Abstract
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)logmlog2 n) and the worst case running time is O(n 22α(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).
This research was partially supported by NSF through a CAREER award CCF-0546509 and grants IIS-0713489 and IIS-1115220.
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Zhu, Y., Xu, J. (2012). On the Central Path Problem. In: Lin, G. (eds) Combinatorial Optimization and Applications. COCOA 2012. Lecture Notes in Computer Science, vol 7402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31770-5_13
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DOI: https://doi.org/10.1007/978-3-642-31770-5_13
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