Approximating minimum manhattan networks in higher dimensions

Authors:
Aparna Das;Emden R. Gansner;Michael Kaufmann;Stephen Kobourov;Joachim Spoerhase;Alexander Wolff
Affiliations:
Dept. of Comp. Sci., University of Arizona, Tucson, AZ;AT&T Labs Research, Florham Park, NJ;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany;Dept. of Comp. Sci., University of Arizona, Tucson, AZ;Institut für Informatik, Universität Würzburg, Germany;Institut für Informatik, Universität Würzburg, Germany
Venue:
ESA'11 Proceedings of the 19th European conference on Algorithms
Year:
2011

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Abstract

We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in Rd, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, L1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless P = NP). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ε 0, an O(nε)- approximation. For 3D, we also give a 4(k - 1)-approximation for the case that the terminals are contained in the union of k ≥ 2 parallel planes.