A friend told me a problem that he had been trying to optimize for a personal graphics project, and it involved having an h × w grid, and wanting to do some pre-processing to answer circle-sum queries. A circle here would be defined as having some center, (y_c, x_c), where 1 ≤ y_c ≤ h and 1 ≤ x_c ≤ w and some radius r, and it is the set of integer grid points (y, x) such that (x - x_c)² + (y - y_c)² ≤ r².

The progress he's made is that in O((hw)²) pre-processing, you can just precompute all the possible queries, so one possible set of complexities is O((hw)²) pre-processing and O(1) queries. Another possibility is to precompute prefix sums in O(hw) time and answer the queries in O(r) time.

Is there any clever way to get a better set of pre-processing and query complexity than the two methods mentioned above?