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 (Formula presented.). 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 (Formula presented.)-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 (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.
| Language | English (US) |
|---|---|
| Pages | 1-21 |
| Number of pages | 21 |
| Journal | Algorithmica |
| DOIs | |
| State | Accepted/In press - Mar 2 2017 |
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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
Approximating the Generalized Minimum Manhattan Network Problem. / Das, Aparna; Fleszar, Krzysztof; Kobourov, Stephen; Spoerhase, Joachim; Veeramoni, Sankar; Wolff, Alexander.
In: Algorithmica, 02.03.2017, p. 1-21.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Approximating the Generalized Minimum Manhattan Network Problem
AU - Das,Aparna
AU - Fleszar,Krzysztof
AU - Kobourov,Stephen
AU - Spoerhase,Joachim
AU - Veeramoni,Sankar
AU - Wolff,Alexander
PY - 2017/3/2
Y1 - 2017/3/2
N2 - 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 (Formula presented.). 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 (Formula presented.)-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 (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.
AB - 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 (Formula presented.). 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 (Formula presented.)-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 (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.
KW - Approximation algorithms
KW - Computational geometry
KW - Minimum Manhattan Network
UR - http://www.scopus.com/inward/record.url?scp=85014075563&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85014075563&partnerID=8YFLogxK
U2 - 10.1007/s00453-017-0298-0
DO - 10.1007/s00453-017-0298-0
M3 - Article
SP - 1
EP - 21
JO - Algorithmica
T2 - Algorithmica
JF - Algorithmica
SN - 0178-4617
ER -