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Algorithmica

, Volume 80, Issue 4, pp 1170–1190 | Cite as

Approximating the Generalized Minimum Manhattan Network Problem

  • Aparna Das
  • Krzysztof Fleszar
  • Stephen Kobourov
  • Joachim Spoerhase
  • Sankar Veeramoni
  • Alexander Wolff
Article
  • 504 Downloads

Abstract

We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in \(\mathbb {R}^2\). The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an \(O(\log n)\)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple \(O(\log ^{d+1} n)\)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best \(O(n^\varepsilon )\)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing \(O(\log n)\)-approximation algorithm for 2D-RSA generalizes to higher dimensions.

Keywords

Approximation algorithms Computational geometry Minimum Manhattan Network 

Notes

Acknowledgements

We thank Michael Kaufmann for his hospitality and his enthusiasm during our respective stays in Tübingen. We thank Esther Arkin, Alon Efrat, Joe Mitchell, and Andreas Spillner for discussions.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceLe Moyne CollegeSyracuseUSA
  2. 2.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  3. 3.Lehrstuhl I, Institut für InformatikUniversität WürzburgWürzburgGermany

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