An (Almost) Optimal Solution for Orthogonal Point Enclosure Query in ℝ3

Abstract

The orthogonal point enclosure query (OPEQ) problem is a fundamental problem in the context of data management for modeling user preferences. Formally, preprocess a set S of n axis-aligned boxes (possibly overlapping) in ℝd into a data structure so that the boxes in S containing a given query point q can be reported efficiently. In the pointer machine model, optimal solutions for the OPEQ in ℝ1 and ℝ2 were discovered in the 1980s: linear-space data structures that can answer the query in O(log n + k) query time, where k is the number of boxes reported. However, for the past three decades, an optimal solution in ℝ3 has been elusive. In this work, we obtain the first data structure that almost optimally solves the OPEQ in ℝ3 in the pointer machine model: an O(n log* n)-space data structure with O(log2 n · log log n + k) query time. Here, log* n is the iterated logarithm of n. This almost matches the lower-bound, which states that any data structure that occupies O(n) space requires Ω(log2 n + k) time to answer an OPEQ in ℝ3. Finally, we also obtain the best known bounds for the OPEQ in higher dimensions (d ≥ 4).