A11280. Pokémon Arena

You are at a dueling arena. You also possess $n$ Pokémons. Initially, only the $1$ -st Pokémon is standing in the arena.

Each Pokémon has $m$ attributes. The $j$ -th attribute of the $i$ -th Pokémon is $a_{i,j}$ . Each Pokémon also has a cost to be hired: the $i$ -th Pokémon's cost is $c_i$ .

You want to have the $n$ -th Pokémon stand in the arena. To do that, you can perform the following two types of operations any number of times in any order:

- Choose three integers $i$ , $j$ , $k$ ( $1 \le i \le n$ , $1 \le j \le m$ , $k > 0$ ), increase $a_{i,j}$ by $k$ permanently. The cost of this operation is $k$ .
- Choose two integers $i$ , $j$ ( $1 \le i \le n$ , $1 \le j \le m$ ) and hire the $i$ -th Pokémon to duel with the current Pokémon in the arena based on the $j$ -th attribute. The $i$ -th Pokémon will win if $a_{i,j}$ is greater than or equal to the $j$ -th attribute of the current Pokémon in the arena (otherwise, it will lose). After the duel, only the winner will stand in the arena. The cost of this operation is $c_i$ .

Find the minimum cost you need to pay to have the $n$ -th Pokémon stand in the arena.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^5$ ). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ( $2 \le n \le 4 \cdot 10^5$ , $1 \le m \le 2 \cdot 10^5$ , $2 \leq n \cdot m \leq 4 \cdot 10^5$ ).

The second line of each test case contains $n$ integers $c_1, c_2, \ldots, c_n$ ( $1 \le c_i \le 10^9$ ).

The $i$ -th of the following $n$ lines contains $m$ integers $a_{i,1}, a_{i,2}, \ldots, a_{i,m}$ ( $1 \le a_{i,j} \le 10^9$ ).

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

For each test case, output the minimum cost to make the $n$ -th Pokémon stand in the arena.

In the first test case, the attribute array of the $1$ -st Pokémon (which is standing in the arena initially) is $[2,9,9]$ .

In the first operation, you can choose $i=3$ , $j=1$ , $k=1$ , and increase $a_{3,1}$ by $1$ permanently. Now the attribute array of the $3$ -rd Pokémon is $[2,2,1]$ . The cost of this operation is $k = 1$ .

In the second operation, you can choose $i=3$ , $j=1$ , and hire the $3$ -rd Pokémon to duel with the current Pokémon in the arena based on the $1$ -st attribute. Since $a_{i,j}=a_{3,1}=2 \ge 2=a_{1,1}$ , the $3$ -rd Pokémon will win. The cost of this operation is $c_3 = 1$ .

Thus, we have made the $3$ -rd Pokémon stand in the arena within the cost of $2$ . It can be proven that $2$ is minimum possible.

In the second test case, the attribute array of the $1$ -st Pokémon in the arena is $[9,9,9]$ .

In the first operation, you can choose $i=2$ , $j=3$ , $k=2$ , and increase $a_{2,3}$ by $2$ permanently. Now the attribute array of the $2$ -nd Pokémon is $[6,1,9]$ . The cost of this operation is $k = 2$ .

In the second operation, you can choose $i=2$ , $j=3$ , and hire the $2$ -nd Pokémon to duel with the current Pokémon in the arena based on the $3$ -rd attribute. Since $a_{i,j}=a_{2,3}=9 \ge 9=a_{1,3}$ , the $2$ -nd Pokémon will win. The cost of this operation is $c_2 = 3$ .

In the third operation, you can choose $i=3$ , $j=2$ , and hire the $3$ -rd Pokémon to duel with the current Pokémon in the arena based on the $2$ -nd attribute. Since $a_{i,j}=a_{1,2}=2 \ge 1=a_{2,2}$ , the $3$ -rd Pokémon can win. The cost of this operation is $c_3 = 1$ .

Thus, we have made the $3$ -rd Pokémon stand in the arena within the cost of $6$ . It can be proven that $6$ is minimum possible.