OH NO1 (-2-3-4)
Description
You are given an undirected graph with $n$ vertices and $3m$ edges. The graph may contains multi-edges, but do not contain self loops.
The graph satisfy the following property: the given edges can be divided into $m$ groups of $3$, such that each group is a triangle.
A triangle are 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 do not matter. If there are multiple valid answers, output any.
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 follows. 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 do not exceed $10^6$ and the sum of $m$ over all test cases do not exceed $4 \cdot 10^5$.
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.
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 follows. 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 do not exceed $10^6$ and the sum of $m$ over all test cases do 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.
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
2 1 3
2 3 3
4 3 3
3 1 2
2 2 3
2 3 4
3 1 1
2 3 4
1 2 4
4 4 3
4 1 1
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 $1$ and $3$
- Added $3$ to vertices $2$ and $3$
The final weights are $[3,5,4,0]$. 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 $[19,16,17,20]$, so all final weights are distinct, which means no two adjacent vertices have the same weight.