In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n+t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log²n / log log n + t) has not been improved for decades.

We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log²n+t) query time in four dimensions. More precisely, we develop a structure that uses O(n (log n /log log n)^d) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O( log n (log n /log log n)^d-4+1/(d-2) + t) time. Ignoring log log n factors, this speeds up the best previous query time by a log^1-1/(d-2)n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n (log n / log h)^d-2 + t) time to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ log^d-2+ε n where ε>0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.