License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.25
URN: urn:nbn:de:0030-drops-146061
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14606/
Cardinal, Jean ;
Iacono, John ;
Koumoutsos, Grigorios
Worst-Case Efficient Dynamic Geometric Independent Set
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.
BibTeX - Entry
@InProceedings{cardinal_et_al:LIPIcs.ESA.2021.25,
author = {Cardinal, Jean and Iacono, John and Koumoutsos, Grigorios},
title = {{Worst-Case Efficient Dynamic Geometric Independent Set}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {25:1--25:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14606},
URN = {urn:nbn:de:0030-drops-146061},
doi = {10.4230/LIPIcs.ESA.2021.25},
annote = {Keywords: Maximum independent set, deamortization, approximation}
}
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Keywords: |
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Maximum independent set, deamortization, approximation |
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Collection: |
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29th Annual European Symposium on Algorithms (ESA 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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31.08.2021 |
