ARC 175 B Parenthesis Arrangement 题解

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Parenthesis Arrangement

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Problem Statement

You are given a string $S$ of length $2 N$ 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