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D - Candidates of No Shortest Paths

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.

3 3
1 2 1
1 3 1
2 3 3

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 $1$.

3 2
1 2 1
2 3 1

0

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