Criticality Considerations in the Design of Geometric Algorithms

We examine a class of geometric problems which all share a feature of considerable relevance: the underlying set is defined implicitly and is at least quadratically larger than the input size. Is efficient computation possible without expanding the set explicitly? If not, are non-trivial lower bounds provable? We give partial answers to these questions by proposing various approaches for attacking these problems. In doing so, we improve a number of complexity bounds for problems of range searching, frame-to-frame coherence, segment stabbing, geometric predicate computation, etc. 1. Triangular Search Let S = {p1, . . . , pn} be a set of n points in E . The triangular search problem is to preprocess S so that, for any query triangle T , the subset of points of S lying inside T can be computed efficiently. An (f(n), g(n)) solution refers to an algorithm for this problem that requires O(f(n)) space and O(g(n)+ output size) query time. The most efficient solutions to date are an (n, n) algorithm by Edelsbrunner and Welzl [EW] and an (n , log n log 1 ) by Cole and Yap [CY] (see also [EKM,W] for other methods). Our first observation is that quadratic space complexity seems to be the price to pay for fast retrieval. This can be understood by looking at the problem of determining whether a query line L passes through any point of S. Obviously, this is a subproblem of the original problem: simply make the triangle T infinitely long and skinny. Searching is (at least conceptually) facilitated by turning to a dual space, whereby lines becomes points and points become lines. The dual of S is an arrangement of n