Sublinear time width-bounded separators and their application to the protein side-chain packing problem

Abstract

Given d>2 and a set of n grid points Q in ℜ d , we design a randomized algorithm that finds a w-wide separator, which is determined by a hyper-plane, in $O(n^{2\over d}\log n)$ sublinear time such that Q has at most $({d\over d+1}+o(1))n$ points on either side of the hyper-plane, and at most $c_{d}wn^{d-1\over d}$ points within $\frac{w}{2}$ distance to the hyper-plane, where c d is a constant for fixed d. In particular, c 3=1.209. To our best knowledge, this is the first sublinear time algorithm for finding geometric separators. Our 3D separator is applied to derive an algorithm for the protein side-chain packing problem, which improves and simplifies the previous algorithm of Xu (Research in computational molecular biology, 9th annual international conference, pp. 408–422, 2005).