201165: [AtCoder]ARC116 F - Deque Game

Problem Statement

We have $K$ sequences. The $i$-th sequence $A_i$ has the length of $N_i$.

Takahashi and Aoki will play a game using these. They will alternately do the following operation until the length of every sequence becomes $1$:

• Choose a sequence of length at least $2$, and delete its first or last element.

Takahashi goes first. Takahashi wants to maximize the sum of the $K$ elements remaining in the end, while Aoki wants to minimize it.

Find the sum of the $K$ elements remaining in the end when both players play optimally.

Constraints

• All values in input are integers.
• $ 1 \leq K \leq 2 \times 10^5$
• $1 \leq N_i$
• $\sum_i N_i \leq 2 \times 10^5$
• $1 \leq A_{i, j} \leq 10^9$

Input

Input is given from Standard Input in the following format:

$K$
$N_1$ $A_{1, 1}$ $A_{1, 2}$ $\cdots$ $A_{1, N_1}$
$N_2$ $A_{2, 1}$ $A_{2, 2}$ $\cdots$ $A_{2, N_2}$
$\vdots$
$N_K$ $A_{K, 1}$ $A_{K, 2}$ $\cdots$ $A_{K, N_K}$

Output

Print the answer.

Sample Input 1

2
3 1 2 3
2 1 10

Sample Output 1

12

Here is one possible progression of the game:

• Takahashi deletes the first element of $A_2$. Now we have $A_1 = \left(1, 2, 3\right), A_2 = \left(10\right)$.
• Aoki deletes the last element of $A_1$. Now we have $A_1 = \left(1, 2\right), A_2 = \left(10\right)$.
• Takahashi deletes the first element of $A_1$. Now we have $A_1 = \left(2\right), A_2 = \left(10\right)$. The length of every sequence has become $1$, so the game ends.

In this case, the sum of the $K$ elements remaining in the end is $12$. Note that the players may have made suboptimal moves here.

Sample Input 2

8
1 2
2 1 2
3 1 2 1
4 1 1 1 2
5 1 1 2 2 1
6 2 2 2 2 1 1
7 1 2 1 1 2 2 2
8 2 2 2 1 1 1 1 2

Sample Output 2

12