Parentheses Flip

First imagine changing every character to ). Let $C$ be the cost of flipping all positions that were originally (.

If position $i$ is then chosen to become (, the additional cost is $-A_i$ when the original character was (, and $+A_i$ otherwise. A correct parenthesis string must contain at least $1,2,3,\ldots,N$ opening parentheses in the odd prefixes of lengths $1,3,5,\ldots,2N-1$.

Scan from left to right and put each available additional cost into a priority queue. Whenever an odd position is reached, choose the cheapest not-yet-chosen position among the prefix and make it an opening parenthesis. Each step satisfies the newly increased prefix requirement as cheaply as possible.

The priority queue gives an $O(N\log N)$ algorithm. Solution written by GPT5.6