The minimum Manhattan network problem: Approximations and exact solutions

作     者：Benkert, Marc Wolff, Alexander Widmann, Florian Shirabe, Takeshi 

作者机构：Univ Karlsruhe Dept Comp Sci D-76128 Karlsruhe Germany Vienna Tech Univ Inst Geoinformat A-1040 Vienna Austria 

出 版 物：《COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS》 (计算几何学)

年 卷 期：2006年第35卷第3期

页      面：188-208页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：German Science Foundation Deutsche Forschungsgemeinschaft, DFG 

主　　题：spanners minimum Manhattan networks approximation algorithm mixed-integer programming 

摘      要：Given a set of points in the plane and a constant t = 1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively. In this paper we study l-spanners under the Manhattan (or L-l-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(n log n) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P. We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets. (c) 2005 Elsevier B.V. All rights reserved.