An Instance-optimal Algorithm for Bichromatic Rectangular Visibility

TL;DR

This paper introduces an instance-optimal algorithm for finding bichromatic rectangle empties in the plane, advancing the theory of order-oblivious algorithms to handle non-local geometric features.

Contribution

It extends the concept of instance-optimal algorithms to a non-local geometric problem, specifically bichromatic rectangle empties, which was not previously achieved.

Findings

Proves the existence of an instance-optimal algorithm for the problem.

Adapts and extends existing methods to handle non-local features.

Enhances understanding of order-oblivious algorithm capabilities.

Abstract

Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the order-oblivious setting. An algorithm A is instance-optimal in the order-oblivious setting for a certain class of algorithms A* if the following hold: - A takes as input a sequence of objects from some domain; - for any instance $\sigma$ and any algorithm A' in A*, the runtime of A on $\sigma$ is at most a constant factor removed from the runtime of A' on the worst possible permutation of $\sigma$. If we identify permutations of a sequence as representing the same instance, this essentially states that A is optimal on every possible input (and not only in the worst case). We design instance-optimal algorithms for the problem of reporting, given a bichromatic set of points in the plane S, all pairs consisting of points of different color which span an empty axis-aligned rectangle (or reporting all points which appear in such a pair). This problem has applications for training-set reduction in nearest-neighbour classifiers. It is also related to the problem consisting of finding the decision boundaries of a euclidean nearest-neighbour classifier, for which Bremner et al. (2005) gave an optimal output-sensitive algorithm. By showing the existence of an instance-optimal algorithm in the order-oblivious setting for this problem we push the methods of Afshani et al. closer to their limits by adapting and extending them to a setting which exhibits highly non-local features. Previous problems for which instance-optimal algorithms were proven to exist were based solely on local relationships between points in a set.