Adaptive Techniques to find Optimal Planar Boxes

TL;DR

This paper introduces an adaptive algorithm for the Optimal Planar Box problem that efficiently finds the best axis-aligned rectangle based on a monotone decomposable score function, with improved data structures for dynamic set management.

Contribution

It presents a new adaptive solution with worst-case $O(n^2\lg n)$ complexity and a novel fully dynamic MCS Splay tree supporting insertions and deletions efficiently.

Findings

Algorithm performs efficiently on various instance classes.

Provides a new dynamic Splay tree with the dynamic finger property.

Improves upon previous data structure results.

Abstract

Given a set $P$ of $n$ planar points, two axes and a real-valued score function $f()$ on subsets of $P$, the Optimal Planar Box problem consists in finding a box (i.e. axis-aligned rectangle) $H$ maximizing $f(H\cap P)$. We consider the case where $f()$ is monotone decomposable, i.e. there exists a composition function $g()$ monotone in its two arguments such that $f(A)=g(f(A_1),f(A_2))$ for every subset $A\subseteq P$ and every partition $\{A_1,A_2\}$ of $A$. In this context we propose a solution for the Optimal Planar Box problem which performs in the worst case $O(n^2\lg n)$ score compositions and coordinate comparisons, and much less on other classes of instances defined by various measures of difficulty. A side result of its own interest is a fully dynamic \textit{MCS Splay tree} data structure supporting insertions and deletions with the \emph{dynamic finger} property, improving upon previous results [Cort\'es et al., J.Alg. 2009].