Optimal Planar Orthogonal Skyline Counting Queries

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′ ≥ x and y′ ≥ 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 ∩ 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) words. 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 matching lower bound in the cell probe model with logarithmic word size: Space usage \(n\lg^{O(1)} n\) implies worst case query time \(\Omega(\lg n/\lg\lg n)\).