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E - Edge Deletion

Score : $500$ points

Problem Statement

You are given a simple connected undirected graph with $N$ vertices and $M$ edges.
Edge $i$ connects Vertex $A_i$ and Vertex $B_i$, and has a length of $C_i$.

Let us delete some number of edges while satisfying the conditions below. Find the maximum number of edges that can be deleted.

• The graph is still connected after the deletion.
• For every pair of vertices $(s, t)$, the distance between $s$ and $t$ remains the same before and after the deletion.

Notes

A simple connected undirected graph is a graph that is simple and connected and has undirected edges.
A graph is simple when it has no self-loops and no multi-edges.
A graph is connected when, for every pair of two vertices $s$ and $t$, $t$ is reachable from $s$ by traversing edges.
The distance between Vertex $s$ and Vertex $t$ is the length of a shortest path between $s$ and $t$.

Constraints

• $2 \leq N \leq 300$
• $N-1 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq A_i \lt B_i \leq N$
• $1 \leq C_i \leq 10^9$
• $(A_i, B_i) \neq (A_j, B_j)$ if $i \neq j$.
• The given graph is connected.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$ $C_1$

$A_2$ $B_2$ $C_2$

$\vdots$

$A_M$ $B_M$ $C_M$

Output

Print the answer.

3 3
1 2 2
2 3 3
1 3 6

1

The distance between each pair of vertices before the deletion is as follows.

• The distance between Vertex $1$ and Vertex $2$ is $2$.
• The distance between Vertex $1$ and Vertex $3$ is $5$.
• The distance between Vertex $2$ and Vertex $3$ is $3$.

Deleting Edge $3$ does not affect the distance between any pair of vertices. It is impossible to delete two or more edges while satisfying the condition, so the answer is $1$.

5 4
1 3 3
2 3 9
3 5 3
4 5 3

0

No edge can be deleted.

5 10
1 2 71
1 3 9
1 4 82
1 5 64
2 3 22
2 4 99
2 5 1
3 4 24
3 5 18
4 5 10

5