# On Some Proximity Problems of Colored Sets     J. Comput. Sci. Technol., pp. 879-886, 2014.

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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

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Introduction
• The paper of  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 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  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 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