Optimal Planar Orthogonal Skyline Counting Queries

TL;DR

This paper introduces an optimal data structure for orthogonal skyline counting queries in the plane, achieving the best possible query time and space efficiency, with proven lower bounds.

Contribution

It presents a space-efficient static data structure with optimal query time for skyline counting, and establishes matching lower bounds in the cell probe model.

Findings

Query time $O(rac{ ext{lg} n}{ ext{lg} ext{lg} n})$ achieved

Space usage is $O(n)$, optimal for the problem

Lower bounds match the upper bounds, proving optimality

Abstract

The skyline of a set of points in the plane is the subset of maximal points, where a point $(x,y)$ is maximal if no other point $(x',y')$ satisfies $x'\ge x$ and $y'\ge Y$. We consider the problem of preprocessing a set $P$ of $n$ points into a space efficient static data structure supporting orthogonal skyline counting queries, i.e. given a query rectangle $R$ to report the size of the skyline of $P$ intersected with $R$. We present a data structure for storing n points with integer coordinates having query time $O(\lg n/\lg\lg n)$ and space usage $O(n)$. The model of computation is a unit cost RAM with logarithmic word size. We prove that these bounds are the best possible by presenting a lower bound in the cell probe model with logarithmic word size: Space usage $n\lg^{O(1)} n$ implies worst case query time $\Omega(\lg n/\lg\lg n)$.