A15252 | Too Many Constraints

You are asked to build an array $a$ , consisting of $n$ integers, each element should be from $1$ to $k$ .

The array should be non-decreasing ( $a_i \le a_{i+1}$ for all $i$ from $1$ to $n-1$ ).

You are also given additional constraints on it. Each constraint is of one of three following types:

- $1~i~x$ : $a_i$ should not be equal to $x$ ;
- $2~i~j~x$ : $a_i + a_j$ should be less than or equal to $x$ ;
- $3~i~j~x$ : $a_i + a_j$ should be greater than or equal to $x$ .

Build any non-decreasing array that satisfies all constraints or report that no such array exists.

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of testcases.

The first line of each testcase contains three integers $n, m$ and $k$ ( $2 \le n \le 2 \cdot 10^4$ ; $0 \le m \le 2 \cdot 10^4$ ; $2 \le k \le 10$ ).

The $i$ -th of the next $m$ lines contains a description of a constraint. Each constraint is of one of three following types:

- $1~i~x$ ( $1 \le i \le n$ ; $1 \le x \le k$ ): $a_i$ should not be equal to $x$ ;
- $2~i~j~x$ ( $1 \le i < j \le n$ ; $2 \le x \le 2 \cdot k$ ): $a_i + a_j$ should be less than or equal to $x$ ;
- $3~i~j~x$ ( $1 \le i < j \le n$ ; $2 \le x \le 2 \cdot k$ ): $a_i + a_j$ should be greater than or equal to $x$ .

The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^4$ . The sum of $m$ over all testcases doesn't exceed $2 \cdot 10^4$ .

For each testcase, determine if there exists a non-decreasing array that satisfies all conditions. If there is no such array, then print -1. Otherwise, print any valid array — $n$ integers from $1$ to $k$ .