LaLa and Spirit Summoning

$\color{blue}{\text{LaLa}}$'s younger sister $\color{purple}{\text{LiLi}}$ is helping $\color{blue}{\text{LaLa}}$ cast the spirit summoning $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$.

While $\color{blue}{\text{LaLa}}$ was asleep, $\color{purple}{\text{LiLi}}$ had already built a prototype of the spirit to summon. The spirit consists of $N$ $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints which allow any $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars attached to them to freely move around them, and $M$ $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars of various colors, each of which connects two $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints and whose length can be adjusted to any non-negative real number before the summoning (but not after).

When it comes to spirit summoning, $\color{blue}{\text{LaLa}}$ has a far higher standard than $\color{purple}{\text{LiLi}}$. Of course, $\color{blue}{\text{LaLa}}$ was not satisfied with $\color{purple}{\text{LiLi}}$'s work whatsoever. $\color{blue}{\text{LaLa}}$ would like to fabulize the prototype by getting rid of some $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars so that

1. the spirit is beautiful, which means there should not be two $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars of the same color present, and
2. the spirit is as easy to control as possible, which means the degree of freedom of the spirit must be minimum over all beautiful spirits obtainable by eliminating some $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars. Note that the minimum always exists as she can always eliminate all $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars to create a beautiful spirit. See the note below for the exact definition of the degree of freedom.

Write a program that computes the degree of freedom of the spirit fabulized by $\color{blue}{\text{LaLa}}$.

The input describes the prototype spirit made by $\color{purple}{\text{LiLi}}$ and is given in the following format:

$N$ $M$

$u\_0$ $v\_0$ $c\_0$

$u\_1$ $v\_1$ $c\_1$

$\vdots$

$u\_{M-1}$ $v\_{M-1}$ $c\_{M-1}$

where $N$ is the number of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints, numbered from $0$ to $N-1$, $M$ is the number of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars, and for each integer $0 \le i < M$, the $i$-th $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bar has color $c\_i$ and connects the $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joint $u\_i$ and $v\_i$.

The input satisfies the following constraints:

• All the numbers in the input are integers.
• $2 \le N \le 200$
• $0 \le M \le 1\\,000$
• $0 \le u\_i < v\_i < N$ and $0 \le c\_i < M$ for all integers $0 \le i < M$

Note that there can be multiple $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars connecting the same pair of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints.

The output should be a single integer equal to the degree of freedom of the spirit fabulized by $\color{blue}{\text{LaLa}}$.

Intuitively, the degree of freedom is the number of axis of motions preserving edge lengths of the spirit embedded on a plane.

More formally, let $E$ be an assignment of planar coordinates (we'll call this an embedding) to all $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints of a spirit. Note that such an embedding can be identified with an element in $\mathbb{R}^{2N}$ by concatenating all coordinates, where $N$ is the number of $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ joints.

Let $C(E)$ be the set of embeddings continuously reachable from $E$ as an element of $\mathbb{R}^{2N}$ while preserving edge lengths. i.e. for each element $E'$ of $C(E)$ and each $\color{red}{\text{m}} \color{brown}{\text{a}} \color{orange}{\text{g}} \color{blue}{\text{i}} \color{magenta} {\text{c}}$ bars of the spirit connecting magic joints $u$ and $v$, the euclidean distance between $u$ and $v$ must be the same in $E$ and $E'$.

The degree of freedom of $E$ is the minimum non-negative integer $k$ such that there exists a continuous bijective mapping $F:D \rightarrow C(E)$ where $D$ is a connected subset of $\mathbb{R}^k$.

The degree of freedom of a spirit is the maximum degree of freedom over all such embeddings $E$.

The following illustrate the spirit fabulized by $\color{blue}{\text{LaLa}}$ along with one of the optimal embedding and the mapping $F$ for each sample tests in order.

1. $k = 5$, $D = \mathbb{R}^2 \times \[0, 2\pi) \times \mathbb{R}^2$

   $F: (x\_0, x\_1, x\_2, x\_3, x\_4) \mapsto \langle(x\_0, x\_1), (x\_0, x\_1) + (\cos{x\_2}, \sin{x\_2}), (x\_3, x\_4)\rangle$

   The following illustrates the 5 degrees of freedom associated with each variables.

   

2. $k = 3$, $D = \mathbb{R}^2 \times \[0, 2\pi)$

   $F: (x\_0, x\_1, x\_2) \mapsto \langle(x\_0, x\_1), (x\_0, x\_1) + (\cos{x\_2}, \sin{x\_2}), (x\_0, x\_1) + (\cos{(\frac{\pi}{3} + x\_2)}, \sin{(\frac{\pi}{3} + x\_2)})\rangle$

   The following illustrates the 3 degrees of freedom associated with each variables.

   

3. $k = 4$, $D = \mathbb{R}^2 \times (((0, 2] \times \[0, 2\pi)) \cup (\lbrace 0 \rbrace \times \[0, \pi)))$

   $F: (x\_0, x\_1, x\_2, x\_3) \mapsto \langle P\_0, P\_0 + \frac{x\_2}{2} P\_1 + \sqrt{1 - \frac{x\_2^2}{4}} P\_2, P\_0 + x\_2 P\_1, P\_0 + \frac{x\_2}{2} P\_1 - \sqrt{1 - \frac{x\_2^2}{4}} P\_2\rangle$

   where $P\_0 = (x\_0, x\_1), P\_1 = (\cos{x\_3}, \sin{x\_3})$ and $P\_2 = (\sin{x\_3}, -\cos{x\_3})$.

   The following figure on the left illustrates the 4 degrees of freedom associated with each variables. Note that the motion associated with the variable $x\_2$ is non-rigid. The one on the right illustrates the motion associated with $x\_2$ in detail.

   

4. $k = 6$, $D = \mathbb{R}^2 \times \[0, 2\pi)^4$

   $F: (x\_0, x\_1, x\_2, x\_3, x\_4, x\_5) \mapsto \langle P\_0, P\_1, P\_2, P\_3, P\_4 \rangle$

   where $P\_0 = (x\_0, x\_1)$ and $P\_i = P\_{i-1} + (\cos{x\_{i + 1}}, \sin{x\_{i + 1}})$ for all integers $1 \le i \le 4$.

   The following illustrates the 6 degrees of freedom associated with each variables.