### Abstract

We consider the problem of Single-Tiered Relay Placement with Basestations, which takes as input a set (Formula presented.) of sensors and a set (Formula presented.) of basestations described as points in a normed space (Formula presented.) , and real numbers (Formula presented.). The objective is to place a minimum cardinality set (Formula presented.) of wireless relay nodes that connects (Formula presented.) and (Formula presented.) according to the following rules. The sensors in (Formula presented.) can communicate within distance (Formula presented.) , relay nodes in (Formula presented.) can communicate within distance (Formula presented.) , and basestations are considered to have an infinite broadcast range. Together the sets (Formula presented.) , and (Formula presented.) induce an undirected graph (Formula presented.) defined as follows: (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.). Then (Formula presented.) connects (Formula presented.) and (Formula presented.) when this induced graph is connected. In the case of the two-dimensional Euclidean plane, we get a (Formula presented.) -approximation algorithm, improving the previous best ratio of 3.11. Let (Formula presented.) be the maximum number of points on a unit ball with pairwise distance strictly bigger than 1. Under certain assumptions, we have a (Formula presented.) -approximation algorithm. When biconnectivity is required, we show that a variant of our previously proposed algorithm has approximation ratio of (Formula presented.). In the case of the two-dimensional Euclidean plane, our ratio of 7 improves our previous bound of 16.

Original language | English (US) |
---|---|

Pages (from-to) | 1280-1297 |

Number of pages | 18 |

Journal | Journal of Combinatorial Optimization |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2016 |

### Fingerprint

### Keywords

- Approximation algorithm
- Biconnectivity
- Steiner points
- Wireless network

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Computational Theory and Mathematics
- Computer Science Applications
- Control and Optimization

### Cite this

*Journal of Combinatorial Optimization*,

*31*(3), 1280-1297. https://doi.org/10.1007/s10878-014-9823-0

**Improved approximation algorithms for single-tiered relay placement.** / Calinescu, Gruia; Grimmer, Benjamin; Misra, Satyajayant; Tongngam, Sutep; Xue, Guoliang; Zhang, Weiyi.

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 31, no. 3, pp. 1280-1297. https://doi.org/10.1007/s10878-014-9823-0

}

TY - JOUR

T1 - Improved approximation algorithms for single-tiered relay placement

AU - Calinescu, Gruia

AU - Grimmer, Benjamin

AU - Misra, Satyajayant

AU - Tongngam, Sutep

AU - Xue, Guoliang

AU - Zhang, Weiyi

PY - 2016/4/1

Y1 - 2016/4/1

N2 - We consider the problem of Single-Tiered Relay Placement with Basestations, which takes as input a set (Formula presented.) of sensors and a set (Formula presented.) of basestations described as points in a normed space (Formula presented.) , and real numbers (Formula presented.). The objective is to place a minimum cardinality set (Formula presented.) of wireless relay nodes that connects (Formula presented.) and (Formula presented.) according to the following rules. The sensors in (Formula presented.) can communicate within distance (Formula presented.) , relay nodes in (Formula presented.) can communicate within distance (Formula presented.) , and basestations are considered to have an infinite broadcast range. Together the sets (Formula presented.) , and (Formula presented.) induce an undirected graph (Formula presented.) defined as follows: (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.). Then (Formula presented.) connects (Formula presented.) and (Formula presented.) when this induced graph is connected. In the case of the two-dimensional Euclidean plane, we get a (Formula presented.) -approximation algorithm, improving the previous best ratio of 3.11. Let (Formula presented.) be the maximum number of points on a unit ball with pairwise distance strictly bigger than 1. Under certain assumptions, we have a (Formula presented.) -approximation algorithm. When biconnectivity is required, we show that a variant of our previously proposed algorithm has approximation ratio of (Formula presented.). In the case of the two-dimensional Euclidean plane, our ratio of 7 improves our previous bound of 16.

AB - We consider the problem of Single-Tiered Relay Placement with Basestations, which takes as input a set (Formula presented.) of sensors and a set (Formula presented.) of basestations described as points in a normed space (Formula presented.) , and real numbers (Formula presented.). The objective is to place a minimum cardinality set (Formula presented.) of wireless relay nodes that connects (Formula presented.) and (Formula presented.) according to the following rules. The sensors in (Formula presented.) can communicate within distance (Formula presented.) , relay nodes in (Formula presented.) can communicate within distance (Formula presented.) , and basestations are considered to have an infinite broadcast range. Together the sets (Formula presented.) , and (Formula presented.) induce an undirected graph (Formula presented.) defined as follows: (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.). Then (Formula presented.) connects (Formula presented.) and (Formula presented.) when this induced graph is connected. In the case of the two-dimensional Euclidean plane, we get a (Formula presented.) -approximation algorithm, improving the previous best ratio of 3.11. Let (Formula presented.) be the maximum number of points on a unit ball with pairwise distance strictly bigger than 1. Under certain assumptions, we have a (Formula presented.) -approximation algorithm. When biconnectivity is required, we show that a variant of our previously proposed algorithm has approximation ratio of (Formula presented.). In the case of the two-dimensional Euclidean plane, our ratio of 7 improves our previous bound of 16.

KW - Approximation algorithm

KW - Biconnectivity

KW - Steiner points

KW - Wireless network

UR - http://www.scopus.com/inward/record.url?scp=84961059767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961059767&partnerID=8YFLogxK

U2 - 10.1007/s10878-014-9823-0

DO - 10.1007/s10878-014-9823-0

M3 - Article

VL - 31

SP - 1280

EP - 1297

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 3

ER -