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



Sample input:
4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1


Sample output:
2

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = deque([S])
    used = set()
    while(que):
        now = que.pop()
        used.add(now)
        for v,c in V[now]:
            if v in used:
                continue
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                que.append(v)
                """
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    """
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = deque([S])
    while(que):
        now = que.pop()
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                que.append(v)
                """
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    """
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = deque([S])
    while(que):
        now = que.pop()
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                que.append(v)
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    """
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    """
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = [(0,S)]
    while(que):
        d, now = heapq.heappop(que)
        if distS[now] < d: continue
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                heappush(que,(distS[v],v))
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    """
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    """
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = [(0,S)]
    """
        while(que):
        d, now = heapq.heappop(que)
        if distS[now] < d: continue
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                heappush(que,(distS[v],v))
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    """
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = [(0,S)]
    while(que):
        d, now = heapq.heappop(que)
        if distS[now] < d: continue
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                heappush(que,(distS[v],v))
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    while(que):
        d, now = heapq.heappop(que)
        if distT[now] < d: continue
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    ans %= mod
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    print(ans)
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
def examC():
    ans = 0
    print(ans)
    return

def examD():
    ans = 0
    print(ans)
    return

def examE():
    def dijkstra(edges, start=0):
        dist = [inf] * len(edges)
        dist[start] = 0
        Q = [(0, start)]  # (dist,vertex)
        while (Q):
            d, v = heapq.heappop(Q)
            if dist[v] < d: continue  # 候補として挙がったd,vだが、他に短いのがある
            for u, cost in edges[v]:
                if dist[u] > dist[v] + cost:
                    dist[u] = dist[v] + cost
                    heapq.heappush(Q, (dist[u], u))
        return dist
    N, M = LI()
    S, T = LI()
    S -= 1; T -= 1
    V = [set()for _ in range(N)]
    for _ in range(M):
        u, v, c = LI()
        u -= 1; v -= 1
        V[u].add((v,c))
        V[v].add((u,c))
    distS = dijkstra(V,S)
    distT = dijkstra(V,T)
    L = distS[T]

    dp1 = [0]*N
    dp2 = [0]*N
    dp1[S] = 1; dp2[T] = 1
    S_collision_points = set()
    S_collision_edges = set()
    que = [(0,S)]
    while(que):
        now = heapq.heappop(que)[1]
        for v,c in V[now]:
            if distT[v]+distS[now]==L-c:
                #print(distS[v],distS[now],now)
                dp1[v] += dp1[now]
                dp1[v] %= mod
                heappush(que,(distS[v],v))
                if distS[v]==L/2:
                    S_collision_points.add(v)
                elif distS[now]<L/2 and L/2<distS[v]:
                    S_collision_edges.add((now,v))
    T_collision_points = set()
    T_collision_edges = set()
    que = [(0,T)]
    while(que):
        now = heapq.heappop(que)[1]
        for v,c in V[now]:
            if distS[v]+distT[now]==L-c:
                dp2[v] += dp2[now]
                dp2[v] %= mod
                heappush(que,(distT[v],v))
                if distT[v]==L/2:
                    T_collision_points.add(v)
                elif distT[now]<L/2 and L/2<distT[v]:
                    T_collision_edges.add((v,now))
    collision_points = S_collision_points & T_collision_points
    collision_edges = S_collision_edges & T_collision_edges
    #print(distS)
    #print(dp1)
    #print(dp2)
    #print(S_collision_edges)
    ans = 0
    ans += dp1[T]**2
    for p in collision_points:
        if distS[p]==distT[p]:
            ans -= dp1[p]*dp2[p]
            ans += mod
            ans %= mod
            #print(ans,p)
    for u,v in collision_edges:
        ans -= dp1[u]*dp2[v]
        ans += mod
        ans %= mod
        #print(ans,u,v)
    ans %= mod
    print(ans)
    return

def examF():
    ans = 0
    print(ans)
    return


from decimal import getcontext,Decimal as dec
import sys,bisect,itertools,heapq,math,random
from copy import deepcopy
from heapq import heappop,heappush,heapify
from collections import Counter,defaultdict,deque
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
def I(): return int(input())
def LI(): return list(map(int,sys.stdin.readline().split()))
def DI(): return dec(input())
def LDI(): return list(map(dec,sys.stdin.readline().split()))
def LSI(): return list(map(str,sys.stdin.readline().split()))
def LS(): return sys.stdin.readline().split()
def SI(): return sys.stdin.readline().strip()
global mod,mod2,inf,alphabet,_ep
mod = 10**9 + 7
mod2 = 998244353
inf = 10**18
_ep = dec("0.000000000001")
alphabet = [chr(ord('a') + i) for i in range(26)]
alphabet_convert = {chr(ord('a') + i): i for i in range(26)}

getcontext().prec = 28

sys.setrecursionlimit(10**7)

if __name__ == '__main__':
    examE()

"""
142
12 9 1445 0 1
asd dfg hj o o
aidn
"""

Submission status: Wrong Answer

Solution code:
# 参考: https://qiita.com/y-tsutsu/items/aa7e8e809d6ac167d6a1

from heapq import heappush, heappop

# 多次元配列
# dp = [[0] * 3 for _ in range(5)


INF = 1 << 28
MOD = int(1e9) + 7


def dijkstra(graph, s: int, t: int):
    n = len(graph)
    dist = [INF] * n
    cnts = [0] * n

    pq = []
    heappush(pq, (0, s))
    dist[s] = 0
    cnts[s] = 1

    while pq:
        (cost, cur) = heappop(pq)

        if dist[cur] != cost:
            continue

        if cur == t:
            return dist, cnts

        for next_idx, next_cost in graph[cur]:
            if dist[next_idx] > cost + next_cost:
                heappush(pq, (cost + next_cost, next_idx))
                dist[next_idx] = cost + next_cost
                cnts[next_idx] = cnts[cur]
            elif dist[next_idx] == cost + next_cost:
                cnts[next_idx] = (cnts[next_idx] + cnts[cur]) % MOD

    return dist, cnts


# 入力を受け取る
n, m = map(int, input().split())

s, t = map(lambda i: int(i) - 1, input().split())

g = [[] for _ in range(n)]

es = [tuple(map(int, input().split())) for _ in range(m)]

for e in es:
    u, v, d = e
    g[u - 1].append((v - 1, d))
    g[v - 1].append((u - 1, d))

dst1, cnt1 = dijkstra(g, s, t)
dst2, cnt2 = dijkstra(g, t, s)

ret = cnt1[t] * cnt1[t] % MOD

for i in range(n):
    if (2 * dst1[i]) == dst1[t] == (2 * dst2[i]):
        sub = cnt1[i] * cnt2[i] % MOD
        sub = sub ** 2 % MOD
        ret = (ret + MOD - sub) % MOD

sum_sub = 0

for e in es:
    u, v, d = e
    u -= 1
    v -= 1

    if cnt1[u] > cnt1[v]:
        u, v = v, u

    if (2 * dst1[u]) < dst1[t] < (2 * dst1[v]) and dst1[u] + d + dst2[v] == dst1[t]:
        sub = cnt1[u] * cnt2[v] % MOD
        sub = sub ** 2 % MOD
        ret = (ret + MOD - sub) % MOD

print(ret)

Submission status: Wrong Answer

Solution code:
def main():
    n, m = map(int, input().split())
    s, t = map(lambda x: int(x)-1, input().split())
    uvd = [list(map(int, input().split())) for _ in [0]*m]
    g = [[] for _ in [0]*n]
    [g[a-1].append((b-1, c)) for a, b, c in uvd]
    [g[b-1].append((a-1, c)) for a, b, c in uvd]

    import heapq
    inf = 10**16
    mod = 10**9+7

    dist_s = [inf]*n
    dp_s = ([0]*n)+[1]
    h = [(0, s, -1)]
    heapq.heapify(h)
    while h:
        d, i, a = heapq.heappop(h)
        if dist_s[i] == inf:
            dist_s[i] = d
            dp_s[i] = dp_s[a]
            for j, k in g[i]:
                if dist_s[j] == inf:
                    heapq.heappush(h, (d+k, j, i))
        elif dist_s[i] == d:
            dp_s[i] = (dp_s[i]+dp_s[a]) % mod

    dist_t = [inf]*n
    dp_t = ([0]*n)+[1]
    h = [(0, t, -1)]
    heapq.heapify(h)
    while h:
        d, i, a = heapq.heappop(h)
        if dist_t[i] == inf:
            dist_t[i] = d
            dp_t[i] = dp_t[a]
            for j, k in g[i]:
                if dist_t[j] == inf:
                    heapq.heappush(h, (d+k, j, i))
        elif dist_t[i] == d:
            dp_t[i] = (dp_t[i]+dp_t[a]) % mod

    ans = dp_s[t]**2 % mod
    short = dist_s[t]

    kouho = []
    point = []
    path = []
    for k, (i, j) in enumerate(zip(dist_s, dist_t)):
        if i+j == short:
            kouho.append(k)

    for i in kouho:
        d = dist_s[i]*2
        if d == short:
            point.append(i)
        elif d < short:
            for j, k in g[i]:
                if d+2*k > short:
                    if d+2*k+2*dist_t[j] == 2*short:
                        path.append((i, j))

    for i in point:
        ans = (ans-dp_s[i]*dp_t[i]) % mod
    for i, j in path:
        ans = (ans-dp_s[i]*dp_t[j]) % mod

    print(ans)


main()