On the Central Path Problem

• Published in: Combinatorial Optimization and Applications Vol. 7402; pp. 138 - 150
• Main Authors: Zhu, Yongding, Xu, Jinhui
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2012; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Binary Search Tree; Central Path; Forward Direction; Voronoi Cell; Voronoi Diagram
• ISBN: 3642317693; 9783642317699
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-31770-5_13

In this paper we consider the following Central Path Problem (CPP): Given a set of m arbitrary (i.e., non-simple) polygonal curves Q = {P1,P2,…,Pm} 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)logmlog2n) and the worst case running time is O(n22α(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(Pi)(i = 1,2,…, m).