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H - Minimum Coloring

Score : $600$ points

Problem Statement

We have a grid with $H$ rows and $W$ columns. Let $(i,j)$ denote the square at the $i$-th row from the top and $j$-th column from the left.

On this grid, there are $N$ white pieces numbered $1$ to $N$. Piece $i$ is on $(A_i,B_i)$.

You can pay the cost of $C_i$ to change Piece $i$ to a black piece.

Find the minimum total cost needed to have at least one black piece in every row and every column.

Constraints

• $1 \leq H,W \leq 10^3$
• $1 \leq N \leq 10^3$
• $1 \leq A_i \leq H$
• $1 \leq B_i \leq W$
• $1 \leq C_i \leq 10^9$
• All pairs $(A_i,B_i)$ are distinct.
• There is at least one white piece in every row and every column.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$H$ $W$ $N$

$A_1$ $B_1$ $C_1$

$\hspace{23pt} \vdots$

$A_N$ $B_N$ $C_N$

Output

Print the answer.

2 3 6
1 1 1
1 2 10
1 3 100
2 1 1000
2 2 10000
2 3 100000

1110

By paying the cost of $1110$ to change Pieces $2, 3, 4$ to black pieces, we can have a black piece in every row and every column.

1 7 7
1 2 200000000
1 7 700000000
1 4 400000000
1 3 300000000
1 6 600000000
1 5 500000000
1 1 100000000

2800000000

3 3 8
3 2 1
3 1 2
2 3 1
2 2 100
2 1 100
1 3 2
1 2 100
1 1 100

6