I/O-Efficient Dynamic Planar Range Skyline Queries

TL;DR

This paper introduces the first fully dynamic I/O-efficient data structures for planar 3-sided range skyline queries, achieving optimal query and update times with space proportional to input size, and establishes lower bounds for related static structures.

Contribution

It extends Sundar's priority queues to support dynamic operations efficiently and proves space lower bounds for static skyline query data structures.

Findings

Supports planar orthogonal 3-sided skyline queries in optimal I/O time.

Supports updates with optimal I/O complexity.

Establishes lower bounds for static skyline query data structures.

Abstract

We present the first fully dynamic worst case I/O-efficient data structures that support planar orthogonal \textit{3-sided range skyline reporting queries} in $\bigO (\log_{2B^\epsilon} n + \frac{t}{B^{1-\epsilon}})$ I/Os and updates in $\bigO (\log_{2B^\epsilon} n)$ I/Os, using $\bigO (\frac{n}{B^{1-\epsilon}})$ blocks of space, for $n$ input planar points, $t$ reported points, and parameter $0 \leq \epsilon \leq 1$. We obtain the result by extending Sundar's priority queues with attrition to support the operations \textsc{DeleteMin} and \textsc{CatenateAndAttrite} in $\bigO (1)$ worst case I/Os, and in $\bigO(1/B)$ amortized I/Os given that a constant number of blocks is already loaded in main memory. Finally, we show that any pointer-based static data structure that supports \textit{dominated maxima reporting queries}, namely the difficult special case of 4-sided skyline queries, in $\bigO(\log^{\bigO(1)}n +t)$ worst case time must occupy $\Omega(n \frac{\log n}{\log \log n})$ space, by adapting a similar lower bounding argument for planar 4-sided range reporting queries.