Probabilistic Algorithms for Eecient Grasping and Xturing

Given an object with n points on its boundary where ngers can be placed, we give algorithms to select a \strong" grasp with a minimal number c of ngers (up to a logarithmic factor) for several measures of goodness. Along similar lines, given an integer c, we nd the \best" c logc nger grasp for a small constant. Furthermore , we generalize existing measures for the case of frictionless assemblies of many objects in contact. Depending on the measure, the algorithms run in expected time O(c 2 n 1+) or O((nc) 1+ + c 4 log n log 3 c). Here is any positive constant. This setting generalizes to higher dimensions in the context of nding sets of xtures. These problems translate into a collection of convex set covering problems. We are given a convex set L, and a set of points U with L conv U in dimension d. There are two basic questions: (1) what is the smallest subset C of U or cover with L conv C, and (2) given an integer c, what is the largest with L conv C among all C U of c points. We present an algorithmic framework which handles these problems in a uniform way and give approximation algorithms for speciic instances of L including convex polytopes and balls. It generalizes an algorithm for polytope covering and approximation by Clarkson Cla93] in several diierent directions: we show it can be used not only for minimizing cover size, but also maximizing the scaling factor (see above), and further more it is valid for smaller cover sizes than previously possible, with appropriate modiications.