Approximating Minimum Manhattan Networks in Higher Dimensions

Aparna Das, Emden R. Gansner, Michael Kaufmann, Stephen Kobourov, Joachim Spoerhase, Alexander Wolff

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in (Formula presented.), 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 (Formula presented.)). 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 algorithm. For 3D, we also give a 4(k−1)-approximation algorithm for the case that the terminals are contained in the union of k≥2 parallel planes.

Original languageEnglish (US)
Pages (from-to)36-52
Number of pages17
JournalAlgorithmica
Volume71
Issue number1
DOIs
StatePublished - 2013

Keywords

  • Approximation algorithms
  • Computational geometry
  • Minimum Manhattan network

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

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    Das, A., Gansner, E. R., Kaufmann, M., Kobourov, S., Spoerhase, J., & Wolff, A. (2013). Approximating Minimum Manhattan Networks in Higher Dimensions. Algorithmica, 71(1), 36-52. https://doi.org/10.1007/s00453-013-9778-z