Space Exploration via Proximity Search
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
374-389
Regular Paper
Sariel
Har-Peled
Sariel Har-Peled
Nirman
Kumar
Nirman Kumar
David M.
Mount
David M. Mount
Benjamin
Raichel
Benjamin Raichel
10.4230/LIPIcs.SOCG.2015.374
L.-E. Andersson and N. F. Stewart. Introduction to the Mathematics of Subdivision Surfaces. SIAM, 2010.
G. Binnig, C. F. Quate, and Ch. Gerber. Atomic force microscope. Phys. Rev. Lett., 56:930-933, Mar 1986.
J. F. Blinn. A generalization of algebraic surface drawing. ACM Trans. Graphics, 1:235-256, 1982.
J.-D. Boissonnat, L. J. Guibas, and S. Oudot. Learning smooth shapes by probing. Comput. Geom. Theory Appl., 37(1):38-58, 2007.
K. L. Clarkson. Coresets, sparse greedy approximation, and the frank-wolfe algorithm. ACM Trans. Algo., 6(4), 2010.
R. Cole and C. K. Yap. Shape from probing. J. Algorithms, 8(1):19-38, 1987.
T. Feder and D. H. Greene. Optimal algorithms for approximate clustering. In Proc. 20th Annu. ACM Sympos. Theory Comput.\CNFSTOC, pages 434-444, 1988.
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.
T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985.
S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. Amer. Math. Soc., 2011.
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.
S. Har-Peled, N. Kumar, D. Mount, and B. Raichel. Space exploration via proximity search. CoRR, abs/1412.1398, 2014.
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.
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.
B. Kalantari. A characterization theorem and an algorithm for A convex hull problem. CoRR, abs/1204.1873, 2012.
B. B. Mandelbrot. The fractal geometry of nature. Macmillan, 1983.
J. M. Mulvey and M. P. Beck. Solving capacitated clustering problems. Euro. J. Oper. Res., 18:339-348, 1984.
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.
S. S. Skiena. Problems in geometric probing. Algorithmica, 4:599-605, 1989.
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.
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.
Wikipedia. Atomic force microscopy - wikipedia, the free encyclopedia, 2014.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode