Lower Bounds for Sorted Geometric Queries in the I/O Model

We study sorted geometric query problems, a class of problems that, to the best of our knowledge and despite their applications, have not received much attention so far. Two of the most prominent problems in this class are angular sorting queries and sorted K-nearest neighbour queries. The former asks us to preprocess an input point set S in the plane so that, given a query point q, the clockwise ordering of the points in S around q can be computed efficiently. In the latter problem, the output is the list of K points in S closest to q, sorted by increasing distance from q. The goal in both problems is to construct a small data structure that can answer queries efficiently. We study sorted geometric query problems in the I/O model and prove that, when limited to linear space, the naive approach of sorting the elements in S in the desired output order from scratch is the best possible. This is highly relevant in an I/O context because storing a massive data set in a superlinear-space data structure is often infeasible. We also prove that answering queries using O(N/B) I/Os requires Omega(N log(M) N) space, where N is the input size, B is the block size, and M is the size of the main memory. This bound is unlikely to be optimal and in fact we can show that, for a particular class of "persistence-based" data structures, the space lower bound can be improved to Omega(N-2/M-O(1)). Both these lower bounds are a first step towards understanding the complexity of sorted geometric query problems. All our lower bounds assume indivisibility of records and hold as long as B = Omega(log(M/B) N).