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Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces

Authors Timothy Carpenter, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Anastasios Sidiropoulos

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Keywords
  • Metric embeddings
  • minimum-distortion embeddings
  • 1-dimensional simplicial complex
  • Fixed-parameter tractable algorithms
  • Approximation algorithms

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Abstract

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give
- an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c);
- an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

Cite As Get BibTex

Timothy Carpenter, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Anastasios Sidiropoulos. Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.21

Author Details

Timothy Carpenter
Fedor V. Fomin
Daniel Lokshtanov
Saket Saurabh
Anastasios Sidiropoulos

References

  1. Sanjeev Arora, James Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. Journal of the American Mathematical Society, 21(1):1-21, 2008. Google Scholar
  2. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):5, 2009. Google Scholar
  3. Mihai Bădoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropoulos. Low-distortion embeddings of general metrics into the line. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 225-233. ACM, 2005. Google Scholar
  4. Mihai Bădoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropoulos. Embedding ultrametrics into low-dimensional spaces. In Proceedings of the 22nd Annual Symposium on Computational Geometry (SoCG), pages 187-196. ACM, 2006. Google Scholar
  5. Mihai Bădoiu, Kedar Dhamdhere, Anupam Gupta, Yuri Rabinovich, Harald Räcke, R. Ravi, and Anastasios Sidiropoulos. Approximation algorithms for low-distortion embeddings into low-dimensional spaces. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 119-128. SIAM, 2005. Google Scholar
  6. Mihai Bădoiu, Piotr Indyk, and Anastasios Sidiropoulos. Approximation algorithms for embedding general metrics into trees. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 512-521. ACM and SIAM, 2007. Google Scholar
  7. Yair Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on Foundations of Computer Science, pages 184-193. IEEE, 1996. Google Scholar
  8. Nishanth Chandran, Ryan Moriarty, Rafail Ostrovsky, Omkant Pandey, Mohammad Ali Safari, and Amit Sahai. Improved algorithms for optimal embeddings. ACM Transactions on Algorithms, 4(4), 2008. URL: http://dx.doi.org/10.1145/1383369.1383376.
  9. Mark de Berg, Krzysztof Onak, and Anastasios Sidiropoulos. Fat polygonal partitions with applications to visualization and embeddings. Journal of Computational Geometry, 4(1):212–239, 2013. Google Scholar
  10. Jeff Edmonds, Anastasios Sidiropoulos, and Anastasios Zouzias. Inapproximability for planar embedding problems. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 222-235. Society for Industrial and Applied Mathematics, 2010. Google Scholar
  11. Martin Farach, Sampath Kannan, and Tandy Warnow. A robust model for finding optimal evolutionary trees. Algorithmica, 13(1-2):155-179, 1995. Google Scholar
  12. Martin Farach-Colton and Piotr Indyk. Approximate nearest neighbor algorithms for hausdorff metrics via embeddings. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 171-179. IEEE, 1999. Google Scholar
  13. Michael Fellows, Fedor V. Fomin, Daniel Lokshtanov, Elena Losievskaja, Frances Rosamond, and Saket Saurabh. Distortion is fixed parameter tractable. ACM Trans. Comput. Theory, 5(4):16:1-16:20, 2013. URL: http://dx.doi.org/10.1145/2489789.
  14. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Elena Losievskaja, Frances A. Rosamond, and Saket Saurabh. Distortion is fixed parameter tractable. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP), volume 5555 of Lecture Notes in Computer Science, pages 463-474. Springer, 2009. Google Scholar
  15. Alexander Hall and Christos Papadimitriou. Approximating the distortion. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 111-122, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  16. Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pages 10-33. IEEE, 2001. Google Scholar
  17. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. Journal of the ACM (JACM), 53(3):307-323, 2006. Google Scholar
  18. Piotr Indyk and Jiri Matousek. Low-distortion embeddings of finite metric spaces. In Handbook of Discrete and Computational Geometry, pages 177-196. CRC Press, 2004. Google Scholar
  19. Claire Kenyon, Yuval Rabani, and Alistair Sinclair. Low distortion maps between point sets. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 272-280. ACM, 2004. Google Scholar
  20. Claire Kenyon, Yuval Rabani, and Alistair Sinclair. Low distortion maps between point sets. SIAM J. Comput., 39(4):1617-1636, 2009. URL: http://dx.doi.org/10.1137/080712921.
  21. Subhash Khot and Rishi Saket. Hardness of embedding metric spaces of equal size. In Approximation, randomization, and combinatorial optimization. Algorithms and techniques, pages 218-227. Springer, 2007. Google Scholar
  22. Nathan Linial. Finite metric-spaces - combinatorics, geometry and algorithms. In Proceedings of the International Congress of Mathematicians, Vol. III, pages 573-586, Beijing, 2002. Higher Ed. Press. Google Scholar
  23. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. Google Scholar
  24. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 760-776. SIAM, 2011. Google Scholar
  25. Jiří Matoušek and Anastasios Sidiropoulos. Inapproximability for metric embeddings into ℝ^d. Transactions of the American Mathematical Society, 362(12):6341-6365, 2010. Google Scholar
  26. Amir Nayyeri and Benjamin Raichel. Reality distortion: Exact and approximate algorithms for embedding into the line. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 729-747. IEEE, 2015. Google Scholar
  27. 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, SODA '17, pages 724-736, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=3039686.3039732.
  28. Christos Papadimitriou and Shmuel Safra. The complexity of low-distortion embeddings between point sets. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), volume 5, pages 112-118. SIAM, 2005. Google Scholar
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