I am currently working on the following problem:

• Given a permutation $$$A$$$ of length $$$N$$$, you have $$$Q$$$ queries specifying 2 numbers $$$X$$$ and $$$Y$$$ that swap the elements at indices $$$X$$$ and $$$Y$$$. After every query, output the number of inversions in the new permutation. All queries must be answered online.

My current solution can process every query in $$$O(log^2 N)$$$ per query and N log N precomputation, by first precomputing the number of inversions in the initial permutation and using a segment tree with a BBST in every node. Each BBST stores all of the elements in the range covered by the segment tree node. We perform a range query on the value at index $$$X$$$ and the value at index $$$Y$$$ to determine the number of numbers smaller than either number in the segment between them, and then compute the change in inversions.

My question is: Is it possible to compute the change in inversions more quickly than log^2 N? eg. computing the change in log N time.