Rectangle stabbing and orthogonal range reporting lower bounds in moderate dimensions

Abstract

We study the orthogonal range reporting and rectangle stabbing problems in moderate dimensions, i.e., when the dimension is clog⁡(n) for some constant c. In orthogonal range reporting, the input is a set of n points in d dimensions, and the goal is to store these n points in a data structure such that given a query rectangle, we can report all the input points contained in the rectangle. The rectangle stabbing problem is the “dual” problem where the input is a set of rectangles, and the query is a point.Our main result is the following: assume using S(n) space, we can solve either problem in d=clog⁡n dimensions, c≥4, using Q(n)+O(t) time in the pointer machine model of computation where t is the output size. Then, we show that if the query time is small, that is, Q(n)=n1−γ, for γ≥22+log⁡c, then the space must be Ω(n1−γncγ/e−o(cγ)). Interestingly, we obtain this lower bound using a non-constructive method, and we show the existence of some codes that generalize a specific aspect of error correction codes. Our result overcomes the shortcomings of the previous lower bounds in the pointer machine model for non-constant dimension [3–5,13], as the previous results could not be extended for d=Ω(log⁡n).The only known lower bounds for rectangle stabbing, when the dimension is non-constant, are based on conditional lower bounds upon the best-known results on CNF-SAT [21]. Therefore, our lower bound is the first non-trivial unconditional lower bound for orthogonal range reporting and rectangle stabbing with non-constant dimension.