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Dynamic Geometric Set Cover and Hitting Set

Authors Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, Jie Xue

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Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Hsien-Chih Chang
  • Department of Computer Science, Duke University, Durham, NC, USA
Subhash Suri
  • Department of Computer Science, University of California at Santa Barbara, CA, USA
Allen Xiao
  • Department of Computer Science, Duke University, Durham, NC, USA
Jie Xue
  • Department of Computer Science, University of California at Santa Barbara, CA, USA

Funding

  • Agarwal, Pankaj K.: NSF grants CCF-15-13816, CCF-15-46392, and IIS-14-08846; ARO grant W911NF-15-1-0408; BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation.
  • Chang, Hsien-Chih: NSF grants CCF-15-13816, CCF-15-46392, and IIS-14-08846; ARO grant W911NF-15-1-0408; BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation.
  • Suri, Subhash: NSF grant CCF-18-14172.
  • Xiao, Allen: NSF grants CCF-15-13816, CCF-15-46392, and IIS-14-08846; ARO grant W911NF-15-1-0408; BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation.
  • Xue, Jie: NSF grant CCF-18-14172.

Cite As Get BibTex

Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic Geometric Set Cover and Hitting Set. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.2

Abstract

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ¹ and ℝ². The first framework uses bootstrapping and gives a (1+ε)-approximate data structure for dynamic interval set cover in ℝ¹ with O(n^α/ε) amortized update time for any constant α > 0; in ℝ², this method gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set with O(n^(1/2+α)) amortized update time. The second framework uses local modification, and leads to a (1+ε)-approximate data structure for dynamic interval hitting set in ℝ¹ with Õ(1/ε) amortized update time; in ℝ², it gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric set cover
  • Geometric hitting set
  • Dynamic data structures

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References

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