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Worst-Case Efficient Dynamic Geometric Independent Set

Authors Jean Cardinal , John Iacono , Grigorios Koumoutsos

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Data structures design and analysis
Keywords
  • Maximum independent set
  • deamortization
  • approximation

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Abstract

We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present a data structure that maintains a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions, where d is assumed to be a constant, in sublinear worst-case update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4+ε)-approximation for squares, with O(log⁴ n) worst-case update time. 
Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with amortized update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. 
Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.

Cite As Get BibTex

Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-Case Efficient Dynamic Geometric Independent Set. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.25

Author Details

Jean Cardinal
  • Université libre de Bruxelles (ULB), Brussels, Belgium
John Iacono
  • Université libre de Bruxelles (ULB), Brussels, Belgium
Grigorios Koumoutsos
  • Université libre de Bruxelles (ULB), Brussels, Belgium

Funding

  • Iacono, John: Supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001 1.
  • Koumoutsos, Grigorios: Supported by the Fonds de la Recherche Scientifique-FNRS under a "Scientific Collaborator" grant.

Acknowledgements

We would like to thank Timothy Chan and Qizheng He for pointing out an error in the previous version of this paper.

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