Minimum Cost to Make All Characters Equal

Given a binary string s (n ≤ 1e5), you may flip any prefix ending at i (cost i+1) or any suffix starting at i (cost n−i); compute the minimum total cost to make all characters identical. The challenge is to choose prefix/suffix flips—which toggle runs and can cancel each other—optimally to remove transitions and reach all 0s or all 1s.