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Orthogonal Emptiness Queries for Random Points

Jonathan E. Dullerud, Sariel Har-Peled

TL;DR

This work addresses the problem of orthogonal emptiness queries for random 2D points, introducing a suite of rank-space reductions and bucketing techniques to achieve near-linear space with constant-time queries. The authors develop constant-time rank structures for random inputs, map point sets to rank space, and derive both rectangle- and quadrant-emptiness data-structures. By combining grid bucketing, crossing-rectangle structures, and hierarchical range-tree methods, they obtain an overall expected preprocessing and space bound of $O\bigl(n \log n (\log \log n)^2\bigr)$ with $O(1)$ query time for random points. These results improve upon prior 2D emptiness structures under the random input model and offer practical benefits for fast, space-efficient orthogonal range searching in random data settings.

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.

Orthogonal Emptiness Queries for Random Points

TL;DR

This work addresses the problem of orthogonal emptiness queries for random 2D points, introducing a suite of rank-space reductions and bucketing techniques to achieve near-linear space with constant-time queries. The authors develop constant-time rank structures for random inputs, map point sets to rank space, and derive both rectangle- and quadrant-emptiness data-structures. By combining grid bucketing, crossing-rectangle structures, and hierarchical range-tree methods, they obtain an overall expected preprocessing and space bound of with query time for random points. These results improve upon prior 2D emptiness structures under the random input model and offer practical benefits for fast, space-efficient orthogonal range searching in random data settings.

Abstract

We present a data-structure for orthogonal range searching for random points in the plane. The new data-structure uses (in expectation) 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 . While the basic idea we use is known [Dev89], we believe our results are still interesting.
Paper Structure (23 sections, 11 theorems, 3 equations, 1 figure)

This paper contains 23 sections, 11 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rank (translation).
  3. Rank space queries in constant time.
  4. Random Point Queries
  5. Previous work.
  6. Preliminaries
  7. Rectangles.
  8. Rank queries for random numbers and points
  9. Rank queries.
  10. Construction.
  11. Emptiness queries in rank space
  12. Rectangle queries
  13. The easy solution: n^4 space.
  14. Reducing the space.
  15. Quadrant queries
  16. ...and 8 more sections

Key Result

Lemma 3.1

Let $S$ be a set of $m$ points chosen uniformly and independently from an internal $I$. Then, one can construct a data-structure of expected size $O(m^2)$, in $O(m^2)$ expected time, such that one can report the rank of a query $x \in I$ in $O(1)$ time.

Figures (1)

  • Figure 1.1: Results on orthogonal range searching in 2d in rank space and random points.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 5 more