Problem:
Score : 600 points

Problem Statement

Kenkoooo found a simple connected graph.
The vertices are numbered 1 through n.
The i-th edge connects Vertex u_i and v_i, and has a fixed integer s_i.

Kenkoooo is trying to write a positive integer in each vertex so that the following condition is satisfied:

For every edge i, the sum of the positive integers written in Vertex u_i and v_i is equal to s_i.

Find the number of such ways to write positive integers in the vertices.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq u_i < v_i \leq n

2 \leq s_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

The graph is connected.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m
u_1 v_1 s_1
:
u_m v_m s_m

Output

Print the number of ways to write positive integers in the vertices so that the condition is satisfied.

Sample Input 1

3 3
1 2 3
2 3 5
1 3 4

Sample Output 1

1

The condition will be satisfied if we write 1,2 and 3 in vertices 1,2 and 3, respectively.
There is no other way to satisfy the condition, so the answer is 1.

Sample Input 2

4 3
1 2 6
2 3 7
3 4 5

Sample Output 2

3

Let a,b,c and d be the numbers to write in vertices 1,2,3 and 4, respectively.
There are three quadruples (a,b,c,d) that satisfy the condition:

(a,b,c,d)=(1,5,2,3)

(a,b,c,d)=(2,4,3,2)

(a,b,c,d)=(3,3,4,1)

Sample Input 3

8 7
1 2 1000000000
2 3 2
3 4 1000000000
4 5 2
5 6 1000000000
6 7 2
7 8 1000000000

Sample Output 3

0

Solution in Python (3.4.3) (judge verdict: Time Limit Exceeded)

N,M=map(int,input().split())
G=[dict() for i in range(N)]
for i in range(M):
    u,v,s=map(int,input().split())
    G[u-1][v-1]=s
    G[v-1][u-1]=s
H=[[] for i in range(N)]
reached=[0 for i in range(N)]
q=[]
q.append(0)
H[0].append((0,1))
while(len(q)>0):
    r=q[-1]
    q.pop()
    reached[r]=1
    for p in G[r]:
        if reached[p]==0:
            q.append(p)
            for s in H[r]:
                H[p].append((G[r][p]-s[0],-s[1]))
flag=0
x=-1
for i in range(N):
    if len(H[i])>=2:
        X=set()
        Y=set()
        for r in H[i]:
            if r[1]==-1:
                X.add(r)
            else:
                Y.add(r)
        if len(X)>=2 or len(Y)>=2:
            print(0)
            exit()
        else:
            a=list(X)[0]
            b=list(Y)[0]
            z=a[0]-b[0]
            if z%2==1:
                print(0)
                exit()
            x=z//2
            if x<0:
                print(0)
                exit()
        flag=1
if flag==1 and x!=-1:
    print(1)
elif flag==1 and x==-1:
    print(0)
else:
    L=[]
    R=[]
    for i in range(N):
        if H[i][0][1]==-1:
            L.append(H[i][0][0])
        else:
            R.append(H[i][0][0])
    print(max([min(L)+min(R)-1,0]))