Orthogonal Emptiness Queries for Random Points

We present a data-structure for orthogonal range searching for random points in the plane. The new data-structure uses (in expectation) $O\bigl(n \log n ( \log \log n)^2 \bigr)$ space, and answers emptiness queries in constant time. As a building block, we construct a data-structure of expected linear size, that can answer predecessor/rank queries, in constant time, for random numbers sampled uniformly from $[0,1]$. While the basic idea we use is known [Dev89], we believe our results are still interesting.