On the central path problem

In this paper we consider the following Central Path Problem (CPP): Given a set of m arbitrary (i.e., non-simple) polygonal curves Q = {P ₁,P ₂,...,P _m} in 2D space, find a curve P, called central path, that best represents all curves in Q. In order for P to best represent Q, P is required to minimize the maximum distance (measured by the directed Hausdorff distance) to all curves in Q and is the locus of the center of minimal spanning disk of Q. For the CPP problem, a direct approach is to first construct the farthest-path Voronoi diagram FPVD(Q) of Q and then derive the central path from it, which could be rather costly. In this paper, we present a novel approach which computes the central path in an "output- sensitive" fashion. Our approach sweeps a minimal spanning disk through Q and computes only a partial structure of the FPVD(Q) directly related to P. The running time of our approach is thus O((H + mk + n + s)logmlog ² n) and the worst case running time is O(n ²2 ^α(n)logn), where n is the size of Q, s is the total number of self-intersecting points of each individual curve in Q, k is the size of the visited portion of FPVD(Q) by the central path algorithm, and H is the number of intersections between the visited portion of FPVD(Q) and VD(P _i )(i = 1,2,..., m).