Orthogonal Emptiness Queries for Random Points

TL;DR

This paper introduces a new data-structure for orthogonal range emptiness queries on random points, achieving constant query time with sub-quadratic expected space, and also presents a linear-size structure for predecessor/rank queries on uniform random numbers.

Contribution

The paper develops a novel data-structure for orthogonal emptiness queries with optimal query time and improved space complexity for random points, and constructs a linear-size structure for predecessor/rank queries on uniform random data.

Findings

Answers emptiness queries in constant time.

Uses expected $O(n \, log n \, (\log \log n)^2)$ space.

Provides a linear-size structure for predecessor/rank queries on uniform random data.

Abstract

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.