An Instance-optimal Algorithm for Bichromatic Rectangular Visibility

Abstract

Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the orderoblivious 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 σ and any algorithm A′ ∈ A, the runtime of A on σ is at most a constant factor removed from the runtime of A′ on the worst possible permutation of σ. 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. 2012 ACM Subject Classification Theory of computation → Computational geometry