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Computing Bi-Lipschitz Outlier Embeddings into the Line

Authors Karine Chubarian, Anastasios Sidiropoulos

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Author Details

Karine Chubarian
  • Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, IL, USA
Anastasios Sidiropoulos
  • Department of Computer Science, University of Illinois at Chicago, IL, USA

Funding

  • Sidiropoulos, Anastasios: Supported by NSF grants CCF-1815145, CCF-1934915, and by NSF CAREER award 1453472.

Cite AsGet BibTex

Karine Chubarian and Anastasios Sidiropoulos. Computing Bi-Lipschitz Outlier Embeddings into the Line. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.36

Abstract

The problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion O(c²), where c denotes the optimal distortion [Bădoiu et al. 2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of outliers. Specifically, we say that a metric space (X,ρ) admits a (k,c)-embedding if there exists K ⊂ X, with |K| = k, such that (X⧵ K, ρ) admits an embedding into the line with distortion at most c. Given k ≥ 0, and a metric space that admits a (k,c)-embedding, for some c ≥ 1, our algorithm computes a (poly(k, c, log n), poly(c))-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • metric embeddings
  • outliers
  • distortion
  • approximation algorithms

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References

  1. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12(3):289-297, 1999. Google Scholar
  2. Karol Borsuk. Drei sätze über die n-dimensionale euklidische sphäre. Fundamenta Mathematicae, 20(1):177-190, 1933. Google Scholar
  3. Mihai Bădoiu, Piotr Indyk, and Anastasios Sidiropoulos. Approximation algorithms for embedding general metrics into trees. In Nikhil Bansal, Kirk Pruhs, and Clifford Stein, editors, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 512-521. SIAM, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283438.
  4. Mihai Bǎdoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropou. Embedding ultrametrics into low-dimensional spaces. In Proceedings of the twenty-second annual symposium on Computational geometry, pages 187-196, 2006. Google Scholar
  5. Mihai Bǎdoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropoulos. Low-distortion embeddings of general metrics into the line. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 225-233, 2005. Google Scholar
  6. Mihai Bǎdoiu, Kedar Dhamdhere, Anupam Gupta, Yuri Rabinovich, Harald Räcke, Ramamoorthi Ravi, and Anastasios Sidiropoulos. Approximation algorithms for low-distortion embeddings into low-dimensional spaces. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 119-128. Society for Industrial and Applied Mathematics, 2005. Google Scholar
  7. Timothy Carpenter, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Anastasios Sidiropoulos. Algorithms for low-distortion embeddings into arbitrary 1-dimensional spaces. In Bettina Speckmann and Csaba D. Tóth, editors, 34th International Symposium on Computational Geometry, SoCG 2018, June 11-14, 2018, Budapest, Hungary, volume 99 of LIPIcs, pages 21:1-21:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.21.
  8. Victor Chepoi, Feodor F Dragan, Ilan Newman, Yuri Rabinovich, and Yann Vaxes. Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs. Discrete & Computational Geometry, 47(1):187-214, 2012. Google Scholar
  9. Vasek Chvatal. A greedy heuristic for the set-covering problem. Mathematics of operations research, 4(3):233-235, 1979. Google Scholar
  10. Mark de Berg, Krzysztof Onak, and Anastasios Sidiropoulos. Fat polygonal partitions with applications to visualization and embeddings. arXiv preprint, 2010. URL: http://arxiv.org/abs/1009.1866.
  11. Uriel Feige, MohammadTaghi Hajiaghayi, and James R Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM Journal on Computing, 38(2):629-657, 2008. Google Scholar
  12. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Elena Losievskaja, Frances A. Rosamond, and Saket Saurabh. Distortion is fixed parameter tractable. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris E. Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, volume 5555 of Lecture Notes in Computer Science, pages 463-474. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_39.
  13. Ian Goodfellow, Patrick McDaniel, and Nicolas Papernot. Making machine learning robust against adversarial inputs. Communications of the ACM, 61(7):56-66, 2018. Google Scholar
  14. Piotr Indyk, Jiří Matoušek, and Anastasios Sidiropoulos. 8: low-distortion embeddings of finite metric spaces. In Handbook of discrete and computational geometry, pages 211-231. Chapman and Hall/CRC, 2017. Google Scholar
  15. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 760-776. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.60.
  16. Jiří Matoušek and Anastasios Sidiropoulos. Inapproximability for metric embeddings into r^d. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 405-413. IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.21.
  17. Amir Nayyeri and Benjamin Raichel. Reality distortion: Exact and approximate algorithms for embedding into the line. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 729-747. IEEE, 2015. Google Scholar
  18. Amir Nayyeri and Benjamin Raichel. A treehouse with custom windows: Minimum distortion embeddings into bounded treewidth graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 724-736. SIAM, 2017. Google Scholar
  19. Anastasios Sidiropoulos, Dingkang Wang, and Yusu Wang. Metric embeddings with outliers. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 670-689. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.43.
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