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 OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

open Scanf
open Printf

let id a = a
let pair a b = a, b
let triplet a b c = a, b, c

let rec scan_list = function
  | 1 -> scanf "%d" (fun k -> [k])
  | n -> scanf "%d " (fun k -> k :: scan_list (n-1))

let g arr u v s = arr.(u-1) <- (v-1,s) :: arr.(u-1);
  arr.(v-1) <- (u-1,s) :: arr.(v-1)

let f n m = let graph = (Array.make n []) in
  for i = 1 to m do
    scanf "%d %d %d\n" (g graph)
  done;
  for i = 0 to n-1 do
    graph.(i) <- List.sort compare graph.(i)
  done;
  graph

let h graph ans = let queue = Queue.create () in
  let rec enqueue a = function
    | (v, c) :: lst -> if ans.(v) = 0 then Queue.add (a, v, c) queue; enqueue a lst
    | [] -> ()
  in let rec f () = if Queue.is_empty queue
       then `Ok
       else let (u, v, c) = Queue.take queue
         in if ans.(v) = 0
         then begin
           ans.(v) <- (if c > ans.(u) then c - ans.(u) else raise @@ failwith "negative answer");
           enqueue v graph.(v);
           f ()
         end
         else if c = ans.(v) + ans.(u)
         then begin
           enqueue v graph.(v);
           f ()
         end
         else raise @@ failwith "incorrect"
  in enqueue 0 graph.(0); f()

let cost lst = List.map (function (_, c) -> c) lst
let minimum lst = List.fold_left min 10000000000000 lst

let compute graph = let n = ref 0 in
  for i = 1 to minimum (cost graph.(0)) - 1 do
    let ans = Array.make (Array.length graph) 0
    in ans.(0) <- i;
    match h graph ans with
    | `Ok -> n := !n + 1
    | exception Failure s -> eprintf "%s\n" s
  done;
  printf "%d\n" !n

let () = compute @@ scanf "%d %d\n" f