Browsing by Subject "Minimum enclosing balls"

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Open Access
An algorithm and a core set result for the weighted euclidean one-center problem
(Institute for Operations Research and the Management Sciences (I N F O R M S), 2009) Kumar, P.; Yıldırım, A. E.
Given a set A of m points in n-dimensional space with corresponding positive weights, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point c A n that minimizes the maximum weighted Euclidean distance from c A to each point in A In this paper, given ε > 0, we propose and analyze an algorithm that computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem. Our algorithm explicitly constructs a small subset X ⊆ A, called an ε-core set of A, for which the optimal solution of the corresponding weighted Euclidean one-center problem is a close approximation to that of A. In addition, we establish that \X\ depends only on ε and on the ratio of the smallest and largest weights, but is independent of the number of points m and the dimension n. This result subsumes and generalizes the previously known core set results for the minimum enclosing ball problem. Our algorithm computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem for A in O(mn\X\) arithmetic operations. Our computational results indicate that the size of the ε-core set computed by the algorithm is, in general, significantly smaller than the theoretical worst-case estimate, which contributes to the efficiency of the algorithm, especially for large-scale instances. We shed some light on the possible reasons for this discrepancy between the theoretical estimate and the practical performance.
Open Access
Identification and elimination of interior points for the minimum enclosing ball problem
(Society for Industrial and Applied Mathematics, 2008) Ahıpaşaoǧlu, S. D.; Yıldırım, E. A.
Given A := {a1,...,am} C ℝn, we consider the problem of reducing the input set for the computation of the minimum enclosing ball of A. In this note, given an approximate solution to the minimum enclosing ball problem, we propose a simple procedure to identify and eliminate points in.A that are guaranteed to lie in the interior of the minimum-radius ball enclosing A. Our computational results reveal that incorporating this procedure into two recent algorithms proposed by Yildirim lead to significant speed-ups in running times especially for randomly generated largescale problems. We also illustrate that the extra overhead due to the elimination procedure remains at an acceptable level for spherical or almost spherical input sets.
Open Access
Two algorithms for the minimum enclosing ball problem
(Society for Industrial and Applied Mathematics, 2008) Yıldırım, E. A.
Given A := {a1.....am} ⊂ ℝn and ε > 0, we propose and analyze two algorithms for the problem of computing a (1 + ε)-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ε) iterations with an overall complexity bound of O(mn/ε) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ε), which is independent of both m and n. The latter algorithm is obtained by incorporating "away" steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice, since, in contrast to the former one, it allows "dropping" points from the working core set at each iteration. Our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small. In contrast to the first algorithm, we also establish that the second algorithm asymptotically exhibits linear convergence, which provides further insight into our computational results, indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size. In particular, we establish the existence of a core set of size O(1/ε) for a much wider class of input sets. We adopt the real number model of computation in our analysis.