A faster dual algorithm for the Euclidean minimum covering ball problem

Abstract

Dearing and Zeck (2009) presented a dual algorithm for the problem of the minimum covering ball in R n . Each iteration of their algorithm has a computational complexity of at least O(n 3 ). In this paper we propose a modification to their algorithm that, together with an implementation that uses updates to the QR factorization of a suitable matrix, achieves a O(n 2 ) iteration.Keywords minimum covering ball · smallest enclosing ball · 1-center · minmax location · computational geometry 1 IntroductionConsider a given set of points P = {p 1 , . . ., p m } in the Euclidean space R n . Let . denote the Euclidean norm. The problem of finding the hypersphere B(x, r) = {y ∈ R n : y − x ≤ r} with minimum radius that covers P, which we will refer to as the minimum covering ball (MB) of P, can be formulated asWe will use the notation MB(P) both to refer to the problem of the minimum covering ball of a set P and, depending on the context, also to the corresponding optimal ball.The MB problem, reported to date back to the 19th century (Sylvester, 1857), is an important and active problem in computational geometry and optimization. Applications include facility location, see e.g. Hale and Moberg (

Primal and dual algorithms for the minimum covering Euclidean ball of a set of Euclidean balls in $\mathbb{R}^n$

“…Dearing and Zeck [6] reported a dual algorithm based on search paths constructed from bisectors of pairs of points. Cavaleiro and Alizadeh [3] present computational improvements to the Dearing and Zeck approach.…”

“…Cavaleiro and Alizadeh [3] present an equivalent procedure for finding α * . The leaving point p i l ∈ S is chosen so that α * = α i l .…”

A dual Simplex-type algorithm for the smallest enclosing ball of balls and related problems

“…Using related ideas, Dearing and Zeck [5] developed a dual algorithm for the MB problem. This algorithm was further improved in [4]. Several approximation algorithms have also been developed focusing on finding an ǫ-core set, [3], that is a subset of S ⊂ P that has the property that the smallest ball containing S once expanded by 1 + ǫ covers P. A surprising fact is the existence of an ǫ-core set of size at most ⌈ 1 ǫ ⌉, independent of the dimension n, for any point set P ⊂ R n , [14,2].…”