101754: [AtCoder]ABC175 E - Picking Goods

Memory Limit:256 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:8 Solved:0

Description

Score : $500$ points

Problem Statement

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.

Constraints

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

Input

Input is given from Standard Input in the following format:

$R$ $C$ $K$
$r_1$ $c_1$ $v_1$
$r_2$ $c_2$ $v_2$
$:$
$r_K$ $c_K$ $v_K$

Output

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


Sample Input 1

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

Sample Output 1

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 Input 2

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

Sample Output 2

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 Input 3

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 Output 3

142

Input

题意翻译

在 $R$ 行 $C$ 列的棋盘格内放置着 $K$ 个物品,第 $i$ 个物品在格子 $(r_i,c_i)$,价值为 $v_i$。 高桥从格子 $(1,1)$ 出发,向格子 $(R,C)$ 移动。高桥君在格子 $(i,j)$ 时,可以走到格子 $(i+1,j)$ 或格子 $(i,j+1)$。 高桥可以拾取路过的格子内的物品(包括起点和终点格子),但是同一行的物品最多只能拾取 $3$ 个。 求高桥可以拾取的物品的总价值的最大值。

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