Indexability of 2D range search revisited: constant redundancy and weak indivisibility

Abstract

In the 2D orthogonal range search problem, we want to preprocess a set of 2D points so that, given any axis-parallel query rectangle, we can report all the data points in the rectangle efficiently. This paper presents a lower bound on the query time that can be achieved by any external memory structure that stores a point at most r times, where r is a constant integer. Previous research has resolved the bound at two extremes: r = 1, and r being arbitrarily large. We, on the other hand, derive the explicit tradeoff at every specific r. A premise that lingers in existing studies is the so-called indivisibility assumption: all the information bits of a point are treated as an atom, i.e., they are always stored together in the same block. We partially remove this assumption by allowing a data structure to freely divide a point into individual bits stored in different blocks. The only assumption is that, those bits must be retrieved for reporting, as opposed to being computed -- we refer to this requirement as the weak indivisibility assumption. We also describe structures to show that our lower bound is tight up to only a small factor.