Problem7032--ABC175 —— E - Picking Goods

7032: ABC175 —— E - Picking Goods

[Creator : ]
Time Limit : 1.000 sec  Memory Limit : 256 MiB

SubmitSolved: 1Submit Num: 1StatisticsShowOff!

Description

There are $K$ items placed on a grid of squares with $R$ rows and $C$ columns. Let $(i, j)$ denote the square at the $i$-th row ($1 \leq i \leq R$) and the $j$-th column ($1 \leq j \leq C$). The $i$-th item is at $(r_i, c_i)$ and has the value $v_i$.
Takahashi will begin at $(1, 1)$, the start, and get to $(R, C)$, the goal. When he is at $(i, j)$, he can move to $(i + 1, j)$ or $(i, j + 1)$ (but cannot move to a non-existent square).
He can pick up items on the squares he visits, including the start and the goal, but at most three for each row. It is allowed to ignore the item on a square he visits.
Find the maximum possible sum of the values of items he picks up.

Input

Input is given from Standard Input in the following format:
$R\ C\ K$
$r_1\ c_1\ v_1$
$r_2\ c_2\ v_2$
$\vdots$
$r_K\ c_K\ v_K$

Output

Print the maximum possible sum of the values of items Takahashi picks up.

Constraints

$1≤R,C≤3000$
$1 \leq K \leq \min(2 \times 10^5, R \times C)$
$1 \leq r_i \leq R$
$1 \leq c_i \leq C$
$(r_i, c_i) \neq (r_j, c_j) (i \neq j)$
$1 \leq v_i \leq 10^9$
All values in input are integers.

Sample 1 Input 

2 2 3
1 1 3
2 1 4
1 2 5

Sample 1 Output 

8

He has two ways to get to the goal:

  • Visit (1, 1), (1, 2), and (2, 2), in this order. In this case, the total value of the items he can pick up is $3 + 5 = 8$.
  • Visit (1, 1), (2, 1), and (2, 2), in this order. In this case, the total value of the items he can pick up is $3 + 4 = 7$.

Thus, the maximum possible sum of the values of items he picks up is $8$.

Sample 2 Input 

2 5 5
1 1 3
2 4 20
1 2 1
1 3 4
1 4 2

Sample 2 Output 

29

We have four items in the 1-st row. The optimal choices are as follows:

  • Visit (1, 1), (1, 2), (1, 3), (1, 4), (2, 4), and (2, 5), in this order, and pick up all items except the one on (1, 2). Then, the total value of the items he picks up will be $3 + 4 + 2 + 20 = 29$.

Sample 3 Input 

4 5 10
2 5 12
1 5 12
2 3 15
1 2 20
1 1 28
2 4 26
3 2 27
4 5 21
3 5 10
1 3 10

Sample 3 Output 

142

Source/Category

 
Submit