Abstract
We study the problem of computing low-distortion embeddings in the streaming model. We present streaming algorithms that, given an n-point metric space M, compute an embedding of M into an n-point metric space M′ that preserves a (1 − σ)-fraction of the distances with small distortion (σ is called the slack). Our algorithms use space polylogarithmic in n and the spread of the metric. Within such space limitations, it is impossible to store the embedding explicitly. We bypass this obstacle by computing a compact representation of M′, without storing the actual bijection from M into M′.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Partially supported by DFG grant So 514/1-2.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abraham, I., Bartal, Y., Chan, T.-H., Dhamdhere, K., Gupta, A., Kleinberg, J., Neiman, O., Slivkins, A.: Metric Embeddings with Relaxed Guarantees. In: Proc. 46th IEEE Sympos. Found. Comput. Sci., pp. 83–100 (2005)
Abraham, I., Bartal, Y., Neiman, O.: Advances in metric embedding theory. In: Proc. 38th ACM Sympos. Theory Comput., pp. 271–286 (2006)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2000)
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest neighbors and n-body potential fields. Journal of the ACM 42(1), 67–90 (1995)
Chan, T.-H.H., Dinitz, M., Gupta, A.: Spanners with slack. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 196–207. Springer, Heidelberg (2006)
Czumaj, A., Sohler, C.: Small Space Representations for Metric Min-Sum k-Clustering and their Applications. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 536–548. Springer, Heidelberg (2007)
Fakcharoenphol, J., Rao, S., Talwar, K.: A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics. In: Proc. ACM Sympos. Theory Comput. (2003)
Frahling, G., Indyk, P., Sohler, C.: Sampling in dynamic data streams and applications. In: Proc. 21st ACM Sympos. Comput. Geom., pp. 142–149 (2005)
Frahling, G., Sohler, C.: Coresets in dynamic geometric data streams. In: Proc. 37th ACM Sympos. Theory Comput., pp. 209–217 (2005)
Indyk, P.: Algorithms for Dynamic Geometric Problems over Data Streams. In: Proc. 36th ACM Sympos. Theory Comput., pp. 373–380 (2004)
Johnson, W., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability (1982)
Kleinberg, J., Slivkins, A., Wexler, T.: Triangulation and Embedding using Small Sets of Beacons. In: Proc. 45th IEEE Sympos. Found. Comput. Sci. (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lammersen, C., Sidiropoulos, A., Sohler, C. (2009). Streaming Embeddings with Slack. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_42
Download citation
DOI: https://doi.org/10.1007/978-3-642-03367-4_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
eBook Packages: Computer ScienceComputer Science (R0)