Two Algorithms for the Minimum Enclosing Ball Problem

Abstract

Abstract. Given A := {a 1 , . . . , a m } ⊂ R n and > 0, we propose and analyze two algorithms for the problem of computing a (1 + )-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/ ) iterations with an overall complexity bound of O(mn/ ) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ ), 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/ ) for a much wider class of input sets. We adopt the real number model of computation in our analysis.

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“…In [1] and [20], algorithms to compute (1 + )-MEBs that scale independently of the dimension of Z and the cardinality of A have been provided. These algorithms are built on the concept of -core set for A, that is, a subset C ⊂ A whose MEB is a (1 + )-MEB of A.…”

“…For this purpose we adopt a variant of the Frank-Wolfe algorithm, recently studied in [3] and [20]. The Frank-Wolfe method (see Algorithm 2, where e i denotes the i-th vector of the canonical basis of R m ) solves the general problem of maximizing a concave function g(α) on the unit simplex.…”

“…Moreover, the updating formula for α shows that only the i * -th entry of α can become nonzero at each iteration. The optimization step 4 is analytic: indeed, it can be shown [3,20] that…”

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