### 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, L_{1}-) 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(n^{ε})-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 language | English (US) |
---|---|

Pages (from-to) | 36-52 |

Number of pages | 17 |

Journal | Algorithmica |

Volume | 71 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

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### Keywords

- Approximation algorithms
- Computational geometry
- Minimum Manhattan network

### ASJC Scopus subject areas

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

### Cite this

*Algorithmica*,

*71*(1), 36-52. https://doi.org/10.1007/s00453-013-9778-z

**Approximating Minimum Manhattan Networks in Higher Dimensions.** / Das, Aparna; Gansner, Emden R.; Kaufmann, Michael; Kobourov, Stephen G; Spoerhase, Joachim; Wolff, Alexander.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 71, no. 1, pp. 36-52. https://doi.org/10.1007/s00453-013-9778-z

}

TY - JOUR

T1 - Approximating Minimum Manhattan Networks in Higher Dimensions

AU - Das, Aparna

AU - Gansner, Emden R.

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Spoerhase, Joachim

AU - Wolff, Alexander

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Computational geometry

KW - Minimum Manhattan network

UR - http://www.scopus.com/inward/record.url?scp=84922000454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922000454&partnerID=8YFLogxK

U2 - 10.1007/s00453-013-9778-z

DO - 10.1007/s00453-013-9778-z

M3 - Article

VL - 71

SP - 36

EP - 52

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -