Summary: On Minimum­Area Hulls \Lambda
Esther M. Arkin y Yi­Jen Chiang z Martin Held x Joseph S. B. Mitchell --
Vera Sacristan k Steven S. Skiena \Lambda\Lambda Tae­Cheon Yang yy
Abstract
We study some minimum­area hull problems that generalize the notion of convex hull to
star­shaped and monotone hulls. Specifically, we consider the minimum­area star­shaped hull
problem: Given an n­vertex simple polygon P , find a minimum­area, star­shaped polygon P \Lambda
containing P . This problem arises in lattice packings of translates of multiple, non­identical
shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed
by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in
which the vertices of P \Lambda are constrained to be vertices of P , and the unrestricted version, in
which the vertices of P \Lambda can be anywhere in the plane. We prove that the restricted problem
falls in the class of ``3sum­hard'' (sometimes called ``n 2 ­hard'') problems, which are suspected
to admit no solutions in o(n 2 ) time. Further, we give an O(n 2 ) time algorithm, improving the
previous bound of O(n 5 ). We also show that the unrestricted problem can be solved in O(n 2 p(n))
time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a
polynomial of degree O(n).
We also consider the case in which P \Lambda is required to be monotone, with respect to an
unspecified direction; we refer to this as the minimum­area monotone hull problem. We give a
matching lower and upper bound of \Theta(n log n) time for computing P \Lambda in the restricted version,

Source: Arkin, Estie - Department of Applied Mathematics and Statistics, SUNY at Stony Brook
Mitchell, Joseph S.B. - Department of Applied Mathematics and Statistics, SUNY at Stony Brook

Collections: Computer Technologies and Information Sciences; Mathematics