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Anonymity-Preserving Space Partitions

Authors Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, Vaishali Surianarayanan

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Author Details

Úrsula Hébert-Johnson
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Chinmay Sonar
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Subhash Suri
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Vaishali Surianarayanan
  • Department of Computer Science, University of California, Santa Barbara, CA, USA

Funding

  • Suri, Subhash: in part supported by NSF grant CCF-1814172.

Acknowledgements

We thank Daniel Lokshtanov for discussions relating to our approximation algorithm for Anonymity-Preserving Partition, as well as an anonymous reviewer, whose comments helped to improve the aforementioned result.

Cite As Get BibTex

Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, and Vaishali Surianarayanan. Anonymity-Preserving Space Partitions. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.32

Abstract

We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ℝ^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ∩ P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ≥ 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Anonymity
  • Hitting Set
  • LP
  • Constant Approximation
  • Fixed-Parameter Tractable
  • Space Partitions
  • Parameterized Complexity

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References

  1. P K Agarwal. Geometric Partitioning and Its Applications. Discrete and Computational Geometry: Papers from the DIMACS Special Year (J. Goodman, R. Pollack, and W. Steiger, eds.), pages 1-37, 1991. Google Scholar
  2. Pankaj K Agarwal and Micha Sharir. Applications of a new space-partitioning technique. Discrete & Computational Geometry, 9(1):11-38, 1993. Google Scholar
  3. Ron Aharoni, Ron Holzman, and Michael Krivelevich. On a Theorem of Lovász on Covers in r-partite Hypergraphs. Combinatorica, 16(2):149-174, 1996. Google Scholar
  4. Walid G Aref and Ihab F Ilyas. Sp-gist: An extensible database index for supporting space partitioning trees. Journal of Intelligent Information Systems, 17(2-3):215-240, 2001. Google Scholar
  5. J. Baltes and J. Anderson. Flexible binary space partitioning for robotic rescue. In Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453), volume 4, pages 3144-3149 vol.3, 2003. Google Scholar
  6. Norbert Beckmann, Hans-Peter Kriegel, Ralf Schneider, and Bernhard Seeger. The R*-tree: an efficient and robust access method for points and rectangles. In Proceedings of the 1990 ACM SIGMOD international conference on Management of data, pages 322-331, 1990. Google Scholar
  7. Jon Louis Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509-517, 1975. Google Scholar
  8. Hervé Brönnimann and Michael T Goodrich. Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995. Google Scholar
  9. J. Cortes, S. Martinez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243-255, 2004. Google Scholar
  10. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. Google Scholar
  11. Pratyush Dayal and Neeldhara Misra. Deleting to Structured Trees. In International Computing and Combinatorics Conference, pages 128-139. Springer, 2019. Google Scholar
  12. Adrian Dumitrescu, Joseph SB Mitchell, and Micha Sharir. Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles. Discrete & Computational Geometry, 31(2):207-227, 2004. Google Scholar
  13. Guy Even, Dror Rawitz, and Shimon Moni Shahar. Hitting sets when the vc-dimension is small. Information Processing Letters, 95(2):358-362, 2005. Google Scholar
  14. Fedor V Fomin, Serge Gaspers, Dieter Kratsch, Mathieu Liedloff, and Saket Saurabh. Iterative compression and exact algorithms. Theoretical Computer Science, 411(7-9):1045-1053, 2010. Google Scholar
  15. Sanjeev Khanna, S. Muthukrishnan, and Mike Paterson. On Approximating Rectangle Tiling and Packing. In Proceedings of the ninth annual ACM-SIAM Symposium On Discrete Algorithms, volume 95, page 384. SIAM, 1998. Google Scholar
  16. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  17. Kristen LeFevre, David J DeWitt, and Raghu Ramakrishnan. Mondrian multidimensional k-anonymity. In 22nd International conference on data engineering (ICDE'06), pages 25-25. IEEE, 2006. Google Scholar
  18. Jiri Matousek. Lectures on discrete geometry, volume 212. Springer Science & Business Media, 2013. Google Scholar
  19. Rolf Niedermeier and Peter Rossmanith. An efficient fixed-parameter algorithm for 3-hitting set. Journal of Discrete Algorithms, 1(1):89-102, 2003. Google Scholar
  20. Hanan Samet. The quadtree and related hierarchical data structures. ACM Computing Surveys (CSUR), 16(2):187-260, 1984. Google Scholar
  21. Adam Smith and Subhash Suri. Rectangular tiling in multidimensional arrays. Journal of Algorithms, 37(2):451-467, 2000. Google Scholar
  22. Marc Van Kreveld, Otfried Schwarzkopf, Mark de Berg, and Mark Overmars. Computational geometry algorithms and applications. Springer, 2000. Google Scholar
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