Computing 52, 269-279(1994) Col~dl~ Springer-Verlag 1994 Printed in Austria Optimal Algorithms for Some Intersection Radius Problems B. K. Bhattacharya*, Burnaby, S. Jadhav t, Kanpur, A. Mukhopadhyay ~, Kanpur, and J.-M. Robert ~, Montreal Received June 1, 1992; revised January 6, 1994 Abstract -- Zusammenfassung Optimal Algorithms for Some Intersection Radius Problems. The intersection radius of a set of n geometrical objects in a d-dimensional Euclidean space, E d, is the radius of the smallest closed hyper- sphere that intersects all the objects of the set. In this paper, we describe optimal algorithms for some intersection radius problems. We first present a linear-time algorithm to determine the smallest closed hypcrsphere that intersects a set of hyperplanes in E~, assuming d to be a fixed parameter. This is done by reducing the problem to a linear programming problem in a (d + 1)-dimensional space, involving 2n linear constraints. We also show how the prune-and-search technique, coupled with the strategy of replacing a ray by a point or a line can be used to solve, in linear time, the intersection radius problem for a set of n line segments in the plane. Currently, no algorithms are known that solve these intersection radius problems within the same time bounds. AMS Subject Classifications: 52.A30, 52.A10 Key words: Intersection radius, prune-and-search, algorithms, complexity, computational geometry. Optimale AIgorithmen fiir den Durchschnitts-Radius. Wit bezeichnen als Radius des Durchschnitts einer Menge yon n geometrischen Objekten im d-dimensionalen Enklidischen Raum E d Radius der kleinsten abgeschlossenen Hyperkugel, welche einen nichtleeren Durchschnitt mit allen Objekten besitzt. In der vorliegenden Arbeit beschreiben wir optimale Algorithmen zuer Bestimmung einiger solcher Radien. Zuerst stellen wir einen Algorithmus mit linearem Zeitbedarf vor, wenn die Objekte Hyperebenen in Ed mit festem d sind. Er beruht auf der Reduktion des Problems aufeine (d + 1)-dimensionale Lineare Optimierungsaufgabe mit 2n linearen Nebenbedingungen. Wir beschreiben auch die L6sung des Durchschnitts-Radius Problems ffir n Strecken in der Ebene. Dazu benutzen wir neben Breitensuche die Ersetzung von Halbstrahlen durch Punkte oder Gerade. Bisher waren keine Algorithmen bekannt, welche diese Probleme in den gleichen Zeitschranken 16sen. I. Introduction Let S be a finite set of objects in a d-dimensional Euclidean space. The stabbing problem consiSts of finding an object (the stabber), which intersects each member of S. Typically, the stabber is a line, or a hyperplane, or a disk etc., and S is a set of * School of Computing Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada. * Department of Computer Science and Engg., Indian Institute ofTechnology, Kanpur, 208016, India. School of Computer Science, McGill University, 3480 University St., Montreal, P.Q., Canada, H3A 2A7.