# On Some Proximity Problems of Colored Sets

J. Comput. Sci. Technol., pp. 879-886, 2014.

Cited by: 4|Views9
EI WOS SCOPUS
Weibo:
Our algorithm can be used to solve the maximum diameter problem of imprecise points modeled as polygons since the candidate pair of points must be vertices of two polygons, and the vertices of each polygons are painted in the same color

Abstract:

The maximum diameter color-spanning set problem (MaxDCS) is defined as follows: given n points with m colors, select m points with m distinct colors such that the diameter of the set of chosen points is maximized. In this paper, we design an optimal O (n log n) time algorithm using rotating calipers for MaxDCS problem in the plane. Our al...More

Code:

Data:

Full Text
Bibtex
Weibo
Introduction
• The paper of [4] is completely devoted to this problem and proposes several efficient algorithms.
• These algorithms are based on the assumption that the positions of input points are precise.
• If a point may randomly appear at one of the many candidate positions, which are painted with the same color, how to compute the maximum possible diameter of the point set with different colors?
• Some uncertain factors interfere with the accuracy of the data and the company wants to know the worst cost based on those imprecise data
Highlights
• Computing the diameter of a set of n points in a ddimensional space (d = 1, 2, 3, . . .) has a long history of research[1,2]
• If a point may randomly appear at one of the many candidate positions, which are painted with the same color, how to compute the maximum possible diameter of the point set with different colors? The problem is called the maximum diameter color-spanning set (MaxDCS) problem
• We proposed an optimal O(n log n) time algorithm for the maximum diameter colorspanning set problem
• Our algorithm can be used to solve the maximum diameter problem of imprecise points modeled as polygons since the candidate pair of points must be vertices of two polygons, and the vertices of each polygons are painted in the same color
• For the query of the farthest foreign neighbor in two dimensions, we proposed O(log n) query time algorithms with O(n log n) preprocessing time and O(n) preprocessing space
Results
• O(n log2 n)[17,18]. O(n log n) AFFN(2) None O(n log n) FFNQ(2) O(log n) FFNQ(3) O(log2 n) CPCS(d)

Tdmin(2, n) log m[17] Tdmin(2, n).
Conclusion
• The authors proposed an optimal O(n log n) time algorithm for the maximum diameter colorspanning set problem.
• For the query of the farthest foreign neighbor in two dimensions, the authors proposed O(log n) query time algorithms with O(n log n) preprocessing time and O(n) preprocessing space.
• For the three-dimensional query problems, the authors gave O(log2 n) query time algorithms with O(f n log n) preprocessing time and O(f n log n) preprocessing space, where f is the size of farthest point Delaunay triangulation of P.
• The authors will focus on the problems of computing the farthest foreign pair in higher dimensional space, and approximate nearest neighbor query of color point set
Summary
• ## Introduction:

The paper of [4] is completely devoted to this problem and proposes several efficient algorithms.
• These algorithms are based on the assumption that the positions of input points are precise.
• If a point may randomly appear at one of the many candidate positions, which are painted with the same color, how to compute the maximum possible diameter of the point set with different colors?
• Some uncertain factors interfere with the accuracy of the data and the company wants to know the worst cost based on those imprecise data
• ## Results:

O(n log2 n)[17,18]. O(n log n) AFFN(2) None O(n log n) FFNQ(2) O(log n) FFNQ(3) O(log2 n) CPCS(d)

Tdmin(2, n) log m[17] Tdmin(2, n).
• ## Conclusion:

The authors proposed an optimal O(n log n) time algorithm for the maximum diameter colorspanning set problem.
• For the query of the farthest foreign neighbor in two dimensions, the authors proposed O(log n) query time algorithms with O(n log n) preprocessing time and O(n) preprocessing space.
• For the three-dimensional query problems, the authors gave O(log2 n) query time algorithms with O(f n log n) preprocessing time and O(f n log n) preprocessing space, where f is the size of farthest point Delaunay triangulation of P.
• The authors will focus on the problems of computing the farthest foreign pair in higher dimensional space, and approximate nearest neighbor query of color point set
Tables
• Table1: Overview of the Time Complexity of Various
Funding
• Regular Paper This research was supported by the International Science and Technology Cooperation Program of China under Grant No 2010DFA92720, and the National Natural Science Foundation of China under Grant Nos. 11271351, 60928006, and 61379087
Reference
• Shamos M I. Computational geometry [Ph.D. Thesis]. Yale University, 1978.
• Toussaint G. Solving geometric problems with the rotating calipers. In Proc. MELECON, May 1983.
• Preparata F P, Shamos M I. Computational Geometry: An Introduction. New York, NY, USA: Springer-Verlag, 1985.
• Malandain G, Boissonnat J. Computing the diameter of a point set. International Journal of Computational Geometry and Applications, 2002, 12(6): 489-509.
• Kreveld M V, Loffler M. Largest bounding box, smallest diameter, and related problems on imprecise points. In Proc. the 10th WADS, Aug. 2007, pp.446-457.
• Kamousi P, Chan T M, Suri S. Stochastic minimum spanning trees in Euclidean spaces. In Proc. the 27th Annual ACM Symp. Computational Geometry, June 2011, pp.65-74.
• Agarwal P K, Efrat A, Sankararaman S et al. Nearest-neighbor searching under uncertainty. In Proc. the 31st Symp. Principles of Database Systems, May 2012, pp.225-236.
• Suri S, Verbeek K, Yildiz H. On the most likely convex hull of uncertain points. In Proc. the 21st European Symp. Algorithms, Sept. 2013, pp.791-802.
• Zhang D, Chee Y M, Mondal A, Tung A K H, Kitsuregawa M. Keyword search in spatial databases: Towards searching by document. In Proc. the 25th IEEE International Conference on Data Engineering, Mar. 29-Apr. 2, 2009, pp.688-699.
• Chen Y, Chen S, Gu Y et al. MarcoPolo: A community system for sharing and integrating travel information on maps. In Proc. the 12th EDBT, Mar. 2009, pp.1148-1151.
• Fleischer R, Xu X. Computing minimum diameter colorspanning sets. In Proc. the 4th FAW, Aug. 2010, pp.285-292.
• Ju W, Fan C, Luo J, Zhu B, Daescu O. On some geometric problems of color-spanning sets. Journal of Combinatorial Optimization, 2013, 26(2): 266-283.
• Vaidya P M. An O(n log n) algorithm for the all-nearest-neighbors problem. Discrete Comput. Geom., 1989, 4(2): 101115.
• Agarwal P K, Matousek J, Suri S. Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Computational Geometry, 1992, 1(4): 189-201.
• Cheong O, Shin C S, Vigneron A. Computing farthest neighbors on a convex polytope. Theor. Comput. Sci.: Computing and Combinatorics, 2003, 296(1): 47-58.
• Agarwal P K, Edelsbrunner H, Schwarzkopf O, Welzl E. Euclidean minimum spanning trees and bichromatic closest pairs. In Proc. the 6th SoCG, June 1990, pp.203-210.
• Dumitrescu A, Guha S. Extreme distances in multicolored point sets. In Proc. Int. Conf. Computational Science, Part III, April 2002, pp.14-25.
• Ramos E A. An optimal deterministic algorithm for computing the diameter of a three-dimensional point set. Discrete and Computational Geometry, 2001, 26(2): 233-244.
• Aggarwal A, Edelsbrunner H, Raghavan P, Tiwari P. Optimal time bounds for some proximity problems in the plane. Information Processing Letters, 1992, 42(1): 55-60.
• de Berg M, Cheong O, van Kreveld M, Overmars M. Computational Geometry (3rd edition), Springer-Verlag, 2008.
• Snoeyink J. Point location. In Handbook of Discrete and Computational Geometry (2nd edition), Goodman J E, O’Rourke J (eds.), 2004, pp.559-574.
• Klee V. On the complexity of d-dimensional Voronoi diagrams. Archiv der Mathematik, 1980, 34: 75-80.
• Chazelle B. An optimal convex hull algorithm and new results on cuttings. In Proc. the 32nd Annu. IEEE Symp. Foundation of Computer Science, Oct. 1991, pp.29-38.
• Dwyer R A. Higher-dimensional Voronoi diagrams in linear expected time. Discrete & Computational Geometry, 1991, 6(1): 343-367.
• Chan T M, Snoeyink J, Yap C K. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 1997, 18(4): 433-454.
• Callahan P B, Kosaraju S R. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 1995, 42(1): 67-90.
• Har-Peled S, Mendel M. Fast construction of nets in low dimensional metrics, and their applications. SIAM J. Comput., 2006, 35(5): 1148-1184.
• Har-Peled S. Geometric Approximation Algorithms. American Mathematical Society, 2011. Cheng-Lin Fan is an assistant researcher in Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences. He received his M.S. degree in computer science and technology from Central South University, Changsha, in 2011. His research interests are in algorithm design and analysis, and computational geometry.
• Jun Luo received his Ph.D. degree in computer science from the University of Texas at Dallas, USA, in 2006. Then he spent two years as a postdoctoral researcher at Utrecht University, the Netherlands. He is currently a researcher in Huawei Noah’s Ark Laboratory in Hong Kong. Before that he was an associate professor in Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences. His research interests are in algorithm design and analysis, computational geometry, GIS, and data mining.
• Wen-Cheng Wang is a professor in the State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, where he leads a research group on computer graphics and image processing. He received his Ph.D. degree in software from the Institute of Software, Chinese Academy of Sciences, in 1998, and the Excellent Ph.D. Dissertation Award from the Degree Committee of State Council of China and Ministry of Education of China in 2001. His research interests include computational geometry, visualization, virtual reality, and image editing. He is a member of ACM and IEEE.
• Fa-Rong Zhong is a professor of computer science at Zhejiang Normal University, Jinhua. He received his Ph.D. degree in computer science from Shanghai Jiao Tong University in 2005. His research interests include BDD-based network reliability analysis and algorithms.
• Binhai Zhu is a professor in computer science at Montana State University, USA. He obtained his Ph.D. degree in computer science from McGill University, Canada, in 1994. He was a postdoctoral research associate at Los Alamos National Laboratory, from 1994 to 1996. From 1996 to 2000, he was an assistant professor at City University of Hong Kong. He has been at Montana State University since 2000 (associate professor until 2006, professor since 2006). Professor Zhu’s research interests are geometric computing, biological/geometric modeling, bioinformatics and combinatorial optimization. He is a member of ACM.