Deque Game

AtCoder

IDarc116_f

Time2000ms

Memory256MB

Difficulty—

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: $K$ $N_1$ $A_{1, 1}$ $A_{1, 2}$ $\cdots$ $A_{1, N_1}$ $N_2$ $A_{2, 1}$ $A_{2, 2}$ $\cdots$ $A_{2, N_2}$ $\vdots$ $N_K$ $A_{K, 1}$ $A_{K, 2}$ $\cdots$ $A_{K, N_K}$ [samples]

Samples

Input #1

2
3 1 2 3
2 1 10

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.

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

Output #2

API Response (JSON)

{
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    "name": "Deque Game",
    "description": {
      "content": "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 s",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc116_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $K$ sequences. The $i$\\-th sequence $A_i$ has the length of $N_i$.\nTakahashi and Aoki will play a game using these. They will alternately do the following operation until the length of every s...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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