On dominance reporting in 3D

Abstract

In this paper, we study the 3D dominance reporting problem in different models of computations and offer optimal results in the pointer machine and the external memory models and a near optimal result in the RAM model; all our results consume linear space. We can answer queries in O(log n + k) time on a pointer machine, with O(log B n + k/B) I/Os in the external memory model and in O((log logn)2 + log log U + k) time in the RAM model and in a U×U×U integer grid. These improve the results of various papers, such as Makris and Tsakalidis (IPL'98), Vengroff and Vitter (STOC'96) and Nekrich (SOCG'07). Here, n, k and B are the input, output and block size respectively. With a log3 n fold increase in the space complexity these can be turned into orthogonal range reporting algorithms with matching query times, improving the previous orthogonal range searching results in the pointer machine and RAM models. Using our 3D results as base cases, we can provide improved orthogonal range reporting algorithms in ℝd,d ≥ 4. We use randomization only in the preprocessing part and our query bounds are all worst case. © 2008 Springer-Verlag Berlin Heidelberg.