ll b) {
	ll ans = 1, x = (a%MOD);
	FOR(i, 0, DigN2(b) - 1) {
		if (Dig2(b, i) == 1) ans = (ans*x) % MOD;
		if (i != (DigN2(b) - 1)) x = (x*x) % MOD;
	}
	return ans;
}

ll Mod(ll a) {
	a %= MOD;
	a = (a + MOD) % MOD;
	return a;
}

ll InvM(ll a) {
	a = Mod(a);
	a = PowM(a, MOD - 2);
	return a;
}

void CFactM(ll n) {
	if (FactM.sz <= n) {
		FOR(i, FactM.sz, n) {
			FactM.pb((FactM[i - 1] * (i%MOD)) % MOD);
		}
	}
	return;
}

void CFactMI(ll n) {
	CFactM(n);
	if (FactMI.sz < (n + 1)) FactMI.res(n + 1, -1);
	if (FactMI[n] == -1) FactMI[n] = PowM(FactM[n], MOD - 2);
	rFOR(i, 1, n - 1) {
		if (FactMI[i] != -1) break;
		FactMI[i] = ((FactMI[i + 1] * ((i + 1) % MOD)) % MOD);
	}
	return;
}

ll CombM(ll n, ll k) {
	if ((n < 0) || (k < 0)) return 0;
	if (n < k) return 0;
	if (n + 1 > FactMI.sz) CFactMI(n);
	return ((((FactMI[k] * FactMI[n - k]) % MOD)*FactM[n]) % MOD);
}

ll LIS(vll v, ll m = 0) {

	if (v.sz > 0) {
		ll ans = 0;
		vll dp(v.sz, INF);
		FOR(i, 0, v.sz - 1) {
			dp[m ? UPB(dp, v[i]) : LOWB(dp, v[i])] = v[i];
		}
		FOR(i, 0, v.sz - 1) {
			if (dp[i] == INF) break;
			ans++;
		}
		return ans;
	}
	else {
		return 0;
	}

}

void cmprs(vll& v) {
	if (v.sz == 0) return;
	vll vv(v);
	IOTA(vv, 0);
	sort(vv.bgn, vv.en, [&](ll v1, ll v2) {return v[v1] < v[v2]; });
	IOTA(v, 0);
	sort(v.bgn, v.en, [&](ll v1, ll v2) {return vv[v1] < vv[v2]; });
	return;
}

ll BblCnt(vll v) {
	cmprs(v);
	SegT<ll> b(v.sz, 0);
	ll ans = 0;
	REP(i, v.sz) {
		ans += (i - b.que(0, v[i]));
		b.add(v[i], 1);
	}
	return ans;
}

ll Bsrch(function<bool(ll x)> f, ll mi, ll ma) {
	ll ng = mi - 1, ok = ma, mid;
	while (ok - ng > 1) {
		mid = (ng + ok) / 2;
		(f(mid) ? ok : ng) = mid;
	}
	return ok;
}

/*
std::cout << std::fixed;
std::cout << std::setprecision(2) << 3.141; // "3.14"
std::cout << std::setprecision(2) << 3.0;   // "3.00"
*/

ll N, M, S, T;
ll u, v, d;
Graph<ll> g;
vector<pair<ll, vll>> Sd, Td;
vpl4 Sre;
vpl4 Tre;

ll Ssr(ll x) {
	if (!Sre[x].se) {
		Sre[x].se = 1;
		ROR(Sd[x].se, i) {
			Sre[x].fi += Ssr(i);
		}
	}
	return Sre[x].fi;
}
ll Tsr(ll x) {
	if (!Tre[x].se) {
		Tre[x].se = 1;
		ROR(Td[x].se, i) {
			Tre[x].fi += Tsr(i);
		}
	}
	return Tre[x].fi;
}

signed main() {

	cin >> N >> M >> S >> T;
	S--;
	T--;
	g.addV(N);
	REP(i, M) {
		cin >> u >> v >> d;
		g.addE(u - 1, v - 1, d);
	}
	Sd = g.Dijk(S);
	Td = g.Dijk(T);
	ll ti = Sd[T].fi;
	Sre.asn(N, mp(0, 0));
	Tre.asn(N, mp(0, 0));
	Sre[S] = mp(1, 1);
	Tre[T] = mp(1, 1);
	Ssr(T);
	Tsr(S);
	ll ans = Sre[T].fi*Tre[S].fi;
	REP(i, N) {
		if (Sd[i].fi * 2 == ti) {
			if (Sre[i].se&&Tre[i].se) ans -= Sre[i].fi * Tre[i].fi;
		}
		else {
			if (Sd[i].fi * 2 < ti) {
				if (Sre[i].se) {
					ROR(Td[i].se, j) {
						if (Sd[j].fi * 2 > ti) {
							if (Tre[j].se) {
								ans -= Sre[i].fi * Tre[j].fi;
							}
						}
					}
				}
			}
		}
	}

	sal(ans);

}