Abstract
Given a point setP of the general dimension we present a recursive algorithm for finding the sphere with the smallest radius which contains all points ofP. For given point setsQ (1), …,Q (l) we extend the algorithm so that it findsl concentric spheres with the smallest sum of radii such that each sphere covers the corresponding point set.
Similar content being viewed by others
References
M. Avriel, Nonlinear Programming: Analysis and Methods. Prentice-Hall, Englewood, Cliffs, 1976.
R. Chandrasekaran, The weighted Euclidean 1-center problem. Oper. Res. Lett.,1 (1982), 111–112.
R. Chandrasekaran and M.J.A.P. Pacca, Weighted min-max location problems: Polynomially bounded algorithm. J. Oper. Res. Soc. India,17 (1980), 172–180.
Z. Drezner and G.O. Wesolowsky, Single facilityl p -distance minmax location. SIAM J. Algebraic Discrete Methods,1 (1980), 315–321.
M.E. Dyer, On a multidimensional search technique and its applications to the euclidean one-center problem. SIAM J. Comput.,15 (1986), 725–738.
J. Elzinga and D.W. Hearn, The minimum covering sphere problem. Management Sci.,19 (1972), 96–104.
J. Elzinga and D.W. Hearn, Geometrical solutions for some minimax location problems. Transportation Sci.,6 (1972), 379–394.
R.L. Francis, Some aspects of a minimax location problem. Oper. Res.,15 (1967), 1163–1169.
N. Megiddo, Linear-time algorithms for linear programming inR 3 and related problems. SIAM J. Comput.,12 (1983), 759–776.
N. Megiddo, The weighted Euclidean 1-center problem. Math. Oper. Res.,8 (1983), 498–504.
R.C. Melville, An implementation study of two algorithms for the minimum spanning circle problem. Computational Geometry (ed. G.T. Toussaint), North-Holland, Amsterdam, 1985, 267–294.
K.P.K. Nair and R. Chandrasekaran, Optimal location of a single service center of certain types. Naval Res. Logist. Quart.,18 (1971), 503–510.
H. Rademacher and O. Toeplitz, The Enjoyment of Mathematics. Princeton University Press, Princeton, New Jersey, 1957.
K. Sekitani and Y. Yamamoto, A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes. Math. Programming, to appear.
M.I. Shamos and D. Hoey, Closest-point problems. Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, 1975, 151–162.
S. Skyum, A simple algorithm for computing the smallest enclosing circle. Inform. Process. Lett.,37 (1991), 121–125.
J.J. Sylvester, A question in the geometry of situation. Quart. J. Pure Appl. Math.,1 (1857), 79.
P. Wolfe, Finding the nearest point in a polytope. Math. Programming,11 (1976), 128–149.
Author information
Authors and Affiliations
About this article
Cite this article
Sekitani, K., Yamamoto, Y. A recursive algorithm for finding the minimum covering sphere of a polytope and the minimum covering concentric spheres of several polytopes. Japan J. Indust. Appl. Math. 10, 255 (1993). https://doi.org/10.1007/BF03167575
Received:
DOI: https://doi.org/10.1007/BF03167575