题目描述

Mr. Pacha, a Bolivian archeologist, discovered an ancient document near Tiwanaku that describes the world during the Tiwanaku Period (300-1000 CE). At that time, there were $N$ countries, numbered from 1 to $N$.

In the document, there is a list of $M$ different pairs of adjacent countries:

$$\begin{aligned}(A[0], B[0]), (A[1], B[1]), \ldots, (A[M - 1], B[M - 1]).\end{aligned} $$

For each $i$ ($0 \leq i < M$), the document states that country $A[i]$ was adjacent to country $B[i]$ and vice versa. Pairs of countries not listed were not adjacent.

Mr. Pacha wants to create a map of the world such that all adjacencies between countries are exactly as they were during the Tiwanaku Period. For this purpose, he first chooses a positive integer $K$. Then, he draws the map as a grid of $K \times K$ square cells, with rows numbered from 0 to $K - 1$ (top to bottom) and columns numbered from 0 to $K - 1$ (left to right).

He wants to color each cell of the map using one of $N$ colors. The colors are numbered from 1 to $N$, and country $j$ ($1 \leq j \leq N$) is represented by color $j$. The coloring must satisfy all of the following conditions:

• For each $j$ ($1 \leq j \leq N$), there is at least one cell with color $j$.
• For each pair of adjacent countries $(A[i], B[i])$, there is at least one pair of adjacent cells such that one of them is colored $A[i]$ and the other is colored $B[i]$. Two cells are adjacent if they share a side.
• For each pair of adjacent cells with different colors, the countries represented by these two colors were adjacent during the Tiwanaku Period.

For example, if $N = 3$, $M = 2$ and the pairs of adjacent countries are $(1, 2)$ and $(2, 3)$, then the pair $(1, 3)$ was not adjacent, and the following map of dimension $K = 3$ satisfies all the conditions.

:::align{center} :::

In particular, a country does not need to form a connected region on the map. In the map above, country 3 forms a connected region, while countries 1 and 2 form disconnected regions.

Your task is to help Mr. Pacha choose a value of $K$ and create a map. The document guarantees that such a map exists. Since Mr. Pacha prefers smaller maps, in the last subtask your score depends on the value of $K$, and lower values of $K$ may result in a better score. However, finding the minimum possible value of $K$ is not required.

Implementation Details

You should implement the following procedure:

std::vector<std::vector<int>> create_map(int N, int M, std::vector<int> A, std::vector<i

• $N$: the number of countries.
• $M$: the number of pairs of adjacent countries.
• $A$ and $B$: arrays of length $M$ describing adjacent countries.
• This procedure is called up to $50$ times for each test case.

The procedure should return an array $C$ that represents the map. Let $K$ be the length of $C$.

• Each element of $C$ must be an array of length $K$, containing integers between $1$ and $N$ inclusive.
• $C[i][j]$ is the color of the cell at row $i$ and column $j$ (for each $i$ and $j$ such that $0 \leq i, j < K$).
• $K$ must be less than or equal to $240$.

输入格式

The first line of the input should contain a single integer $T$, the number of scenarios. A description of $T$ scenarios should follow, each in the format specified below.

N M
A[0] B[0]
:
A[M-1] B[M-1]

输出格式

P
Q[0] Q[1] ... Q[P-1]

C[0][0] ... C[0][Q[0]-1]
:
C[P-1][0] ... C[P-1][Q[P-1]-1]

Here, $P$ is the length of the array $C$ returned by create_map, and $Q[i]$ ($0 \leq i < P$) is the length of $C[i]$. Note that line 3 in the output format is intentionally left blank.

2
3 2
1 2
2 3
4 4
1 2
1 3
2 4
3 4

3
3 3 3
2 3 3
2 3 2
1 2 1
7
7 7 7 7 7 7 7
2 1 3 3 4 3 4
2 1 3 3 3 3 3
2 1 1 1 3 4 4
2 2 2 1 3 4 3
1 1 1 2 4 4 4
2 2 1 2 2 4 3
2 2 1 2 2 4 4

提示

Example 1

Consider the following call:

create_map(3, 2, [1, 2], [2, 3])

This is the example from the task description, so the procedure can return the following map.

[
[2, 3, 3],
[2, 3, 2],
[1, 2, 1]
]

Example 2

Consider the following call:

create_map(4, 4, [1, 1, 2, 3], [2, 3, 4, 4])

In this example, $N = 4$, $M = 4$ and the country pairs $(1, 2)$, $(1, 3)$, $(2, 4)$, and $(3, 4)$ are adjacent. Consequently, the pairs $(1, 4)$ and $(2, 3)$ are not adjacent.

The procedure can return the following map of dimension $K = 7$, which satisfies all the conditions.

[
[2, 1, 3, 3, 4, 3, 4],
[2, 1, 3, 3, 3, 3, 3],
[2, 1, 1, 1, 3, 4, 4],
[2, 2, 2, 1, 3, 4, 3],
[1, 1, 1, 2, 4, 4, 4],
[2, 2, 1, 2, 2, 4, 3],
[2, 2, 1, 2, 2, 4, 4]
]

The map could be smaller; for example, the procedure can return the following map of dimension $K = 2$.

[
[3, 1],
[4, 2]
]

Note that both maps satisfy $K / N \leq 2$.

Constraints

• $1 \leq N \leq 40$
• $0 \leq M \leq \frac{N \cdot (N-1)}{2}$
• $1 \leq A[i] < B[i] \leq N$ for each $i$ such that $0 \leq i < M$.
• The pairs $(A[0], B[0]), \ldots, (A[M-1], B[M-1])$ are distinct.
• There exists at least one map which satisfies all the conditions.

Subtasks and Scoring

Subtask	Score	Additional Constraints
1	5	$M = N - 1$, $A[i] = i + 1$, $B[i] = i + 2$ for each $0 \leq i < M$.
2	10	$M = N - 1$
3	7	$M = \frac{N \cdot (N - 1)}{2}$
4	8	Country 1 is adjacent to all other countries. Some other pairs of countries may also be adjacent.
5	14	$N \leq 15$
6	56	No additional constraints.

In subtask 6, your score depends on the value of $K$.

• If any map returned by create_map does not satisfy all the conditions, your score for the subtask will be $0$.
• Otherwise, let $R$ be the maximum value of $K/N$ over all calls to create_map. Then, you receive a partial score according to the following table:

Limits	Score
$6 < R$	$0$
$4 < R \leq 6$	$14$
$3 < R \leq 4$	$28$
$2.5 < R \leq 3$	$42$
$2 < R \leq 2.5$	$49$
$R \leq 2$	$56$