Efficient non-intersection queries on aggregated geometric data

Let S be a set of geometric objects that are aggregated into disjoint groups. The problem considered is that of preprocessing S so that for any query object, q, the distinct groups such that no objects from those groups are intersected by q can be reported efficiently. The goal is to devise solutions where the query time is sensitive to the output size, i.e., the number of groups reported. Unfortunately, the obvious approaches of (i) solving the corresponding intersection problem for aggregated data and reporting the complement, or (ii) querying with the complement of q are either expensive or incorrect. Efficient, output-sensitive solutions are given to several non-intersection searching problems on aggregated data, using methods such as geometric duality, sparsification, persistence, filtering search, and pruning.