Geometric retrieval problems

A large class of geometric retrieval problems has the following form. Given a set X of geometric objects, preprocess to obtain a data structure D(X). Now use D(X) to rapidly answer queries on X. We say an algorithm for such a problem has (worst-case) space-time complexity O(f(n),g(n)) if the space requirement for D(X) is O(f) and the 'locate run-time' required for each retrieval is O(g). We show three techniques which can consistently be exploited in solving such problems. For instance, using our techniques, we obtain an O(n2+e, lognlog(l/∈)) spacetime algorithm for the polygon retrieval problem, for arbitrarily small ∈, improving on the previous solution having complexity O(n7,logn).