In this work, we study the problem of reporting and counting maximal points in a query rectangle for a set of n integer points that lie on an n×n grid A point is said to be maximal inside a query rectangle if it is not dominated by any other point inside the query rectangle Our model of computation is unit-cost RAM model with word size of O(logn) bits For the reporting version of the problem, we present a data structure of size $O(n\frac{\log n}{\log\log n})$ words and support querying in $O(\frac{\log n}{\log\log n}+k)$ time where k is the size of the output For the counting version, we present a data structure of size $O(n\frac{\log^{2} n}{\log\log n})$ words which supports querying in $O(\frac{\log^{\frac{3}{2}}n} {\log\log n})$  Both the data structures are static in nature The reporting version of the problem has been studied in [1] and [5] To the best of our knowledge, this is the first sub-logarithmic result for the reporting version and the first work for the counting version of the problem

Range aggregate maximal points in the plane

In the planar range skyline reporting problem, we store a set P of n 2D points in a structure such that, given a query rectangle Q = [a_1, a_2] x [b_1, b_2], the maxima (a.k.a. skyline) of P \cap Q can be reported efficiently. The query is 3-sided if an edge of Q is grounded, giving rise to two variants: top-open (b_2 = \infty) and left-open (a_1 = -\infty) queries. 
All our results are in external memory under the O(n/B) space budget, for both the static and dynamic settings: 
* For static P, we give structures that answer top-open queries in O(log_B n + k/B), O(loglog_B U + k/B), and O(1 + k/B) I/Os when the universe is R^2, a U x U grid, and a rank space grid [O(n)]^2, respectively (where k is the number of reported points). The query complexity is optimal in all cases. 
* We show that the left-open case is harder, such that any linear-size structure must incur \Omega((n/B)^e + k/B) I/Os for a query. We show that this case is as difficult as the general 4-sided queries, for which we give a static structure with the optimal query cost O((n/B)^e + k/B). 
* We give a dynamic structure that supports top-open queries in O(log_2B^e (n/B) + k/B^1-e) I/Os, and updates in O(log_2B^e (n/B)) I/Os, for any e satisfying 0 \le e \le 1. This leads to a dynamic structure for 4-sided queries with optimal query cost O((n/B)^e + k/B), and amortized update cost O(log (n/B)). 
As a contribution of independent interest, we propose an I/O-efficient version of the fundamental structure priority queue with attrition (PQA). Our PQA supports FindMin, DeleteMin, and InsertAndAttrite all in O(1) worst case I/Os, and O(1/B) amortized I/Os per operation. 
We also add the new CatenateAndAttrite operation that catenates two PQAs in O(1) worst case and O(1/B) amortized I/Os. This operation is a non-trivial extension to the classic PQA of Sundar, even in internal memory.

I/O-Efficient Planar Range Skyline and Attrition Priority Queues

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 $\Omega(\lg n/\lg\lg n)$.

Optimal Planar Orthogonal Skyline Counting Queries

We study the static and dynamic planar range skyline reporting problem in the external memory model with block size B, under a linear space budget. The problem asks for an O(n/B) space data structure that stores n points in the plane, and supports reporting the k maximal input points (a.k.a.skyline) among the points that lie within a given query rectangle Q = [α1[α2] × [β1β2. When Q is 3-sided, i.e. one of its edges is grounded, two variants arise: top-open for β2 = ∞ and left-open for α1 = - ∞ (symmetrically bottom-open and right-open) queries.We present optimal static data structures for top-open queries, for the cases where the universe is R2, a U × U grid, and rank space [O(n)]2. We also show that left-open queries are harder, as they require Ω((n/B)e + k/B) I/Os for e > 0, when only linear space is allowed. We show that the lower bound is tight, by a structure that supports 4-sided queries in matching complexities. Interestingly, these lower and upper bounds coincide with those of the planar orthogonal range reporting problem, i.e., the skyline requirement does not alter the problem difficulty at all!Finally, we present the first dynamic linear space data structure that supports top-open queries in O(log2Ben + k/B1 e > and updates in O(log2Ben) worst case I/Os, for e ∈ [0, 1]. This also yields a linear space data structure for 4-sided queries with optimal query I/Os and O(log(n/B)) amortized update I/Os. We consider of independent interest the main component of our dynamic structures, a new real-time I/O-efficient and catenable variant of the fundamental structure priority queue with attrition by Sundar.

https://www.cse.cuhk.edu.hk/~taoyf/paper/pods13.pdf

I/O-efficient planar range skyline and attrition priority queues

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 $\Omega(\lg n/\lg\lg n)$.

/pdf/optimal-planar-orthogonal-skyline-counting-queries-sz6x3o88a4.pdf

We consider the dominating point set reporting problem in two-dimension. We propose a data structure for nding the set of dominating points inside a given orthogonal query rectangle. Given a set of n points in the plane, it supports 4-sided queries in O(logn +k), where k is size of the output, usingO(n logn) space. This work can be of application when range queries are generated using mobile devices with limited display capacity.

/pdf/on-finding-skyline-points-for-range-queries-in-plane-41qwh9pbze.pdf

On Finding Skyline Points for Range Queries in Plane.

We consider the problem of reporting and counting maximal points in a given orthogonal query range in two-dimensions. Our model of computation is the pointer machine model. Let P be a static set of n points in ℝ2. A point is maximal in P if it is not dominated by any other point in P. We propose a linear space data structure that can support counting the number of maximal points inside a 3-sided orthogonal query rectangle unbounded on its right in O(logn) time. For counting the number of maximal points in a 4-sided orthogonal query rectangle, we propose an O(n logn) space data structure that can be constructed in O(n logn) time and queried upon in O(logn) time. This work proposes the first data structure for counting the number of maximal points in a query range. Das et al. proposed a data structure for the counting version in the word RAM model [WALCOM 2012].

Anil Kishore Kalavagattu

Papers

On Finding Skyline Points for Range Queries in Plane.

Range aggregate maximal points in the plane

On Counting Range Maxima Points in Plane