The maximin line problem with regional demand

Given a familyP 1⁄4 fP 1; . . . ; Pmg of m polygonal regions (possibly intersecting) in the plane, we consider the problem of locating a straight line ‘ intersecting the convex hull of P and such that minkd(Pk,‘) is maximal. We give an algorithm that solves the problem in time O((m + n logm) logn) using O(m + n) space, where n is the total number of vertices of P1, . . . ,Pm. The previous best running time for this problem was O(n ) [J. Janardan, F.P. Preparata, Widest-corridor problems, Nordic Journal of Computing 1 (1994) 231–245]. We also improve the known complexity for several variants of this problem which include a three dimensional version – the maximin plane problem –, the weighted problem and considering measuring distance different to the Euclidean one. 2006 Elsevier B.V. All rights reserved.