Inplace Algorithm for Priority Search Tree and its use in Computing Largest Empty Axis-Parallel Rectangle

TL;DR

This paper introduces space-efficient algorithms for constructing priority search trees and computing the largest empty axis-parallel rectangle among points, with applications in embedded systems and geometric optimization.

Contribution

It presents an in-place priority search tree algorithm with minimal extra space and applies it to find maximum empty rectangles, including arbitrarily oriented ones, efficiently.

Findings

In-place priority search tree construction in O(n log n) time

Efficient computation of largest empty axis-parallel rectangle in O(n log^2 n + m) time

An O(n^3 log n) in-place algorithm for maximum area empty rectangle of arbitrary orientation

Abstract

There is a high demand of space-efficient algorithms in built-in or embedded softwares. In this paper, we consider the problem of designing space-efficient algorithms for computing the maximum area empty rectangle (MER) among a set of points inside a rectangular region $\cal R$ in 2D. We first propose an inplace algorithm for computing the priority search tree with a set of $n$ points in $\cal R$ using $O(\log n)$ extra bit space in $O(n\log n)$ time. It supports all the standard queries on priority search tree in $O(\log^2n)$ time. We also show an application of this algorithm in computing the largest empty axis-parallel rectangle. Our proposed algorithm needs $O(n\log^2n +m)$ time and $O(\log n)$ work-space apart from the array used for storing $n$ input points. Here $m$ is the number of maximal empty rectangles present in $\cal R$. Finally, we consider the problem of locating the maximum area empty rectangle of arbitrary orientation among a set of $n$ points, and propose an $O(n^3\log n)$ time in-place algorithm for that problem.