Problem:
Score : 400 points

Problem Statement

You are given an undirected connected weighted graph with N vertices and M edges that contains neither self-loops nor double edges.

The i-th (1≤i≤M) edge connects vertex a_i and vertex b_i with a distance of c_i.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

A connected graph is a graph where there is a path between every pair of different vertices.

Find the number of the edges that are not contained in any shortest path between any pair of different vertices.

Constraints

2≤N≤100

N-1≤M≤min(N(N-1)/2,1000)

1≤a_i,b_i≤N

1≤c_i≤1000

c_i is an integer.

The given graph contains neither self-loops nor double edges.

The given graph is connected.

Input

The input is given from Standard Input in the following format:

N M
a_1 b_1 c_1
a_2 b_2 c_2
:
a_M b_M c_M

Output

Print the number of the edges in the graph that are not contained in any shortest path between any pair of different vertices.

Sample Input 1

3 3
1 2 1
1 3 1
2 3 3

Sample Output 1

1

In the given graph, the shortest paths between all pairs of different vertices are as follows:

The shortest path from vertex 1 to vertex 2 is: vertex 1 → vertex 2, with the length of 1.

The shortest path from vertex 1 to vertex 3 is: vertex 1 → vertex 3, with the length of 1.

The shortest path from vertex 2 to vertex 1 is: vertex 2 → vertex 1, with the length of 1.

The shortest path from vertex 2 to vertex 3 is: vertex 2 → vertex 1 → vertex 3, with the length of 2.

The shortest path from vertex 3 to vertex 1 is: vertex 3 → vertex 1, with the length of 1.

The shortest path from vertex 3 to vertex 2 is: vertex 3 → vertex 1 → vertex 2, with the length of 2.

Thus, the only edge that is not contained in any shortest path, is the edge of length 3 connecting vertex 2 and vertex 3, hence the output should be 1.

Sample Input 2

3 2
1 2 1
2 3 1

Sample Output 2

0

Every edge is contained in some shortest path between some pair of different vertices.

Solution in OCaml (4.02.3) (judge verdict: Accepted)

module IntSet = Set.Make (struct
  type t = int
  let compare = compare
end)
 
let warshall_floyd ( + ) min n d =
  for i = 0 to n - 1 do
    for j = 0 to n - 1 do
      for k = 0 to n - 1 do
        d.(j).(k) <- min d.(j).(k) (d.(j).(i) + d.(i).(k))
      done
    done
  done
 
let () =
  let n, m = Scanf.scanf "%d %d\n" (fun n m -> n, m) in
  let es = Array.init m (fun _ ->
    Scanf.scanf "%d %d %d\n" (fun a b c -> (a, b, c))) in
  let dp = Array.make_matrix n n infinity in
  for i = 0 to n - 1 do
    dp.(i).(i) <- 0.
  done;
  Array.iter (fun (a, b, c) ->
    dp.(a - 1).(b - 1) <- float_of_int c;
    dp.(b - 1).(a - 1) <- float_of_int c) es;
  warshall_floyd ( +. ) min n dp;
  Array.to_list es
  |> List.filter (fun (a, b, c) ->
      dp.(a - 1).(b - 1) < float_of_int c)
  |> List.length
  |> Printf.printf "%d\n"