Bags Game

AtCoder

IDabc303_g

Time2500ms

Memory256MB

Difficulty—

$N$ bags are arranged in a row. The $i$\-th bag contains $x_i$ yen (the currency in Japan). Takahashi and Aoki, who have sufficient money, take alternating turns to perform the following action: * choose one of the following three actions and perform it. * choose the leftmost or rightmost bag and take it. * pay $A$ yen to Snuke. Then, repeat the following action $\min(B,n)$ times (where $n$ is the number of the remaining bags): choose the leftmost or rightmost bag and take it. * pay $C$ yen to Snuke. Then, repeat the following action $\min(D,n)$ times (where $n$ is the number of the remaining bags): choose the leftmost or rightmost bag and take it. When all the bags are taken, Takahashi's benefit is defined by "(total amount of money in the bags that Takahashi took) - (total amount of money that Takahashi paid to snuke)"; let this amount be $X$ yen. We similarly define Aoki's benefit, denoting the amount by $Y$ yen. Find $X-Y$ if Takahashi and Aoki make optimal moves to respectively maximize and minimize $X-Y$. ## Constraints * $1 \leq N \leq 3000$ * $1 \leq x_i \leq 10^9$ * $1 \leq A,C \leq 10^9$ * $1 \leq B,D \leq N$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $A$ $B$ $C$ $D$ $x_1$ $x_2$ $\ldots$ $x_N$ [samples]

Samples

Input #1

5 10 2 1000000000 1
1 100 1 1 1

Output #1

90

If Takahashi and Aoki make optimal moves, it ends up being $X=92$ and $Y=2$.

Input #2

10 45 3 55 4
5 10 15 20 25 30 35 40 45 50

Output #2

Input #3

15 796265 10 165794055 1
18804175 185937909 1934689 18341 68370722 1653 1 2514380 31381214 905 754483 11 5877098 232 31600

Output #3

302361955

API Response (JSON)

{
  "problem": {
    "name": "Bags Game",
    "description": {
      "content": "$N$ bags are arranged in a row. The $i$\\-th bag contains $x_i$ yen (the currency in Japan). Takahashi and Aoki, who have sufficient money, take alternating turns to perform the following action: *   ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2500,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc303_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "$N$ bags are arranged in a row. The $i$\\-th bag contains $x_i$ yen (the currency in Japan).\nTakahashi and Aoki, who have sufficient money, take alternating turns to perform the following action:\n\n*   ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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