# On Some Proximity Problems of Colored Sets

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

EI WOS SCOPUS

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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

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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

- 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

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- 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.

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