Space Exploration via Proximity Search

Authors Sariel Har-Peled, Nirman Kumar, David M. Mount, Benjamin Raichel

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Sariel Har-Peled
Nirman Kumar
David M. Mount
Benjamin Raichel

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Sariel Har-Peled, Nirman Kumar, David M. Mount, and Benjamin Raichel. Space Exploration via Proximity Search. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 374-389, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.
  • Proximity search
  • implicit point set
  • probing


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  1. L.-E. Andersson and N. F. Stewart. Introduction to the Mathematics of Subdivision Surfaces. SIAM, 2010. Google Scholar
  2. G. Binnig, C. F. Quate, and Ch. Gerber. Atomic force microscope. Phys. Rev. Lett., 56:930-933, Mar 1986. Google Scholar
  3. J. F. Blinn. A generalization of algebraic surface drawing. ACM Trans. Graphics, 1:235-256, 1982. Google Scholar
  4. J.-D. Boissonnat, L. J. Guibas, and S. Oudot. Learning smooth shapes by probing. Comput. Geom. Theory Appl., 37(1):38-58, 2007. Google Scholar
  5. K. L. Clarkson. Coresets, sparse greedy approximation, and the frank-wolfe algorithm. ACM Trans. Algo., 6(4), 2010. Google Scholar
  6. R. Cole and C. K. Yap. Shape from probing. J. Algorithms, 8(1):19-38, 1987. Google Scholar
  7. T. Feder and D. H. Greene. Optimal algorithms for approximate clustering. In Proc. 20th Annu. ACM Sympos. Theory Comput.\CNFSTOC, pages 434-444, 1988. Google Scholar
  8. A. Goel, P. Indyk, and K. R. Varadarajan. Reductions among high dimensional proximity problems. In Proc. 12th ACM-SIAM Sympos. Discrete Algs.\CNFSODA, pages 769-778, 2001. Google Scholar
  9. T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985. Google Scholar
  10. S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. Amer. Math. Soc., 2011. Google Scholar
  11. S. Har-Peled, P. Indyk, and R. Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. Theory Comput., 8:321-350, 2012. Special issue in honor of Rajeev Motwani. Google Scholar
  12. S. Har-Peled, N. Kumar, D. Mount, and B. Raichel. Space exploration via proximity search. CoRR, abs/1412.1398, 2014. Google Scholar
  13. S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. SIAM J. Comput., 35(5):1148-1184, 2006. Google Scholar
  14. P. Indyk. Nearest neighbors in high-dimensional spaces. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 39, pages 877-892. CRC Press LLC, 2nd edition, 2004. Google Scholar
  15. B. Kalantari. A characterization theorem and an algorithm for A convex hull problem. CoRR, abs/1204.1873, 2012. Google Scholar
  16. B. B. Mandelbrot. The fractal geometry of nature. Macmillan, 1983. Google Scholar
  17. J. M. Mulvey and M. P. Beck. Solving capacitated clustering problems. Euro. J. Oper. Res., 18:339-348, 1984. Google Scholar
  18. F. Panahi, A. Adler, A. F. van der Stappen, and K. Goldberg. An efficient proximity probing algorithm for metrology. In Proc. IEEE Int. Conf. Autom. Sci. Engin. (CASE), pages 342-349, 2013. Google Scholar
  19. S. S. Skiena. Problems in geometric probing. Algorithmica, 4:599-605, 1989. Google Scholar
  20. S. S. Skiena. Geometric reconstruction problems. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 26, pages 481-490. CRC Press LLC, Boca Raton, FL, 1997. Google Scholar
  21. R. M. Smelik, K. J. De Kraker, S. A. Groenewegen, T. Tutenel, and R. Bidarra. A survey of procedural methods for terrain modelling. In Proc. of the CASA Work. 3D Adv. Media Gaming Simul., 2009. Google Scholar
  22. Wikipedia. Atomic force microscopy - wikipedia, the free encyclopedia, 2014. Google Scholar
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