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 $c\log \left(n\right)$ 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 $S\left(n\right)$ space, we can solve either problem in $d=c\log n$ dimensions, $c\ge 4$, using $Q\left(n\right)+O\left(t\right)$ 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, $Q\left(n\right)=n^{1-\gamma }$, for $\gamma \ge \frac{2}{2+\log c}$, then the space must be $\Omega \left(n^{1-\gamma }{n}^{\sqrt{c\gamma }/e-o\left(\sqrt{c\gamma }\right)}\right)$. 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], [4], [5], [13], as the previous results could not be extended for $d=\Omega \left(\sqrt{\log n}\right)$.

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.