ABSTRACT
A low-distortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Low-distortion embeddings have recently found numerous applications in computer science.Most of the known embedding results are "absolute",that is, of the form: any metric Y from a given class of metrics C can be embedded into a metric X with low distortion c. This is beneficial if one can guarantee low distortion for all metrics Y in C. However, in any situations, the worst-case distortion is too large to be meaningful. For example, if X is a line metric, then even very simple metrics (an n - point star or an n -point cycle) are embeddable into X only with distortion linear in n. Nevertheless, embeddings into the line (or into low-dimensional spaces) are important for many applications.A solution to this issue is to consider "relative" (or "approximation") embedding problems, where the goal is to design an (a-approxiation) algorithm which, given any metric X from C as an input, finds an embedding of X into Y which has distortion a *cY (X), where cY (X)is the best possible distortion of an embedding of X into Y.In this paper we show algorithms and hardness results for relative embedding problems.In particular we give: •an algorith that, given a general metric M, finds an embedding with distortion O (Δ3⁄4 poly(c line (M))), where Δ is the spread of M•an algorithm that,given a weighted tree etric M, finds an embedding with distortion poly(c line (M)) •a hardness result, showing that computing minimum line distortion is hard to approximate up to a factor polynomial in n,even for weighted tree metrics with spread Δ=n O (1).
- R. Agarwala, V. Bafna, M. Farach-Colton, B. Narayanan, M. Paterson, and M. Thorup. On the approximability of numerical taxonomy: (fitting distances by tree metrics. 7th Symposium on Discrete Algorithms, 1996.]] Google ScholarDigital Library
- M. Bǎdoiu. Approximation algorithm for embedding metrics into a two-dimensional space. 14th Annual ACM-SIAM Symposium on Discrete Algorithms, 2003.]] Google ScholarDigital Library
- M. Bǎdoiu, E. Demaine, M. Hajiaghai, and P. Indyk. Embeddings with extra information. Proceedings of the ACM Symposium on Computational Geometry, 2004.]] Google ScholarDigital Library
- M. Bǎdoiu, K. Dhamdhere, A. Gupta, Y. Rabinovich, H. Raecke, R. Ravi, and A. Sidiropoulos. Approximation algorithms for low-distortion embeddings into low-dimensional spaces. Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 2005.]] Google ScholarDigital Library
- M. Bǎdoiu, P. Indyk, and A. Sidiropoulos. A constant-factor approximation algorithm for embedding unweighted graphs into trees. AI Lab Technical Memo AIM-2004-015, 2004.]]Google Scholar
- K. Dhamdhere. Approximating additive distortion of line embeddings. Proceedings of RANDOM-APPROX, 2004.]]Google Scholar
- K. Dhamdhere, A. Gupta, and R. Ravi. Approximating average distortion for embeddings into line. Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), 2004.]]Google Scholar
- J. Dunagan and S. Vempala. On euclidean embeddings and bandwidth minimization. Proceedings of the 5th Workshop on Randomization and Approximation, 2001.]] Google ScholarDigital Library
- Y. Emek and D. Peleg. Approximating minimum max-stretch spanning trees on unweighted graphs. Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 2004.]] Google ScholarDigital Library
- M. Farach-Colton, S. Kannan, and T. Warnow. A robust model for finding optimal evolutionary tree. Annual ACM Symposium on Theory of Computing, 1993.]] Google ScholarDigital Library
- U. Feige. Approximating the bandwidth via volume respecting embeddings. Journal of Computer and System Sciences, 60(3):510--539, 2000.]] Google ScholarDigital Library
- J. Hastad, L. Ivansson, and J. Lagergren. Fitting points on the real line and its application to rh mapping. Lecture Notes in Computer Science, 1461:465--467, 1998.]] Google ScholarDigital Library
- P. Indyk. Tutorial: Algorithmic applications of low-distortion geometric embeddings. Annual Symposium on Foundations of Computer Science, 2001.]] Google ScholarDigital Library
- C. Kenyon, Y. Rabani, and A. Sinclair. Low distortion maps between point sets. Annual ACM Symposium on Theory of Computing, 2004.]] Google ScholarDigital Library
- N. Linial. Finite metric spaces - combinatorics, geometry and algorithms. Proceedings of the International Congress of Mathematicians III, pages 573--586, 2002.]]Google Scholar
- N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Proceedings of 35th Annual IEEE Symposium on Foundations of Computer Science, pages 577--591, 1994.]]Google ScholarDigital Library
- J. Matoušek. Bi-lipschitz embeddings into low-dimensional euclidean spaces. Comment. Math. Univ. Carolinae, 31:589--600, 1990.]]Google Scholar
- MDS: Working Group on Algorithms for Multidimensional Scaling. Algorithms for multidimensional scaling. DIMACS Web Page.]]Google Scholar
- C. Papadimitriou and S. Safra. The complexity of low-distortion embeddings between point sets. Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 112--118, 2005.]] Google ScholarDigital Library
- J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. http://isomap.stanford.edu/.]]Google Scholar
- W. Unger. The complexity of the approximation of the bandwidth problem. Annual Symposium on Foundations of Computer Science, 1998.]] Google ScholarDigital Library
Index Terms
- Low-distortion embeddings of general metrics into the line
Recommendations
Clan embeddings into trees, and low treewidth graphs
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of ComputingIn low distortion metric embeddings, the goal is to embed a host “hard” metric space into a “simpler” target space while approximately preserving pairwise distances. A highly desirable target space is that of a tree metric. Unfortunately, such embedding ...
On Average Distortion of Embedding Metrics into the Line
We introduce and study the notion of the average distortion of a nonexpanding embedding of one metric space into another. Less sensitive than the multiplicative metric distortion, the average distortion captures well the global picture and, overall, is ...
Approximation Algorithms for Low-Distortion Embeddings into Low-Dimensional Spaces
We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an $O(\sqrt{n})$-approximation algorithm for the problem of finding a line embedding of a ...
Comments