Approximation algorithms for finding maximum containing circle and sphere

We first study maximum containing circle problem. The input to the problem is a weighted set of points and a circle of fixed radius, and the output is a suitable location of the circle such that the sum of the weights of the points covered by the circle is maximized. We propose a special polygon, called symmetrical rectilinear polygon (SRP). In this paper, we give a method for constructing the circumscribed SRP of a circle and prove the area relationship between this polygon and the circle. We solve the maximum containing SRP problem exactly, and based on this, give an algorithm for solving the (1 - E )-approximation of maximum containing circle problem. We also show that the algorithm is valid for most inputs. It only needs ( ( 1 )) O nE -1 log n + nE -1 log time for unit points and o (nE -2 logn) n) time for weighted points. Due E to its low time complexity, the algorithm can run as a stand-alone algorithm or as a preprocessor for other algorithms. As an extension of our work, we discuss a 3D version of the unweighted maximum containing circle problem, i.e., containing the maximum number of points with a given sphere. We give a (1 - E )-approximation algorithm for this problem that returns correct results in (( ) ( max{O { O n 2 3 E -3 / log2 2 3 (nE-2) nE -2 ) (loglog (nE-2))O(1)) nE -2 )) O (1) ) , o n 2 (E -1 log n)2) 2 ) } time for most cases.