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# An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility

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## Cite As

Jean Cardinal, Justin Dallant, and John Iacono. An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.24

## 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 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• computational geometry
• instance-optimality
• colored point sets
• empty rectangles
• visibility

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## References

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