OH NO1 (-2-3-4)

4000ms

524288K

Description:

You are given an undirected graph with $$$n$$$ vertices and $$$3m$$$ edges. The graph may contain multi-edges, but does not contain self-loops.

The graph satisfies the following property: the given edges can be divided into $$$m$$$ groups of $$$3$$$, such that each group is a triangle.

A triangle is defined as three edges $$$(a,b)$$$, $$$(b,c)$$$ and $$$(c,a)$$$ for some three distinct vertices $$$a,b,c$$$ ($$$1 \leq a,b,c \leq n$$$).

Initially, each vertex $$$v$$$ has a non-negative integer weight $$$a_v$$$. For every edge $$$(u,v)$$$ in the graph, you should perform the following operation exactly once:

Choose an integer $$$x$$$ between $$$1$$$ and $$$4$$$. Then increase both $$$a_u$$$ and $$$a_v$$$ by $$$x$$$.

After performing all operations, the following requirement should be satisfied: if $$$u$$$ and $$$v$$$ are connected by an edge, then $$$a_u \ne a_v$$$.

It can be proven this is always possible under the constraints of the task. Output a way to do so, by outputting the choice of $$$x$$$ for each edge. It is easy to see that the order of operations does not matter. If there are multiple valid answers, output any.

Input:

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$3 \le n \le 10^6$$$, $$$1 \le m \le 4 \cdot 10^5$$$) — denoting the graph have $$$n$$$ vertices and $$$3m$$$ edges.

The second line of each test case contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$0 \leq a_i \leq 10^6$$$) — the initial weights of each vertex.

Then $$$m$$$ lines follow. The $$$i$$$-th line contains three integers $$$a_i$$$, $$$b_i$$$, $$$c_i$$$ ($$$1 \leq a_i < b_i < c_i \leq n$$$) — denotes that three edges $$$(a_i,b_i)$$$, $$$(b_i,c_i)$$$ and $$$(c_i,a_i)$$$.

Note that the graph may contain multi-edges: a pair $$$(x,y)$$$ may appear in multiple triangles.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^6$$$ and the sum of $$$m$$$ over all test cases does not exceed $$$4 \cdot 10^5$$$.

Output:

For each test case, output $$$m$$$ lines of $$$3$$$ integers each.

The $$$i$$$-th line should contains three integers $$$e_{ab},e_{bc},e_{ca}$$$ ($$$1 \leq e_{ab}, e_{bc} , e_{ca} \leq 4$$$), denoting the choice of value $$$x$$$ for edges $$$(a_i, b_i)$$$, $$$(b_i,c_i)$$$ and $$$(c_i, a_i)$$$ respectively.

Sample Input:

4
4 1
0 0 0 0
1 2 3
5 2
0 0 0 0 0
1 2 3
1 4 5
4 4
3 4 5 6
1 2 3
1 2 4
1 3 4
2 3 4
5 4
0 1000000 412 412 412
1 2 3
1 4 5
2 4 5
3 4 5

Sample Output:

Note:

In the first test case, the initial weights are $$$[0,0,0,0]$$$. We have added values as follows:

Added $$$2$$$ to vertices $$$1$$$ and $$$2$$$
Added $$$1$$$ to vertices $$$2$$$ and $$$3$$$
Added $$$3$$$ to vertices $$$3$$$ and $$$1$$$

The final weights are $$$[5,3,4,0]$$$. The output is valid because $$$a_1 \neq a_2$$$, $$$a_1 \neq a_3$$$, $$$a_2 \neq a_3$$$, and that all chosen values are between $$$1$$$ and $$$4$$$.

In the second test case, the initial weights are $$$[0,0,0,0,0]$$$. The weights after the operations are $$$[12,5,6,7,6]$$$. The output is valid because $$$a_1 \neq a_2$$$, $$$a_1 \neq a_3$$$, $$$a_2 \neq a_3$$$, and that $$$a_1 \neq a_4$$$, $$$a_1 \neq a_5$$$, $$$a_4 \neq a_5$$$, and that all chosen values are between $$$1$$$ and $$$4$$$.

In the third test case, the initial weights are $$$[3,4,5,6]$$$. The weights after the operations are $$$[19,16,17,20]$$$, and all final weights are distinct, which means no two adjacent vertices have the same weight.

Informação

Provedor Codeforces

Código CF1815F