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An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility

Authors Jean Cardinal , Justin Dallant, John Iacono

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Author Details

Jean Cardinal
  • Université libre de Bruxelles (ULB), Brussels, Belgium
Justin Dallant
  • Université libre de Bruxelles (ULB), Brussels, Belgium
John Iacono
  • Université libre de Bruxelles (ULB), Brussels, Belgium

Funding

  • Dallant, Justin: This work was supported by the French Community of Belgium via the funding of a FRIA grant.
  • Iacono, John: Supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001 1.

Cite As Get BibTex

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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