Orthogonal Queries in Segments

We consider the orthgonal clipping problem in a set of segments: Given a set of n segments in d-dimensional space, we preprocess them into a data structure such that given an orthogonal query window, the segments intersecting it can be counted/reported efficiently. We show that the efficiency of the data structure significantly depends on a geometric discrete parameter K named the projected-image complexity, which becomes Θ (n²) in the worst case but practically much smaller. If we use O(m) space, where K log^4d-7 n ≥ m ≥ n log^4d-7 n, the query time is O((K/m)^1/2 log^max{4,4d-5} n). This is near to an Ω((K/m)^1/2) lower bound.