Problem

At Stripe, multiple workers acquire locks in sequences called plans. To avoid deadlocks, all workers must follow a single global lock order (a total order of lock IDs). You may fix each plan using adjacent swaps only.

Return the minimum total number of adjacent swaps needed across all plans so that there exists one global order in which every plan becomes increasing (i.e., each plan is a subsequence of that global order).

Formal specification

• Input: plans: list[list[int]] where each inner list contains distinct integers (lock IDs).
• Output: int = minimum total adjacent swaps across all plans.

Examples

1. plans = [[3, 1, 2], [1, 2, 3]] → 2

   • A valid global order is [1, 2, 3]. First plan needs 2 adjacent swaps to become [1,2,3]; second needs 0.

2. plans = [[2, 5, 1, 3, 4]] → 3

   • One valid global order is [2, 5, 1, 3, 4] itself. Minimum adjacent swaps to match that order equals the inversion count relative to it, which is 3.

Constraints

• 0 <= len(plans) <= 2 * 10^4
• sum(len(plans[i])) <= 2 * 10^5
• 1 <= plans[i][j] <= 10^9
• Values are distinct within each plan