Parenthesis Arrangement

AtCoder

IDarc175_b

Time2000ms

Memory256MB

Difficulty—

You are given a string $S$ of length $2N$ consisting of the characters `(` and `)`. Let $S_i$ denote the $i$\-th character from the left of $S$. You can perform the following two types of operations zero or more times in any order: * Choose a pair of integers $(i,j)$ such that $1\leq i < j \leq 2N$. Swap $S_i$ and $S_j$. The cost of this operation is $A$. * Choose an integer $i$ such that $1\leq i \leq 2N$. Replace $S_i$ with `(` or `)`. The cost of this operation is $B$. Your goal is to make $S$ a correct parenthesis sequence. Find the minimum total cost required to achieve this goal. It can be proved that the goal can always be achieved with a finite number of operations. What is a correct parenthesis sequence? A correct parenthesis sequence is a string that satisfies any of the following conditions: * It is an empty string. * It is formed by concatenating `(`, $A$, `)` in this order where $A$ is a correct parenthesis sequence. * It is formed by concatenating $A$ and $B$ in this order where $A$ and $B$ are correct parenthesis sequences. ## Constraints * All input values are integers. * $1 \leq N \leq 5\times 10^5$ * $1\leq A,B\leq 10^9$ * $S$ is a string of length $2N$ consisting of the characters `(` and `)`. ## Input The Input is given from Standard Input in the following format: $N$ $A$ $B$ $S$ [samples]

Samples

Input #1

3 3 2
)))(()

Output #1

5

Here is one way to operate:

*   Swap $S_3$ and $S_4$. $S$ becomes `))()()`. The cost is $3$.
*   Replace $S_1$ with `(`. $S$ becomes `()()()`, which is a correct parentheses sequence. The cost is $2$.

In this case, we have made $S$ a correct bracket sequence for a total cost of $5$. There is no way to make $S$ a correct bracket sequence for less than $5$.

Input #2

1 175 1000000000
()

Output #2

0

The given $S$ is already a correct bracket sequence, so no operation is needed.

Input #3

7 2622 26092458
))()((((()()((

Output #3

52187538

API Response (JSON)

{
  "problem": {
    "name": "Parenthesis Arrangement",
    "description": {
      "content": "You are given a string $S$ of length $2N$ consisting of the characters `(` and `)`. Let $S_i$ denote the $i$\\-th character from the left of $S$. You can perform the following two types of operations z",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc175_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a string $S$ of length $2N$ consisting of the characters `(` and `)`. Let $S_i$ denote the $i$\\-th character from the left of $S$.\nYou can perform the following two types of operations z...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments